From 745f9a9bfb6864e84f90830909d10bb9529e3a7b Mon Sep 17 00:00:00 2001 From: fjosw Date: Sun, 21 Jan 2024 11:42:22 +0000 Subject: [PATCH] Documentation updated --- docs/pyerrors.html | 2 +- docs/pyerrors/correlators.html | 2 +- docs/pyerrors/covobs.html | 2 +- docs/pyerrors/dirac.html | 2 +- docs/pyerrors/fits.html | 2 +- docs/pyerrors/input.html | 2 +- docs/pyerrors/input/bdio.html | 2 +- docs/pyerrors/input/dobs.html | 2 +- docs/pyerrors/input/hadrons.html | 2 +- docs/pyerrors/input/json.html | 2 +- docs/pyerrors/input/misc.html | 2 +- docs/pyerrors/input/openQCD.html | 2 +- docs/pyerrors/input/pandas.html | 2 +- docs/pyerrors/input/sfcf.html | 2 +- docs/pyerrors/input/utils.html | 2 +- docs/pyerrors/integrate.html | 2 +- docs/pyerrors/linalg.html | 2 +- docs/pyerrors/misc.html | 2 +- docs/pyerrors/mpm.html | 2 +- docs/pyerrors/obs.html | 2 +- docs/pyerrors/roots.html | 2 +- docs/pyerrors/special.html | 159 ++++++++++++++++--------------- docs/pyerrors/version.html | 2 +- docs/search.js | 2 +- 24 files changed, 105 insertions(+), 100 deletions(-) diff --git a/docs/pyerrors.html b/docs/pyerrors.html index a0feb78e..4f8d09b5 100644 --- a/docs/pyerrors.html +++ b/docs/pyerrors.html @@ -3,7 +3,7 @@ - + pyerrors API documentation diff --git a/docs/pyerrors/correlators.html b/docs/pyerrors/correlators.html index 059a8207..0cd371d4 100644 --- a/docs/pyerrors/correlators.html +++ b/docs/pyerrors/correlators.html @@ -3,7 +3,7 @@ - + pyerrors.correlators API documentation diff --git a/docs/pyerrors/covobs.html b/docs/pyerrors/covobs.html index 3be5a0a8..6139a409 100644 --- a/docs/pyerrors/covobs.html +++ b/docs/pyerrors/covobs.html @@ -3,7 +3,7 @@ - + pyerrors.covobs API documentation diff --git a/docs/pyerrors/dirac.html b/docs/pyerrors/dirac.html index 9d12a125..ae5750b9 100644 --- a/docs/pyerrors/dirac.html +++ b/docs/pyerrors/dirac.html @@ -3,7 +3,7 @@ - + pyerrors.dirac API documentation diff --git a/docs/pyerrors/fits.html b/docs/pyerrors/fits.html index ab8874cd..187473a9 100644 --- a/docs/pyerrors/fits.html +++ b/docs/pyerrors/fits.html @@ -3,7 +3,7 @@ - + pyerrors.fits API documentation diff --git a/docs/pyerrors/input.html b/docs/pyerrors/input.html index 04759c83..d1e18b99 100644 --- a/docs/pyerrors/input.html +++ b/docs/pyerrors/input.html @@ -3,7 +3,7 @@ - + pyerrors.input API documentation diff --git a/docs/pyerrors/input/bdio.html b/docs/pyerrors/input/bdio.html index 071d152a..a05e13a5 100644 --- a/docs/pyerrors/input/bdio.html +++ b/docs/pyerrors/input/bdio.html @@ -3,7 +3,7 @@ - + pyerrors.input.bdio API documentation diff --git a/docs/pyerrors/input/dobs.html b/docs/pyerrors/input/dobs.html index 6ebf1419..c43367be 100644 --- a/docs/pyerrors/input/dobs.html +++ b/docs/pyerrors/input/dobs.html @@ -3,7 +3,7 @@ - + pyerrors.input.dobs API documentation diff --git a/docs/pyerrors/input/hadrons.html b/docs/pyerrors/input/hadrons.html index cb0dc83a..de866fa2 100644 --- a/docs/pyerrors/input/hadrons.html +++ b/docs/pyerrors/input/hadrons.html @@ -3,7 +3,7 @@ - + pyerrors.input.hadrons API documentation diff --git a/docs/pyerrors/input/json.html b/docs/pyerrors/input/json.html index 55eac157..279d500c 100644 --- a/docs/pyerrors/input/json.html +++ b/docs/pyerrors/input/json.html @@ -3,7 +3,7 @@ - + pyerrors.input.json API documentation diff --git a/docs/pyerrors/input/misc.html b/docs/pyerrors/input/misc.html index d2a9ff38..17b0f688 100644 --- a/docs/pyerrors/input/misc.html +++ b/docs/pyerrors/input/misc.html @@ -3,7 +3,7 @@ - + pyerrors.input.misc API documentation diff --git a/docs/pyerrors/input/openQCD.html b/docs/pyerrors/input/openQCD.html index 2435cc2a..2b6b3496 100644 --- a/docs/pyerrors/input/openQCD.html +++ b/docs/pyerrors/input/openQCD.html @@ -3,7 +3,7 @@ - + pyerrors.input.openQCD API documentation diff --git a/docs/pyerrors/input/pandas.html b/docs/pyerrors/input/pandas.html index 9913e883..6ec235ea 100644 --- a/docs/pyerrors/input/pandas.html +++ b/docs/pyerrors/input/pandas.html @@ -3,7 +3,7 @@ - + pyerrors.input.pandas API documentation diff --git a/docs/pyerrors/input/sfcf.html b/docs/pyerrors/input/sfcf.html index 2a5c0436..41a28897 100644 --- a/docs/pyerrors/input/sfcf.html +++ b/docs/pyerrors/input/sfcf.html @@ -3,7 +3,7 @@ - + pyerrors.input.sfcf API documentation diff --git a/docs/pyerrors/input/utils.html b/docs/pyerrors/input/utils.html index 84e64325..d67c384d 100644 --- a/docs/pyerrors/input/utils.html +++ b/docs/pyerrors/input/utils.html @@ -3,7 +3,7 @@ - + pyerrors.input.utils API documentation diff --git a/docs/pyerrors/integrate.html b/docs/pyerrors/integrate.html index 064bab1c..61492434 100644 --- a/docs/pyerrors/integrate.html +++ b/docs/pyerrors/integrate.html @@ -3,7 +3,7 @@ - + pyerrors.integrate API documentation diff --git a/docs/pyerrors/linalg.html b/docs/pyerrors/linalg.html index 89c30624..bef9f46e 100644 --- a/docs/pyerrors/linalg.html +++ b/docs/pyerrors/linalg.html @@ -3,7 +3,7 @@ - + pyerrors.linalg API documentation diff --git a/docs/pyerrors/misc.html b/docs/pyerrors/misc.html index 5a79c807..f40428f9 100644 --- a/docs/pyerrors/misc.html +++ b/docs/pyerrors/misc.html @@ -3,7 +3,7 @@ - + pyerrors.misc API documentation diff --git a/docs/pyerrors/mpm.html b/docs/pyerrors/mpm.html index 1f149cf8..c2079441 100644 --- a/docs/pyerrors/mpm.html +++ b/docs/pyerrors/mpm.html @@ -3,7 +3,7 @@ - + pyerrors.mpm API documentation diff --git a/docs/pyerrors/obs.html b/docs/pyerrors/obs.html index 113e4ba6..0b47e142 100644 --- a/docs/pyerrors/obs.html +++ b/docs/pyerrors/obs.html @@ -3,7 +3,7 @@ - + pyerrors.obs API documentation diff --git a/docs/pyerrors/roots.html b/docs/pyerrors/roots.html index 54223adb..72f83f66 100644 --- a/docs/pyerrors/roots.html +++ b/docs/pyerrors/roots.html @@ -3,7 +3,7 @@ - + pyerrors.roots API documentation diff --git a/docs/pyerrors/special.html b/docs/pyerrors/special.html index 569ac24f..c830731b 100644 --- a/docs/pyerrors/special.html +++ b/docs/pyerrors/special.html @@ -3,7 +3,7 @@ - + pyerrors.special API documentation @@ -351,6 +351,9 @@
Examples

for \( 0 \leq x \leq 1 \).

+

This function is the cumulative distribution function for the beta +distribution; its range is [0, 1].

+
Parameters
+
See Also
+ +

gammaincc: regularized upper incomplete gamma function
+gammaincinv: inverse of the regularized lower incomplete gamma function
+gammainccinv: inverse of the regularized upper incomplete gamma function

+
Notes

The function satisfies the relation gammainc(a, x) + @@ -1167,12 +1178,6 @@

Notes

The implementation largely follows that of [boost]_.

-
See also
- -

gammaincc : regularized upper incomplete gamma function -gammaincinv : inverse of the regularized lower incomplete gamma function -gammainccinv : inverse of the regularized upper incomplete gamma function

-
References

.. [dlmf] NIST Digital Library of Mathematical functions @@ -1268,6 +1273,12 @@

Returns
  • scalar or ndarray: Values of the upper incomplete gamma function
    • +
    See Also
    + +

    gammainc: regularized lower incomplete gamma function
    +gammaincinv: inverse of the regularized lower incomplete gamma function
    +gammainccinv: inverse of the regularized upper incomplete gamma function

    +
    Notes

    The function satisfies the relation gammainc(a, x) + @@ -1276,12 +1287,6 @@

    Notes

    The implementation largely follows that of [boost]_.

    -
    See also
    - -

    gammainc : regularized lower incomplete gamma function -gammaincinv : inverse of the regularized lower incomplete gamma function -gammainccinv : inverse of the regularized upper incomplete gamma function

    -
    References

    .. [dlmf] NIST Digital Library of Mathematical functions @@ -1382,17 +1387,17 @@

    Returns
  • scalar or ndarray: Sign of the gamma function
    • -
    Notes
    - -

    The gamma function can be computed as gammasgn(x) * -np.exp(gammaln(x)).

    -
    See Also

    gamma: the gamma function
    gammaln: log of the absolute value of the gamma function
    loggamma: analytic continuation of the log of the gamma function

    +
    Notes
    + +

    The gamma function can be computed as gammasgn(x) * +np.exp(gammaln(x)).

    +
    References

    .. [dlmf] NIST Digital Library of Mathematical Functions @@ -1489,6 +1494,10 @@

    Returns
  • scalar or ndarray: Function results
    • +
    See Also
    + +

    gamma,, gammaln,, loggamma

    +
    Notes

    The gamma function has no zeros and has simple poles at @@ -1496,10 +1505,6 @@

    Notes
    at the nonpositive integers. See the discussion in [dlmf]_ for more details.

    -
    See Also
    - -

    gamma,, gammaln,, loggamma

    -
    References

    .. [dlmf] Nist, Digital Library of Mathematical functions, @@ -1719,6 +1724,11 @@

    Returns
    Value of the Bessel function of the first kind of order 0 at x. +
    See Also
    + +

    jv: Bessel function of real order and complex argument.
    +spherical_jn: spherical Bessel functions.

    +
    Notes

    The domain is divided into the intervals [0, 5] and (5, infinity). In the @@ -1737,11 +1747,6 @@

    Notes
    It should not be confused with the spherical Bessel functions (see spherical_jn).

    -
    See also
    - -

    jv : Bessel function of real order and complex argument. -spherical_jn : spherical Bessel functions.

    -
    References
    Examples
    @@ -1840,6 +1845,11 @@
    Returns
    Value of the Bessel function of the second kind of order 0 at x. +
    See Also
    + +

    j0: Bessel function of the first kind of order 0
    +yv: Bessel function of the first kind

    +
    Notes

    The domain is divided into the intervals [0, 5] and (5, infinity). In the @@ -1855,11 +1865,6 @@

    Notes

    This function is a wrapper for the Cephes 1 routine y0.

    -
    See also
    - -

    j0: Bessel function of the first kind of order 0 -yv: Bessel function of the first kind

    -
    References
    Examples
    @@ -1958,6 +1963,11 @@
    Returns
    Value of the Bessel function of the first kind of order 1 at x. +
    See Also
    + +

    jv: Bessel function of the first kind
    +spherical_jn: spherical Bessel functions.

    +
    Notes

    The domain is divided into the intervals [0, 8] and (8, infinity). In the @@ -1969,11 +1979,6 @@

    Notes
    It should not be confused with the spherical Bessel functions (see spherical_jn).

    -
    See also
    - -

    jv: Bessel function of the first kind -spherical_jn: spherical Bessel functions.

    -
    References
    Examples
    @@ -2072,6 +2077,12 @@
    Returns
    Value of the Bessel function of the second kind of order 1 at x. +
    See Also
    + +

    j1: Bessel function of the first kind of order 1
    +yn: Bessel function of the second kind
    +yv: Bessel function of the second kind

    +
    Notes

    The domain is divided into the intervals [0, 8] and (8, infinity). In the @@ -2082,12 +2093,6 @@

    Notes

    This function is a wrapper for the Cephes 1 routine y1.

    -
    See also
    - -

    j1: Bessel function of the first kind of order 1 -yn: Bessel function of the second kind -yv: Bessel function of the second kind

    -
    References
    Examples
    @@ -2188,12 +2193,12 @@
    Returns
    Value of the Bessel function, \( J_v(z) \). -
    See also
    +
    See Also
    -

    jve : \( J_v \) with leading exponential behavior stripped off. -spherical_jn : spherical Bessel functions. -j0 : faster version of this function for order 0. -j1 : faster version of this function for order 1.

    +

    jve: \( J_v \) with leading exponential behavior stripped off.
    +spherical_jn: spherical Bessel functions.
    +j0: faster version of this function for order 0.
    +j1: faster version of this function for order 1.

    Notes
    @@ -2355,6 +2360,12 @@
    Returns
    Value of the Bessel function, \( Y_n(x) \). +
    See Also
    + +

    yv: For real order and real or complex argument.
    +y0: faster implementation of this function for order 0
    +y1: faster implementation of this function for order 1

    +
    Notes

    Wrapper for the Cephes 1 routine yn.

    @@ -2363,12 +2374,6 @@
    Notes
    values computed by the Cephes routines y0 and y1. If n = 0 or 1, the routine for y0 or y1 is called directly.

    -
    See also
    - -

    yv : For real order and real or complex argument. -y0: faster implementation of this function for order 0 -y1: faster implementation of this function for order 1

    -
    References
    Examples
    @@ -2512,6 +2517,11 @@
    Returns
    Value of the modified Bessel function of order 0 at x. +
    See Also
    + +

    iv: Modified Bessel function of any order
    +i0e: Exponentially scaled modified Bessel function of order 0

    +
    Notes

    The range is partitioned into the two intervals [0, 8] and (8, infinity). @@ -2519,11 +2529,6 @@

    Notes

    This function is a wrapper for the Cephes 1 routine i0.

    -
    See also
    - -

    iv: Modified Bessel function of any order -i0e: Exponentially scaled modified Bessel function of order 0

    -
    References
    Examples
    @@ -2629,6 +2634,11 @@
    Returns
    Value of the modified Bessel function of order 1 at x. +
    See Also
    + +

    iv: Modified Bessel function of the first kind
    +i1e: Exponentially scaled modified Bessel function of order 1

    +
    Notes

    The range is partitioned into the two intervals [0, 8] and (8, infinity). @@ -2636,11 +2646,6 @@

    Notes

    This function is a wrapper for the Cephes 1 routine i1.

    -
    See also
    - -

    iv: Modified Bessel function of the first kind -i1e: Exponentially scaled modified Bessel function of order 1

    -
    References
    Examples
    @@ -2741,6 +2746,12 @@
    Returns
  • scalar or ndarray: Values of the modified Bessel function.
    • +
    See Also
    + +

    ive: This function with leading exponential behavior stripped off.
    +i0: Faster version of this function for order 0.
    +i1: Faster version of this function for order 1.

    +
    Notes

    For real z and \( v \in [-50, 50] \), the evaluation is carried out @@ -2768,12 +2779,6 @@

    Notes

    is used, where \( K_v(z) \) is the modified Bessel function of the second kind, evaluated using the AMOS routine zbesk.

    -
    See also
    - -

    ive : This function with leading exponential behavior stripped off. -i0 : Faster version of this function for order 0. -i1 : Faster version of this function for order 1.

    -
    References
    Examples
    @@ -2924,6 +2929,12 @@
    Returns
  • scalar or ndarray: Values of the exponentially scaled modified Bessel function.
    • +
    See Also
    + +

    iv: Modified Bessel function of the first kind
    +i0e: Faster implementation of this function for order 0
    +i1e: Faster implementation of this function for order 1

    +
    Notes

    For positive v, the AMOS 1 zbesi routine is called. It uses a @@ -2950,12 +2961,6 @@

    Notes

    ive is useful for large arguments z: for these, iv easily overflows, while ive does not due to the exponential scaling.

    -
    See also
    - -

    iv: Modified Bessel function of the first kind -i0e: Faster implementation of this function for order 0 -i1e: Faster implementation of this function for order 1

    -
    References
    Examples
    diff --git a/docs/pyerrors/version.html b/docs/pyerrors/version.html index 6b3bf7bf..4b82a109 100644 --- a/docs/pyerrors/version.html +++ b/docs/pyerrors/version.html @@ -3,7 +3,7 @@ - + pyerrors.version API documentation diff --git a/docs/search.js b/docs/search.js index 4700a81a..1a1fef64 100644 --- a/docs/search.js +++ b/docs/search.js @@ -1,6 +1,6 @@ window.pdocSearch = (function(){ /** elasticlunr - http://weixsong.github.io * Copyright (C) 2017 Oliver Nightingale * Copyright (C) 2017 Wei Song * MIT Licensed */!function(){function e(e){if(null===e||"object"!=typeof e)return e;var t=e.constructor();for(var n in e)e.hasOwnProperty(n)&&(t[n]=e[n]);return t}var t=function(e){var n=new t.Index;return n.pipeline.add(t.trimmer,t.stopWordFilter,t.stemmer),e&&e.call(n,n),n};t.version="0.9.5",lunr=t,t.utils={},t.utils.warn=function(e){return function(t){e.console&&console.warn&&console.warn(t)}}(this),t.utils.toString=function(e){return void 0===e||null===e?"":e.toString()},t.EventEmitter=function(){this.events={}},t.EventEmitter.prototype.addListener=function(){var e=Array.prototype.slice.call(arguments),t=e.pop(),n=e;if("function"!=typeof t)throw new TypeError("last argument must be a function");n.forEach(function(e){this.hasHandler(e)||(this.events[e]=[]),this.events[e].push(t)},this)},t.EventEmitter.prototype.removeListener=function(e,t){if(this.hasHandler(e)){var n=this.events[e].indexOf(t);-1!==n&&(this.events[e].splice(n,1),0==this.events[e].length&&delete this.events[e])}},t.EventEmitter.prototype.emit=function(e){if(this.hasHandler(e)){var t=Array.prototype.slice.call(arguments,1);this.events[e].forEach(function(e){e.apply(void 0,t)},this)}},t.EventEmitter.prototype.hasHandler=function(e){return e in this.events},t.tokenizer=function(e){if(!arguments.length||null===e||void 0===e)return[];if(Array.isArray(e)){var n=e.filter(function(e){return null===e||void 0===e?!1:!0});n=n.map(function(e){return t.utils.toString(e).toLowerCase()});var i=[];return n.forEach(function(e){var n=e.split(t.tokenizer.seperator);i=i.concat(n)},this),i}return e.toString().trim().toLowerCase().split(t.tokenizer.seperator)},t.tokenizer.defaultSeperator=/[\s\-]+/,t.tokenizer.seperator=t.tokenizer.defaultSeperator,t.tokenizer.setSeperator=function(e){null!==e&&void 0!==e&&"object"==typeof e&&(t.tokenizer.seperator=e)},t.tokenizer.resetSeperator=function(){t.tokenizer.seperator=t.tokenizer.defaultSeperator},t.tokenizer.getSeperator=function(){return t.tokenizer.seperator},t.Pipeline=function(){this._queue=[]},t.Pipeline.registeredFunctions={},t.Pipeline.registerFunction=function(e,n){n in t.Pipeline.registeredFunctions&&t.utils.warn("Overwriting existing registered function: "+n),e.label=n,t.Pipeline.registeredFunctions[n]=e},t.Pipeline.getRegisteredFunction=function(e){return e in t.Pipeline.registeredFunctions!=!0?null:t.Pipeline.registeredFunctions[e]},t.Pipeline.warnIfFunctionNotRegistered=function(e){var n=e.label&&e.label in this.registeredFunctions;n||t.utils.warn("Function is not registered with pipeline. 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this.config},t.Configuration.prototype.reset=function(){this.config={}},lunr.SortedSet=function(){this.length=0,this.elements=[]},lunr.SortedSet.load=function(e){var t=new this;return t.elements=e,t.length=e.length,t},lunr.SortedSet.prototype.add=function(){var e,t;for(e=0;e1;){if(r===e)return o;e>r&&(t=o),r>e&&(n=o),i=n-t,o=t+Math.floor(i/2),r=this.elements[o]}return r===e?o:-1},lunr.SortedSet.prototype.locationFor=function(e){for(var t=0,n=this.elements.length,i=n-t,o=t+Math.floor(i/2),r=this.elements[o];i>1;)e>r&&(t=o),r>e&&(n=o),i=n-t,o=t+Math.floor(i/2),r=this.elements[o];return r>e?o:e>r?o+1:void 0},lunr.SortedSet.prototype.intersect=function(e){for(var t=new lunr.SortedSet,n=0,i=0,o=this.length,r=e.length,s=this.elements,u=e.elements;;){if(n>o-1||i>r-1)break;s[n]!==u[i]?s[n]u[i]&&i++:(t.add(s[n]),n++,i++)}return t},lunr.SortedSet.prototype.clone=function(){var e=new lunr.SortedSet;return e.elements=this.toArray(),e.length=e.elements.length,e},lunr.SortedSet.prototype.union=function(e){var t,n,i;this.length>=e.length?(t=this,n=e):(t=e,n=this),i=t.clone();for(var o=0,r=n.toArray();oWhat is pyerrors?\n\n

    pyerrors is a python package for error computation and propagation of Markov chain Monte Carlo data.\nIt is based on the gamma method arXiv:hep-lat/0306017. Some of its features are:

    \n\n
      \n
    • automatic differentiation for exact linear error propagation as suggested in arXiv:1809.01289 (partly based on the autograd package).
      • \n
    • treatment of slow modes in the simulation as suggested in arXiv:1009.5228.
      • \n
    • coherent error propagation for data from different Markov chains.
      • \n
    • non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in arXiv:1809.01289.
      • \n
    • real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).
      • \n
    \n\n

    More detailed examples can found in the GitHub repository \"badge\".

    \n\n

    If you use pyerrors for research that leads to a publication please consider citing:

    \n\n
      \n
    • Fabian Joswig, Simon Kuberski, Justus T. Kuhlmann, Jan Neuendorf, pyerrors: a python framework for error analysis of Monte Carlo data. Comput.Phys.Commun. 288 (2023) 108750.
      • \n
    • Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
      • \n
    • Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
      • \n
    \n\n

    and

    \n\n
      \n
    • Stefan Schaefer, Rainer Sommer, Francesco Virotta, Critical slowing down and error analysis in lattice QCD simulations. Nucl.Phys.B 845 (2011) 93-119.
      • \n
    \n\n

    where applicable.

    \n\n

    There exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.

    \n\n

    Installation

    \n\n

    Install the most recent release using pip and pypi:

    \n\n
    \n
    python -m pip install pyerrors     # Fresh install\npython -m pip install -U pyerrors  # Update\n
    \n
    \n\n

    Install the most recent release using conda and conda-forge:

    \n\n
    \n
    conda install -c conda-forge pyerrors  # Fresh install\nconda update -c conda-forge pyerrors   # Update\n
    \n
    \n\n

    Install the current develop version:

    \n\n
    \n
    python -m pip install -U --no-deps --force-reinstall git+https://github.com/fjosw/pyerrors.git@develop\n
    \n
    \n\n

    (Also works for any feature branch).

    \n\n

    Basic example

    \n\n
    \n
    import numpy as np\nimport pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object\nmy_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object\nmy_new_obs.gamma_method()                     # Estimate the statistical error\nprint(my_new_obs)                             # Print the result to stdout\n> 0.31498(72)\n
    \n
    \n\n

    The Obs class

    \n\n

    pyerrors introduces a new datatype, Obs, which simplifies error propagation and estimation for auto- and cross-correlated data.\nAn Obs object can be initialized with two arguments, the first is a list containing the samples for an observable from a Monte Carlo chain.\nThe samples can either be provided as python list or as numpy array.\nThe second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble. It is crucial for the correct error propagation that observations from the same Monte Carlo history are labeled with the same name. See Multiple ensembles/replica for details.

    \n\n
    \n
    import pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name'])\n
    \n
    \n\n

    Error propagation

    \n\n

    When performing mathematical operations on Obs objects the correct error propagation is intrinsically taken care of using a first order Taylor expansion\n$$\\delta_f^i=\\sum_\\alpha \\bar{f}_\\alpha \\delta_\\alpha^i\\,,\\quad \\delta_\\alpha^i=a_\\alpha^i-\\bar{a}_\\alpha\\,,$$\nas introduced in arXiv:hep-lat/0306017.\nThe required derivatives $\\bar{f}_\\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in arXiv:1809.01289.

    \n\n

    The Obs class is designed such that mathematical numpy functions can be used on Obs just as for regular floats.

    \n\n
    \n
    import numpy as np\nimport pyerrors as pe\n\nmy_obs1 = pe.Obs([samples1], ['ensemble_name'])\nmy_obs2 = pe.Obs([samples2], ['ensemble_name'])\n\nmy_sum = my_obs1 + my_obs2\n\nmy_m_eff = np.log(my_obs1 / my_obs2)\n\niamzero = my_m_eff - my_m_eff\n# Check that value and fluctuations are zero within machine precision\nprint(iamzero == 0.0)\n> True\n
    \n
    \n\n

    Error estimation

    \n\n

    The error estimation within pyerrors is based on the gamma method introduced in arXiv:hep-lat/0306017.\nAfter having arrived at the derived quantity of interest the gamma_method can be called as detailed in the following example.

    \n\n
    \n
    my_sum.gamma_method()\nprint(my_sum)\n> 1.70(57)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%)\n>  t_int         2.71422900e+00 +/- 6.40320983e-01 S = 2.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n

    The gamma_method is not automatically called after every intermediate step in order to prevent computational overhead.

    \n\n

    We use the following definition of the integrated autocorrelation time established in Madras & Sokal 1988\n$$\\tau_\\mathrm{int}=\\frac{1}{2}+\\sum_{t=1}^{W}\\rho(t)\\geq \\frac{1}{2}\\,.$$\nThe window $W$ is determined via the automatic windowing procedure described in arXiv:hep-lat/0306017.\nThe standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the gamma_method as parameter.

    \n\n
    \n
    my_sum.gamma_method(S=3.0)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%)\n>  t_int         3.29909703e+00 +/- 9.77310102e-01 S = 3.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n

    The integrated autocorrelation time $\\tau_\\mathrm{int}$ and the autocorrelation function $\\rho(W)$ can be monitored via the methods pyerrors.obs.Obs.plot_tauint and pyerrors.obs.Obs.plot_rho.

    \n\n

    If the parameter $S$ is set to zero it is assumed that the dataset does not exhibit any autocorrelation and the window size is chosen to be zero.\nIn this case the error estimate is identical to the sample standard error.

    \n\n

    Exponential tails

    \n\n

    Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\\rho$ as suggested in arXiv:1009.5228. The longest autocorrelation time in the history, $\\tau_\\mathrm{exp}$, can be passed to the gamma_method as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.

    \n\n
    \n
    my_sum.gamma_method(tau_exp=7.2)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%)\n>  t_int         3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20,  N_sigma = 1\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n

    For the full API see pyerrors.obs.Obs.gamma_method.

    \n\n

    Multiple ensembles/replica

    \n\n

    Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their name.

    \n\n
    \n
    obs1 = pe.Obs([samples1], ['ensemble1'])\nobs2 = pe.Obs([samples2], ['ensemble2'])\n\nmy_sum = obs1 + obs2\nmy_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 2 ensembles:\n>   \u00b7 Ensemble 'ensemble1' : 1000 configurations (from 1 to 1000)\n>   \u00b7 Ensemble 'ensemble2' : 500 configurations (from 1 to 500)\n
    \n
    \n\n

    Observables from the same Monte Carlo chain have to be initialized with the same name for correct error propagation. If different names were used in this case the data would be treated as statistically independent resulting in loss of relevant information and a potential over or under estimate of the statistical error.

    \n\n

    pyerrors identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar | in the name of the data set.

    \n\n
    \n
    obs1 = pe.Obs([samples1], ['ensemble1|r01'])\nobs2 = pe.Obs([samples2], ['ensemble1|r02'])\n\n> my_sum = obs1 + obs2\n> my_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1'\n>     \u00b7 Replicum 'r01' : 1000 configurations (from 1 to 1000)\n>     \u00b7 Replicum 'r02' : 500 configurations (from 1 to 500)\n
    \n
    \n\n

    Error estimation for multiple ensembles

    \n\n

    In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.

    \n\n
    \n
    pe.Obs.S_dict['ensemble1'] = 2.5\npe.Obs.tau_exp_dict['ensemble2'] = 8.0\npe.Obs.tau_exp_dict['ensemble3'] = 2.0\n
    \n
    \n\n

    In case the gamma_method is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.\nPassing arguments to the gamma_method still dominates over the dictionaries.

    \n\n

    Irregular Monte Carlo chains

    \n\n

    Obs objects defined on irregular Monte Carlo chains can be initialized with the parameter idl.

    \n\n
    \n
    # Observable defined on configurations 20 to 519\nobs1 = pe.Obs([samples1], ['ensemble1'], idl=[range(20, 520)])\nobs1.details()\n> Result         9.98319881e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 20 to 519)\n\n# Observable defined on every second configuration between 5 and 1003\nobs2 = pe.Obs([samples2], ['ensemble1'], idl=[range(5, 1005, 2)])\nobs2.details()\n> Result         9.99100712e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 5 to 1003 in steps of 2)\n\n# Observable defined on configurations 2, 9, 28, 29 and 501\nobs3 = pe.Obs([samples3], ['ensemble1'], idl=[[2, 9, 28, 29, 501]])\nobs3.details()\n> Result         1.01718064e+00\n> 5 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 5 configurations (irregular range)\n
    \n
    \n\n

    Obs objects defined on regular and irregular histories of the same ensemble can be combined with each other and the correct error propagation and estimation is automatically taken care of.

    \n\n

    Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.\nMake sure to check the autocorrelation time with e.g. pyerrors.obs.Obs.plot_rho or pyerrors.obs.Obs.plot_tauint.

    \n\n

    For the full API see pyerrors.obs.Obs.

    \n\n

    Correlators

    \n\n

    When one is not interested in single observables but correlation functions, pyerrors offers the Corr class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize a Corr objects one needs to arrange the data as a list of Obs

    \n\n
    \n
    my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0      0.7957(80)\n> 1      0.5156(51)\n> 2      0.3227(33)\n> 3      0.2041(21)\n
    \n
    \n\n

    In case the correlation functions are not defined on the outermost timeslices, for example because of fixed boundary conditions, a padding can be introduced.

    \n\n
    \n
    my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3], padding=[1, 1])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0\n> 1      0.7957(80)\n> 2      0.5156(51)\n> 3      0.3227(33)\n> 4      0.2041(21)\n> 5\n
    \n
    \n\n

    The individual entries of a correlator can be accessed via slicing

    \n\n
    \n
    print(my_corr[3])\n> 0.3227(33)\n
    \n
    \n\n

    Error propagation with the Corr class works very similar to Obs objects. Mathematical operations are overloaded and Corr objects can be computed together with other Corr objects, Obs objects or real numbers and integers.

    \n\n
    \n
    my_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr\n
    \n
    \n\n

    pyerrors provides the user with a set of regularly used methods for the manipulation of correlator objects:

    \n\n
      \n
    • Corr.gamma_method applies the gamma method to all entries of the correlator.
      • \n
    • Corr.m_eff to construct effective masses. Various variants for periodic and fixed temporal boundary conditions are available.
      • \n
    • Corr.deriv returns the first derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
      • \n
    • Corr.second_deriv returns the second derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
      • \n
    • Corr.symmetric symmetrizes parity even correlations functions, assuming periodic boundary conditions.
      • \n
    • Corr.anti_symmetric anti-symmetrizes parity odd correlations functions, assuming periodic boundary conditions.
      • \n
    • Corr.T_symmetry averages a correlator with its time symmetry partner, assuming fixed boundary conditions.
      • \n
    • Corr.plateau extracts a plateau value from the correlator in a given range.
      • \n
    • Corr.roll periodically shifts the correlator.
      • \n
    • Corr.reverse reverses the time ordering of the correlator.
      • \n
    • Corr.correlate constructs a disconnected correlation function from the correlator and another Corr or Obs object.
      • \n
    • Corr.reweight reweights the correlator.
      • \n
    \n\n

    pyerrors can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (see pyerrors.correlators.Corr.GEVP).

    \n\n

    For the full API see pyerrors.correlators.Corr.

    \n\n

    Complex valued observables

    \n\n

    pyerrors can handle complex valued observables via the class pyerrors.obs.CObs.\nCObs are initialized with a real and an imaginary part which both can be Obs valued.

    \n\n
    \n
    my_real_part = pe.Obs([samples1], ['ensemble1'])\nmy_imag_part = pe.Obs([samples2], ['ensemble1'])\n\nmy_cobs = pe.CObs(my_real_part, my_imag_part)\nmy_cobs.gamma_method()\nprint(my_cobs)\n> (0.9959(91)+0.659(28)j)\n
    \n
    \n\n

    Elementary mathematical operations are overloaded and samples are properly propagated as for the Obs class.

    \n\n
    \n
    my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs)\nmy_derived_cobs.gamma_method()\nprint(my_derived_cobs)\n> (1.668(23)+0.0j)\n
    \n
    \n\n

    The Covobs class

    \n\n

    In many projects, auxiliary data that is not based on Monte Carlo chains enters. Examples are experimentally determined mesons masses which are used to set the scale or renormalization constants. These numbers come with an error that has to be propagated through the analysis. The Covobs class allows to define such quantities in pyerrors. Furthermore, external input might consist of correlated quantities. An example are the parameters of an interpolation formula, which are defined via mean values and a covariance matrix between all parameters. The contribution of the interpolation formula to the error of a derived quantity therefore might depend on the complete covariance matrix.

    \n\n

    This concept is built into the definition of Covobs. In pyerrors, external input is defined by $M$ mean values, a $M\\times M$ covariance matrix, where $M=1$ is permissible, and a name that uniquely identifies the covariance matrix. Below, we define the pion mass, based on its mean value and error, 134.9768(5). Note, that the square of the error enters cov_Obs, since the second argument of this function is the covariance matrix of the Covobs.

    \n\n
    \n
    import pyerrors.obs as pe\n\nmpi = pe.cov_Obs(134.9768, 0.0005**2, 'pi^0 mass')\nmpi.gamma_method()\nmpi.details()\n> Result         1.34976800e+02 +/- 5.00000000e-04 +/- 0.00000000e+00 (0.000%)\n>  pi^0 mass     5.00000000e-04\n> 0 samples in 1 ensemble:\n>   \u00b7 Covobs   'pi^0 mass'\n
    \n
    \n\n

    The resulting object mpi is an Obs that contains a Covobs. In the following, it may be handled as any other Obs. The contribution of the covariance matrix to the error of an Obs is determined from the $M \\times M$ covariance matrix $\\Sigma$ and the gradient of the Obs with respect to the external quantities, which is the $1\\times M$ Jacobian matrix $J$, via\n$$s = \\sqrt{J^T \\Sigma J}\\,,$$\nwhere the Jacobian is computed for each derived quantity via automatic differentiation.

    \n\n

    Correlated auxiliary data is defined similarly to above, e.g., via

    \n\n
    \n
    RAP = pe.cov_Obs([16.7457, -19.0475], [[3.49591, -6.07560], [-6.07560, 10.5834]], 'R_AP, 1906.03445, (5.3a)')\nprint(RAP)\n> [Obs[16.7(1.9)], Obs[-19.0(3.3)]]\n
    \n
    \n\n

    where RAP now is a list of two Obs that contains the two correlated parameters.

    \n\n

    Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the Covobs class allows to quote the derivative of a result with respect to the external quantities. If these derivatives are published together with the result, small shifts in the definition of external quantities, e.g., the definition of the physical point, can be performed a posteriori based on the published information. This may help to compare results of different groups. The gradient of an Obs o with respect to a covariance matrix with the identifying string k may be accessed via

    \n\n
    \n
    o.covobs[k].grad\n
    \n
    \n\n

    Error propagation in iterative algorithms

    \n\n

    pyerrors supports exact linear error propagation for iterative algorithms like various variants of non-linear least squares fits or root finding. The derivatives required for the error propagation are calculated as described in arXiv:1809.01289.

    \n\n

    Least squares fits

    \n\n

    Standard non-linear least square fits with errors on the dependent but not the independent variables can be performed with pyerrors.fits.least_squares. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.

    \n\n

    Fit functions have to be of the following form

    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n    return a[1] * anp.exp(-a[0] * x)\n
    \n
    \n\n

    It is important that numerical functions refer to autograd.numpy instead of numpy for the automatic differentiation in iterative algorithms to work properly.

    \n\n

    Fits can then be performed via

    \n\n
    \n
    fit_result = pe.fits.least_squares(x, y, func)\nprint("\\n", fit_result)\n> Fit with 2 parameters\n> Method: Levenberg-Marquardt\n> `ftol` termination condition is satisfied.\n> chisquare/d.o.f.: 0.9593035785160936\n\n>  Goodness of fit:\n> \u03c7\u00b2/d.o.f. = 0.959304\n> p-value   = 0.5673\n> Fit parameters:\n> 0      0.0548(28)\n> 1      1.933(64)\n
    \n
    \n\n

    where x is a list or numpy.array of floats and y is a list or numpy.array of Obs.

    \n\n

    Data stored in Corr objects can be fitted directly using the Corr.fit method.

    \n\n
    \n
    my_corr = pe.Corr(y)\nfit_result = my_corr.fit(func, fitrange=[12, 25])\n
    \n
    \n\n

    this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.

    \n\n

    For fit functions with multiple independent variables the fit function can be of the form

    \n\n
    \n
    def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n

    pyerrors also supports correlated fits which can be triggered via the parameter correlated_fit=True.\nDetails about how the required covariance matrix is estimated can be found in pyerrors.obs.covariance.\nDirect visualizations of the performed fits can be triggered via resplot=True or qqplot=True.

    \n\n

    For all available options including combined fits to multiple datasets see pyerrors.fits.least_squares.

    \n\n

    Total least squares fits

    \n\n

    pyerrors can also fit data with errors on both the dependent and independent variables using the total least squares method also referred to as orthogonal distance regression as implemented in scipy, see pyerrors.fits.least_squares. The syntax is identical to the standard least squares case, the only difference being that x also has to be a list or numpy.array of Obs.

    \n\n

    For the full API see pyerrors.fits for fits and pyerrors.roots for finding roots of functions.

    \n\n

    Matrix operations

    \n\n

    pyerrors provides wrappers for Obs- and CObs-valued matrix operations based on numpy.linalg. The supported functions include:

    \n\n
      \n
    • inv for the matrix inverse.
      • \n
    • cholseky for the Cholesky decomposition.
      • \n
    • det for the matrix determinant.
      • \n
    • eigh for eigenvalues and eigenvectors of hermitean matrices.
      • \n
    • eig for eigenvalues of general matrices.
      • \n
    • pinv for the Moore-Penrose pseudoinverse.
      • \n
    • svd for the singular-value-decomposition.
      • \n
    \n\n

    For the full API see pyerrors.linalg.

    \n\n

    Export data

    \n\n

    \n\n

    The preferred exported file format within pyerrors is json.gz. Files written to this format are valid JSON files that have been compressed using gzip. The structure of the content is inspired by the dobs format of the ALPHA collaboration. The aim of the format is to facilitate the storage of data in a self-contained way such that, even years after the creation of the file, it is possible to extract all necessary information:

    \n\n
      \n
    • What observables are stored? Possibly: How exactly are they defined.
      • \n
    • How does each single ensemble or external quantity contribute to the error of the observable?
      • \n
    • Who did write the file when and on which machine?
      • \n
    \n\n

    This can be achieved by storing all information in one single file. The export routines of pyerrors are written such that as much information as possible is written automatically as described in the following example

    \n\n
    \n
    my_obs = pe.Obs([samples], ["test_ensemble"])\nmy_obs.tag = "My observable"\n\npe.input.json.dump_to_json(my_obs, "test_output_file", description="This file contains a test observable")\n# For a single observable one can equivalently use the class method dump\nmy_obs.dump("test_output_file", description="This file contains a test observable")\n\ncheck = pe.input.json.load_json("test_output_file")\n\nprint(my_obs == check)\n> True\n
    \n
    \n\n

    The format also allows to directly write out the content of Corr objects or lists and arrays of Obs objects by passing the desired data to pyerrors.input.json.dump_to_json.

    \n\n

    json.gz format specification

    \n\n

    The first entries of the file provide optional auxiliary information:

    \n\n
      \n
    • program is a string that indicates which program was used to write the file.
      • \n
    • version is a string that specifies the version of the format.
      • \n
    • who is a string that specifies the user name of the creator of the file.
      • \n
    • date is a string and contains the creation date of the file.
      • \n
    • host is a string and contains the hostname of the machine where the file has been written.
      • \n
    • description contains information on the content of the file. This field is not filled automatically in pyerrors. The user is advised to provide as detailed information as possible in this field. Examples are: Input files of measurements or simulations, LaTeX formulae or references to publications to specify how the observables have been computed, details on the analysis strategy, ... This field may be any valid JSON type. Strings, arrays or objects (equivalent to dicts in python) are well suited to provide information.
      • \n
    \n\n

    The only necessary entry of the file is the field\n-obsdata, an array that contains the actual data.

    \n\n

    Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of Obs, list, numpy.ndarray, Corr. All Obs inside a structure (with dimension > 0) have to be defined on the same set of configurations. Different structures, that are represented by entries of the array obsdata, are treated independently. Each entry of the array obsdata has the following required entries:

    \n\n
      \n
    • type is a string that specifies the type of the structure. This allows to parse the content to the correct form after reading the file. It is always possible to interpret the content as list of Obs.
      • \n
    • value is an array that contains the mean values of the Obs inside the structure.\nThe following entries are optional:
      • \n
    • layout is a string that specifies the layout of multi-dimensional structures. Examples are \"2, 2\" for a 2x2 dimensional matrix or \"64, 4, 4\" for a Corr with $T=64$ and 4x4 matrices on each time slices. \"1\" denotes a single Obs. Multi-dimensional structures are stored in row-major format (see below).
      • \n
    • tag is any JSON type. It contains additional information concerning the structure. The tag of an Obs in pyerrors is written here.
      • \n
    • reweighted is a Bool that may be used to specify, whether the Obs in the structure have been reweighted.
      • \n
    • data is an array that contains the data from MC chains. We will define it below.
      • \n
    • cdata is an array that contains the data from external quantities with an error (Covobs in pyerrors). We will define it below.
      • \n
    \n\n

    The array data contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:

    \n\n
      \n
    • id, a string that contains the name of the ensemble
      • \n
    • replica, an array that contains an entry per replica of the ensemble.
      • \n
    \n\n

    Each entry of replica contains\nname, a string that contains the name of the replica\ndeltas, an array that contains the actual data.

    \n\n

    Each entry in deltas corresponds to one configuration of the replica and has $1+N$ many entries. The first entry is an integer that specifies the configuration number that, together with ensemble and replica name, may be used to uniquely identify the configuration on which the data has been obtained. The following N entries specify the deltas, i.e., the deviation of the observable from the mean value on this configuration, of each Obs inside the structure. Multi-dimensional structures are stored in a row-major format. For primary observables, such as correlation functions, $value + delta_i$ matches the primary data obtained on the configuration.

    \n\n

    The array cdata contains information about the contribution of auxiliary observables, represented by Covobs in pyerrors, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:

    \n\n
      \n
    • id, a string that identifies the covariance matrix
      • \n
    • layout, a string that defines the dimensions of the $M\\times M$ covariance matrix (has to be \"M, M\" or \"1\").
      • \n
    • cov, an array that contains the $M\\times M$ many entries of the covariance matrix, stored in row-major format.
      • \n
    • grad, an array that contains N entries, one for each Obs inside the structure. Each entry itself is an array, that contains the M gradients of the Nth observable with respect to the quantity that corresponds to the Mth diagonal entry of the covariance matrix.
      • \n
    \n\n

    A JSON schema that may be used to verify the correctness of a file with respect to the format definition is stored in ./examples/json_schema.json. The schema is a self-descriptive format definition and contains an exemplary file.

    \n\n

    Julia I/O routines for the json.gz format, compatible with ADerrors.jl, can be found here.

    \n"}, "pyerrors.correlators": {"fullname": "pyerrors.correlators", "modulename": "pyerrors.correlators", "kind": "module", "doc": "

    \n"}, "pyerrors.correlators.Corr": {"fullname": "pyerrors.correlators.Corr", "modulename": "pyerrors.correlators", "qualname": "Corr", "kind": "class", "doc": "

    The class for a correlator (time dependent sequence of pe.Obs).

    \n\n

    Everything, this class does, can be achieved using lists or arrays of Obs.\nBut it is simply more convenient to have a dedicated object for correlators.\nOne often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient\nto iterate over all timeslices for every operation. This is especially true, when dealing with matrices.

    \n\n

    The correlator can have two types of content: An Obs at every timeslice OR a matrix at every timeslice.\nOther dependency (eg. spatial) are not supported.

    \n\n

    The Corr class can also deal with missing measurements or paddings for fixed boundary conditions.\nThe missing entries are represented via the None object.

    \n\n
    Initialization
    \n\n

    A simple correlator can be initialized with a list or a one-dimensional array of Obs or Cobs

    \n\n
    \n
    corr11 = pe.Corr([obs1, obs2])\ncorr11 = pe.Corr(np.array([obs1, obs2]))\n
    \n
    \n\n

    A matrix-valued correlator can either be initialized via a two-dimensional array of Corr objects

    \n\n
    \n
    matrix_corr = pe.Corr(np.array([[corr11, corr12], [corr21, corr22]]))\n
    \n
    \n\n

    or alternatively via a three-dimensional array of Obs or CObs of shape (T, N, N) where T is\nthe temporal extent of the correlator and N is the dimension of the matrix.

    \n"}, "pyerrors.correlators.Corr.__init__": {"fullname": "pyerrors.correlators.Corr.__init__", "modulename": "pyerrors.correlators", "qualname": "Corr.__init__", "kind": "function", "doc": "

    Initialize a Corr object.

    \n\n
    Parameters
    \n\n
      \n
    • data_input (list or array):\nlist of Obs or list of arrays of Obs or array of Corrs (see class docstring for details).
      • \n
    • padding (list, optional):\nList with two entries where the first labels the padding\nat the front of the correlator and the second the padding\nat the back.
      • \n
    • prange (list, optional):\nList containing the first and last timeslice of the plateau\nregion identified for this correlator.
      • \n
    \n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "pyerrors.correlators.Corr.tag": {"fullname": "pyerrors.correlators.Corr.tag", "modulename": "pyerrors.correlators", "qualname": "Corr.tag", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.content": {"fullname": "pyerrors.correlators.Corr.content", "modulename": "pyerrors.correlators", "qualname": "Corr.content", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.T": {"fullname": "pyerrors.correlators.Corr.T", "modulename": "pyerrors.correlators", "qualname": "Corr.T", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.prange": {"fullname": "pyerrors.correlators.Corr.prange", "modulename": "pyerrors.correlators", "qualname": "Corr.prange", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.reweighted": {"fullname": "pyerrors.correlators.Corr.reweighted", "modulename": "pyerrors.correlators", "qualname": "Corr.reweighted", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.gamma_method": {"fullname": "pyerrors.correlators.Corr.gamma_method", "modulename": "pyerrors.correlators", "qualname": "Corr.gamma_method", "kind": "function", "doc": "

    Apply the gamma method to the content of the Corr.

    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.gm": {"fullname": "pyerrors.correlators.Corr.gm", "modulename": "pyerrors.correlators", "qualname": "Corr.gm", "kind": "function", "doc": "

    Apply the gamma method to the content of the Corr.

    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.projected": {"fullname": "pyerrors.correlators.Corr.projected", "modulename": "pyerrors.correlators", "qualname": "Corr.projected", "kind": "function", "doc": "

    We need to project the Correlator with a Vector to get a single value at each timeslice.

    \n\n

    The method can use one or two vectors.\nIf two are specified it returns v1@G@v2 (the order might be very important.)\nBy default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to

    \n", "signature": "(self, vector_l=None, vector_r=None, normalize=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.item": {"fullname": "pyerrors.correlators.Corr.item", "modulename": "pyerrors.correlators", "qualname": "Corr.item", "kind": "function", "doc": "

    Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.

    \n\n
    Parameters
    \n\n
      \n
    • i (int):\nFirst index to be picked.
      • \n
    • j (int):\nSecond index to be picked.
      • \n
    \n", "signature": "(self, i, j):", "funcdef": "def"}, "pyerrors.correlators.Corr.plottable": {"fullname": "pyerrors.correlators.Corr.plottable", "modulename": "pyerrors.correlators", "qualname": "Corr.plottable", "kind": "function", "doc": "

    Outputs the correlator in a plotable format.

    \n\n

    Outputs three lists containing the timeslice index, the value on each\ntimeslice and the error on each timeslice.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.symmetric": {"fullname": "pyerrors.correlators.Corr.symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.symmetric", "kind": "function", "doc": "

    Symmetrize the correlator around x0=0.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.anti_symmetric": {"fullname": "pyerrors.correlators.Corr.anti_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.anti_symmetric", "kind": "function", "doc": "

    Anti-symmetrize the correlator around x0=0.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.is_matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.is_matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.is_matrix_symmetric", "kind": "function", "doc": "

    Checks whether a correlator matrices is symmetric on every timeslice.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.trace": {"fullname": "pyerrors.correlators.Corr.trace", "modulename": "pyerrors.correlators", "qualname": "Corr.trace", "kind": "function", "doc": "

    Calculates the per-timeslice trace of a correlator matrix.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.matrix_symmetric", "kind": "function", "doc": "

    Symmetrizes the correlator matrices on every timeslice.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.GEVP": {"fullname": "pyerrors.correlators.Corr.GEVP", "modulename": "pyerrors.correlators", "qualname": "Corr.GEVP", "kind": "function", "doc": "

    Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.

    \n\n

    The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the\nlargest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing

    \n\n
    \n
    C.GEVP(t0=2)[0]  # Ground state vector(s)\nC.GEVP(t0=2)[:3]  # Vectors for the lowest three states\n
    \n
    \n\n
    Parameters
    \n\n
      \n
    • t0 (int):\nThe time t0 for the right hand side of the GEVP according to $G(t)v_i=\\lambda_i G(t_0)v_i$
      • \n
    • ts (int):\nfixed time $G(t_s)v_i=\\lambda_i G(t_0)v_i$ if sort=None.\nIf sort=\"Eigenvector\" it gives a reference point for the sorting method.
      • \n
    • sort (string):\nIf this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.\n
        \n
      • \"Eigenvalue\": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice. (default)
        • \n
      • \"Eigenvector\": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.\nThe reference state is identified by its eigenvalue at $t=t_s$.
        • \n
      • None: The GEVP is solved only at ts, no sorting is necessary
        • \n
      • \n
    • vector_obs (bool):\nIf True, uncertainties are propagated in the eigenvector computation (default False).
      • \n
    \n\n
    Other Parameters
    \n\n
      \n
    • state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
      • \n
    • method (str):\nMethod used to solve the GEVP.\n
        \n
      • \"eigh\": Use scipy.linalg.eigh to solve the GEVP. (default for vector_obs=False)
        • \n
      • \"cholesky\": Use manually implemented solution via the Cholesky decomposition. Automatically chosen if vector_obs==True.
        • \n
      • \n
    \n", "signature": "(self, t0, ts=None, sort='Eigenvalue', vector_obs=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Eigenvalue": {"fullname": "pyerrors.correlators.Corr.Eigenvalue", "modulename": "pyerrors.correlators", "qualname": "Corr.Eigenvalue", "kind": "function", "doc": "

    Determines the eigenvalue of the GEVP by solving and projecting the correlator

    \n\n
    Parameters
    \n\n
      \n
    • state (int):\nThe state one is interested in ordered by energy. The lowest state is zero.
      • \n
    • All other parameters are identical to the ones of Corr.GEVP.
      • \n
    \n", "signature": "(self, t0, ts=None, state=0, sort='Eigenvalue', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Hankel": {"fullname": "pyerrors.correlators.Corr.Hankel", "modulename": "pyerrors.correlators", "qualname": "Corr.Hankel", "kind": "function", "doc": "

    Constructs an NxN Hankel matrix

    \n\n

    C(t) c(t+1) ... c(t+n-1)\nC(t+1) c(t+2) ... c(t+n)\n.................\nC(t+(n-1)) c(t+n) ... c(t+2(n-1))

    \n\n
    Parameters
    \n\n
      \n
    • N (int):\nDimension of the Hankel matrix
      • \n
    • periodic (bool, optional):\ndetermines whether the matrix is extended periodically
      • \n
    \n", "signature": "(self, N, periodic=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.roll": {"fullname": "pyerrors.correlators.Corr.roll", "modulename": "pyerrors.correlators", "qualname": "Corr.roll", "kind": "function", "doc": "

    Periodically shift the correlator by dt timeslices

    \n\n
    Parameters
    \n\n
      \n
    • dt (int):\nnumber of timeslices
      • \n
    \n", "signature": "(self, dt):", "funcdef": "def"}, "pyerrors.correlators.Corr.reverse": {"fullname": "pyerrors.correlators.Corr.reverse", "modulename": "pyerrors.correlators", "qualname": "Corr.reverse", "kind": "function", "doc": "

    Reverse the time ordering of the Corr

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.thin": {"fullname": "pyerrors.correlators.Corr.thin", "modulename": "pyerrors.correlators", "qualname": "Corr.thin", "kind": "function", "doc": "

    Thin out a correlator to suppress correlations

    \n\n
    Parameters
    \n\n
      \n
    • spacing (int):\nKeep only every 'spacing'th entry of the correlator
      • \n
    • offset (int):\nOffset the equal spacing
      • \n
    \n", "signature": "(self, spacing=2, offset=0):", "funcdef": "def"}, "pyerrors.correlators.Corr.correlate": {"fullname": "pyerrors.correlators.Corr.correlate", "modulename": "pyerrors.correlators", "qualname": "Corr.correlate", "kind": "function", "doc": "

    Correlate the correlator with another correlator or Obs

    \n\n
    Parameters
    \n\n
      \n
    • partner (Obs or Corr):\npartner to correlate the correlator with.\nCan either be an Obs which is correlated with all entries of the\ncorrelator or a Corr of same length.
      • \n
    \n", "signature": "(self, partner):", "funcdef": "def"}, "pyerrors.correlators.Corr.reweight": {"fullname": "pyerrors.correlators.Corr.reweight", "modulename": "pyerrors.correlators", "qualname": "Corr.reweight", "kind": "function", "doc": "

    Reweight the correlator.

    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl.
      • \n
    \n", "signature": "(self, weight, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.T_symmetry": {"fullname": "pyerrors.correlators.Corr.T_symmetry", "modulename": "pyerrors.correlators", "qualname": "Corr.T_symmetry", "kind": "function", "doc": "

    Return the time symmetry average of the correlator and its partner

    \n\n
    Parameters
    \n\n
      \n
    • partner (Corr):\nTime symmetry partner of the Corr
      • \n
    • parity (int):\nParity quantum number of the correlator, can be +1 or -1
      • \n
    \n", "signature": "(self, partner, parity=1):", "funcdef": "def"}, "pyerrors.correlators.Corr.deriv": {"fullname": "pyerrors.correlators.Corr.deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.deriv", "kind": "function", "doc": "

    Return the first derivative of the correlator with respect to x0.

    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, forward, backward, improved, log, default: symmetric
      • \n
    \n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.second_deriv": {"fullname": "pyerrors.correlators.Corr.second_deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.second_deriv", "kind": "function", "doc": "

    Return the second derivative of the correlator with respect to x0.

    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice:\n - symmetric (default)\n $$\\tilde{\\partial}^2_0 f(x_0) = f(x_0+1)-2f(x_0)+f(x_0-1)$$\n - big_symmetric\n $$\\partial^2_0 f(x_0) = \\frac{f(x_0+2)-2f(x_0)+f(x_0-2)}{4}$$\n - improved\n $$\\partial^2_0 f(x_0) = \\frac{-f(x_0+2) + 16 * f(x_0+1) - 30 * f(x_0) + 16 * f(x_0-1) - f(x_0-2)}{12}$$\n - log\n $$f(x) = \\tilde{\\partial}^2_0 log(f(x_0))+(\\tilde{\\partial}_0 log(f(x_0)))^2$$
      • \n
    \n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.m_eff": {"fullname": "pyerrors.correlators.Corr.m_eff", "modulename": "pyerrors.correlators", "qualname": "Corr.m_eff", "kind": "function", "doc": "

    Returns the effective mass of the correlator as correlator object

    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\nlog : uses the standard effective mass log(C(t) / C(t+1))\ncosh, periodic : Use periodicity of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.\nsinh : Use anti-periodicity of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.\nSee, e.g., arXiv:1205.5380\narccosh : Uses the explicit form of the symmetrized correlator (not recommended)\nlogsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
      • \n
    • guess (float):\nguess for the root finder, only relevant for the root variant
      • \n
    \n", "signature": "(self, variant='log', guess=1.0):", "funcdef": "def"}, "pyerrors.correlators.Corr.fit": {"fullname": "pyerrors.correlators.Corr.fit", "modulename": "pyerrors.correlators", "qualname": "Corr.fit", "kind": "function", "doc": "

    Fits function to the data

    \n\n
    Parameters
    \n\n
      \n
    • function (obj):\nfunction to fit to the data. See fits.least_squares for details.
      • \n
    • fitrange (list):\nTwo element list containing the timeslices on which the fit is supposed to start and stop.\nCaution: This range is inclusive as opposed to standard python indexing.\nfitrange=[4, 6] corresponds to the three entries 4, 5 and 6.\nIf not specified, self.prange or all timeslices are used.
      • \n
    • silent (bool):\nDecides whether output is printed to the standard output.
      • \n
    \n", "signature": "(self, function, fitrange=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.plateau": {"fullname": "pyerrors.correlators.Corr.plateau", "modulename": "pyerrors.correlators", "qualname": "Corr.plateau", "kind": "function", "doc": "

    Extract a plateau value from a Corr object

    \n\n
    Parameters
    \n\n
      \n
    • plateau_range (list):\nlist with two entries, indicating the first and the last timeslice\nof the plateau region.
      • \n
    • method (str):\nmethod to extract the plateau.\n 'fit' fits a constant to the plateau region\n 'avg', 'average' or 'mean' just average over the given timeslices.
      • \n
    • auto_gamma (bool):\napply gamma_method with default parameters to the Corr. Defaults to None
      • \n
    \n", "signature": "(self, plateau_range=None, method='fit', auto_gamma=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.set_prange": {"fullname": "pyerrors.correlators.Corr.set_prange", "modulename": "pyerrors.correlators", "qualname": "Corr.set_prange", "kind": "function", "doc": "

    Sets the attribute prange of the Corr object.

    \n", "signature": "(self, prange):", "funcdef": "def"}, "pyerrors.correlators.Corr.show": {"fullname": "pyerrors.correlators.Corr.show", "modulename": "pyerrors.correlators", "qualname": "Corr.show", "kind": "function", "doc": "

    Plots the correlator using the tag of the correlator as label if available.

    \n\n
    Parameters
    \n\n
      \n
    • x_range (list):\nlist of two values, determining the range of the x-axis e.g. [4, 8].
      • \n
    • comp (Corr or list of Corr):\nCorrelator or list of correlators which are plotted for comparison.\nThe tags of these correlators are used as labels if available.
      • \n
    • logscale (bool):\nSets y-axis to logscale.
      • \n
    • plateau (Obs):\nPlateau value to be visualized in the figure.
      • \n
    • fit_res (Fit_result):\nFit_result object to be visualized.
      • \n
    • fit_key (str):\nKey for the fit function in Fit_result.fit_function (for combined fits).
      • \n
    • ylabel (str):\nLabel for the y-axis.
      • \n
    • save (str):\npath to file in which the figure should be saved.
      • \n
    • auto_gamma (bool):\nApply the gamma method with standard parameters to all correlators and plateau values before plotting.
      • \n
    • hide_sigma (float):\nHides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
      • \n
    • references (list):\nList of floating point values that are displayed as horizontal lines for reference.
      • \n
    • title (string):\nOptional title of the figure.
      • \n
    \n", "signature": "(\tself,\tx_range=None,\tcomp=None,\ty_range=None,\tlogscale=False,\tplateau=None,\tfit_res=None,\tfit_key=None,\tylabel=None,\tsave=None,\tauto_gamma=False,\thide_sigma=None,\treferences=None,\ttitle=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.spaghetti_plot": {"fullname": "pyerrors.correlators.Corr.spaghetti_plot", "modulename": "pyerrors.correlators", "qualname": "Corr.spaghetti_plot", "kind": "function", "doc": "

    Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.

    \n\n
    Parameters
    \n\n
      \n
    • logscale (bool):\nDetermines whether the scale of the y-axis is logarithmic or standard.
      • \n
    \n", "signature": "(self, logscale=True):", "funcdef": "def"}, "pyerrors.correlators.Corr.dump": {"fullname": "pyerrors.correlators.Corr.dump", "modulename": "pyerrors.correlators", "qualname": "Corr.dump", "kind": "function", "doc": "

    Dumps the Corr into a file of chosen type

    \n\n
    Parameters
    \n\n
      \n
    • filename (str):\nName of the file to be saved.
      • \n
    • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n", "signature": "(self, filename, datatype='json.gz', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.print": {"fullname": "pyerrors.correlators.Corr.print", "modulename": "pyerrors.correlators", "qualname": "Corr.print", "kind": "function", "doc": "

    \n", "signature": "(self, print_range=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.sqrt": {"fullname": "pyerrors.correlators.Corr.sqrt", "modulename": "pyerrors.correlators", "qualname": "Corr.sqrt", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.log": {"fullname": "pyerrors.correlators.Corr.log", "modulename": "pyerrors.correlators", "qualname": "Corr.log", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.exp": {"fullname": "pyerrors.correlators.Corr.exp", "modulename": "pyerrors.correlators", "qualname": "Corr.exp", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sin": {"fullname": "pyerrors.correlators.Corr.sin", "modulename": "pyerrors.correlators", "qualname": "Corr.sin", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cos": {"fullname": "pyerrors.correlators.Corr.cos", "modulename": "pyerrors.correlators", "qualname": "Corr.cos", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tan": {"fullname": "pyerrors.correlators.Corr.tan", "modulename": "pyerrors.correlators", "qualname": "Corr.tan", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sinh": {"fullname": "pyerrors.correlators.Corr.sinh", "modulename": "pyerrors.correlators", "qualname": "Corr.sinh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cosh": {"fullname": "pyerrors.correlators.Corr.cosh", "modulename": "pyerrors.correlators", "qualname": "Corr.cosh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tanh": {"fullname": "pyerrors.correlators.Corr.tanh", "modulename": "pyerrors.correlators", "qualname": "Corr.tanh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsin": {"fullname": "pyerrors.correlators.Corr.arcsin", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsin", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccos": {"fullname": "pyerrors.correlators.Corr.arccos", "modulename": "pyerrors.correlators", "qualname": "Corr.arccos", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctan": {"fullname": "pyerrors.correlators.Corr.arctan", "modulename": "pyerrors.correlators", "qualname": "Corr.arctan", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsinh": {"fullname": "pyerrors.correlators.Corr.arcsinh", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsinh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccosh": {"fullname": "pyerrors.correlators.Corr.arccosh", "modulename": "pyerrors.correlators", "qualname": "Corr.arccosh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctanh": {"fullname": "pyerrors.correlators.Corr.arctanh", "modulename": "pyerrors.correlators", "qualname": "Corr.arctanh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.real": {"fullname": "pyerrors.correlators.Corr.real", "modulename": "pyerrors.correlators", "qualname": "Corr.real", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.imag": {"fullname": "pyerrors.correlators.Corr.imag", "modulename": "pyerrors.correlators", "qualname": "Corr.imag", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.prune": {"fullname": "pyerrors.correlators.Corr.prune", "modulename": "pyerrors.correlators", "qualname": "Corr.prune", "kind": "function", "doc": "

    Project large correlation matrix to lowest states

    \n\n

    This method can be used to reduce the size of an (N x N) correlation matrix\nto (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise\nis still small.

    \n\n
    Parameters
    \n\n
      \n
    • Ntrunc (int):\nRank of the target matrix.
      • \n
    • tproj (int):\nTime where the eigenvectors are evaluated, corresponds to ts in the GEVP method.\nThe default value is 3.
      • \n
    • t0proj (int):\nTime where the correlation matrix is inverted. Choosing t0proj=1 is strongly\ndiscouraged for O(a) improved theories, since the correctness of the procedure\ncannot be granted in this case. The default value is 2.
      • \n
    • basematrix (Corr):\nCorrelation matrix that is used to determine the eigenvectors of the\nlowest states based on a GEVP. basematrix is taken to be the Corr itself if\nis is not specified.
      • \n
    \n\n
    Notes
    \n\n

    We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving\nthe GEVP $$C(t) v_n(t, t_0) = \\lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \\equiv t_\\mathrm{proj}$\nand $t_0 \\equiv t_{0, \\mathrm{proj}}$. The target matrix is projected onto the subspace of the\nresulting eigenvectors $v_n, n=1,\\dots,N_\\mathrm{trunc}$ via\n$$G^\\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large\ncorrelation matrix and to remove some noise that is added by irrelevant operators.\nThis may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated\nbound $t_0 \\leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.

    \n", "signature": "(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.N": {"fullname": "pyerrors.correlators.Corr.N", "modulename": "pyerrors.correlators", "qualname": "Corr.N", "kind": "variable", "doc": "

    \n"}, "pyerrors.covobs": {"fullname": "pyerrors.covobs", "modulename": "pyerrors.covobs", "kind": "module", "doc": "

    \n"}, "pyerrors.covobs.Covobs": {"fullname": "pyerrors.covobs.Covobs", "modulename": "pyerrors.covobs", "qualname": "Covobs", "kind": "class", "doc": "

    \n"}, "pyerrors.covobs.Covobs.__init__": {"fullname": "pyerrors.covobs.Covobs.__init__", "modulename": "pyerrors.covobs", "qualname": "Covobs.__init__", "kind": "function", "doc": "

    Initialize Covobs object.

    \n\n
    Parameters
    \n\n
      \n
    • mean (float):\nMean value of the new Obs
      • \n
    • cov (list or array):\n2d Covariance matrix or 1d diagonal entries
      • \n
    • name (str):\nidentifier for the covariance matrix
      • \n
    • pos (int):\nPosition of the variance belonging to mean in cov.\nIs taken to be 1 if cov is 0-dimensional
      • \n
    • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
      • \n
    \n", "signature": "(mean, cov, name, pos=None, grad=None)"}, "pyerrors.covobs.Covobs.name": {"fullname": "pyerrors.covobs.Covobs.name", "modulename": "pyerrors.covobs", "qualname": "Covobs.name", "kind": "variable", "doc": "

    \n"}, "pyerrors.covobs.Covobs.value": {"fullname": "pyerrors.covobs.Covobs.value", "modulename": "pyerrors.covobs", "qualname": "Covobs.value", "kind": "variable", "doc": "

    \n"}, "pyerrors.covobs.Covobs.errsq": {"fullname": "pyerrors.covobs.Covobs.errsq", "modulename": "pyerrors.covobs", "qualname": "Covobs.errsq", "kind": "function", "doc": "

    Return the variance (= square of the error) of the Covobs

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.covobs.Covobs.cov": {"fullname": "pyerrors.covobs.Covobs.cov", "modulename": "pyerrors.covobs", "qualname": "Covobs.cov", "kind": "variable", "doc": "

    \n"}, "pyerrors.covobs.Covobs.grad": {"fullname": "pyerrors.covobs.Covobs.grad", "modulename": "pyerrors.covobs", "qualname": "Covobs.grad", "kind": "variable", "doc": "

    \n"}, "pyerrors.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "

    \n"}, "pyerrors.dirac.gammaX": {"fullname": "pyerrors.dirac.gammaX", "modulename": "pyerrors.dirac", "qualname": "gammaX", "kind": "variable", "doc": "

    \n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+1.j],\n [ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, -0.-1.j, 0.+0.j, 0.+0.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaY": {"fullname": "pyerrors.dirac.gammaY", "modulename": "pyerrors.dirac", "qualname": "gammaY", "kind": "variable", "doc": "

    \n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j],\n [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [-1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaZ": {"fullname": "pyerrors.dirac.gammaZ", "modulename": "pyerrors.dirac", "qualname": "gammaZ", "kind": "variable", "doc": "

    \n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -0.-1.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+1.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaT": {"fullname": "pyerrors.dirac.gammaT", "modulename": "pyerrors.dirac", "qualname": "gammaT", "kind": "variable", "doc": "

    \n", "default_value": "array([[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gamma": {"fullname": "pyerrors.dirac.gamma", "modulename": "pyerrors.dirac", "qualname": "gamma", "kind": "variable", "doc": "

    \n", "default_value": "array([[[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+1.j],\n [ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, -0.-1.j, 0.+0.j, 0.+0.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j],\n [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [-1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -0.-1.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+1.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n [ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]]])"}, "pyerrors.dirac.gamma5": {"fullname": "pyerrors.dirac.gamma5", "modulename": "pyerrors.dirac", "qualname": "gamma5", "kind": "variable", "doc": "

    \n", "default_value": "array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, -1.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j]])"}, "pyerrors.dirac.identity": {"fullname": "pyerrors.dirac.identity", "modulename": "pyerrors.dirac", "qualname": "identity", "kind": "variable", "doc": "

    \n", "default_value": "array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]])"}, "pyerrors.dirac.epsilon_tensor": {"fullname": "pyerrors.dirac.epsilon_tensor", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor", "kind": "function", "doc": "

    Rank-3 epsilon tensor

    \n\n

    Based on https://codegolf.stackexchange.com/a/160375

    \n\n
    Returns
    \n\n
      \n
    • elem (int):\nElement (i,j,k) of the epsilon tensor of rank 3
      • \n
    \n", "signature": "(i, j, k):", "funcdef": "def"}, "pyerrors.dirac.epsilon_tensor_rank4": {"fullname": "pyerrors.dirac.epsilon_tensor_rank4", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor_rank4", "kind": "function", "doc": "

    Rank-4 epsilon tensor

    \n\n

    Extension of https://codegolf.stackexchange.com/a/160375

    \n\n
    Returns
    \n\n
      \n
    • elem (int):\nElement (i,j,k,o) of the epsilon tensor of rank 4
      • \n
    \n", "signature": "(i, j, k, o):", "funcdef": "def"}, "pyerrors.dirac.Grid_gamma": {"fullname": "pyerrors.dirac.Grid_gamma", "modulename": "pyerrors.dirac", "qualname": "Grid_gamma", "kind": "function", "doc": "

    Returns gamma matrix in Grid labeling.

    \n", "signature": "(gamma_tag):", "funcdef": "def"}, "pyerrors.fits": {"fullname": "pyerrors.fits", "modulename": "pyerrors.fits", "kind": "module", "doc": "

    \n"}, "pyerrors.fits.Fit_result": {"fullname": "pyerrors.fits.Fit_result", "modulename": "pyerrors.fits", "qualname": "Fit_result", "kind": "class", "doc": "

    Represents fit results.

    \n\n
    Attributes
    \n\n
      \n
    • fit_parameters (list):\nresults for the individual fit parameters,\nalso accessible via indices.
      • \n
    • chisquare_by_dof (float):\nreduced chisquare.
      • \n
    • p_value (float):\np-value of the fit
      • \n
    • t2_p_value (float):\nHotelling t-squared p-value for correlated fits.
      • \n
    \n", "bases": "collections.abc.Sequence"}, "pyerrors.fits.Fit_result.fit_parameters": {"fullname": "pyerrors.fits.Fit_result.fit_parameters", "modulename": "pyerrors.fits", "qualname": "Fit_result.fit_parameters", "kind": "variable", "doc": "

    \n"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "

    Apply the gamma method to all fit parameters

    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.Fit_result.gm": {"fullname": "pyerrors.fits.Fit_result.gm", "modulename": "pyerrors.fits", "qualname": "Fit_result.gm", "kind": "function", "doc": "

    Apply the gamma method to all fit parameters

    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.least_squares": {"fullname": "pyerrors.fits.least_squares", "modulename": "pyerrors.fits", "qualname": "least_squares", "kind": "function", "doc": "

    Performs a non-linear fit to y = func(x).\n ```

    \n\n
    Parameters
    \n\n
      \n
    • For an uncombined fit:
      • \n
    • x (list):\nlist of floats.
      • \n
    • y (list):\nlist of Obs.
      • \n

    • func (object):\nfit function, has to be of the form

      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(a, x):\n   return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
      \n
      \n\n

      For multiple x values func can be of the form

      \n\n
      \n
      def func(a, x):\n   (x1, x2) = x\n   return a[0] * x1 ** 2 + a[1] * x2\n
      \n
      \n\n

      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.

      • \n
    • OR For a combined fit:
      • \n
    • x (dict):\ndict of lists.
      • \n
    • y (dict):\ndict of lists of Obs.
      • \n

    • funcs (dict):\ndict of objects\nfit functions have to be of the form (here a[0] is the common fit parameter)\n```python\nimport autograd.numpy as anp\nfuncs = {\"a\": func_a,\n \"b\": func_b}

      \n\n

      def func_a(a, x):\n return a[1] * anp.exp(-a[0] * x)

      \n\n

      def func_b(a, x):\n return a[2] * anp.exp(-a[0] * x)

      \n\n

      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.

      • \n
    • priors (dict or list, optional):\npriors can either be a dictionary with integer keys and the corresponding priors as values or\na list with an entry for every parameter in the fit. The entries can either be\nObs (e.g. results from a previous fit) or strings containing a value and an error formatted like\n0.548(23), 500(40) or 0.5(0.4)
      • \n
    • silent (bool, optional):\nIf true all output to the console is omitted (default False).
      • \n
    • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for\nnon-linear fits with many parameters. In case of correlated fits the guess is used to perform\nan uncorrelated fit which then serves as guess for the correlated fit.
      • \n
    • method (str, optional):\ncan be used to choose an alternative method for the minimization of chisquare.\nThe possible methods are the ones which can be used for scipy.optimize.minimize and\nmigrad of iminuit. If no method is specified, Levenberg-Marquard is used.\nReliable alternatives are migrad, Powell and Nelder-Mead.
      • \n
    • tol (float, optional):\ncan be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence\nto a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly\ninvalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values\nThe stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
      • \n
    • correlated_fit (bool):\nIf True, use the full inverse covariance matrix in the definition of the chisquare cost function.\nFor details about how the covariance matrix is estimated see pyerrors.obs.covariance.\nIn practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).\nThis procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
      • \n
    • expected_chisquare (bool):\nIf True estimates the expected chisquare which is\ncorrected by effects caused by correlated input data (default False).
      • \n
    • resplot (bool):\nIf True, a plot which displays fit, data and residuals is generated (default False).
      • \n
    • qqplot (bool):\nIf True, a quantile-quantile plot of the fit result is generated (default False).
      • \n
    • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • output (Fit_result):\nParameters and information on the fitted result.
      • \n
    \n", "signature": "(x, y, func, priors=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.total_least_squares": {"fullname": "pyerrors.fits.total_least_squares", "modulename": "pyerrors.fits", "qualname": "total_least_squares", "kind": "function", "doc": "

    Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.

    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nlist of Obs, or a tuple of lists of Obs
      • \n
    • y (list):\nlist of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
      • \n

    • func (object):\nfunc has to be of the form

      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(a, x):\n   return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
      \n
      \n\n

      For multiple x values func can be of the form

      \n\n
      \n
      def func(a, x):\n   (x1, x2) = x\n   return a[0] * x1 ** 2 + a[1] * x2\n
      \n
      \n\n

      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.

      • \n
    • silent (bool, optional):\nIf true all output to the console is omitted (default False).
      • \n
    • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for non-linear\nfits with many parameters.
      • \n
    • expected_chisquare (bool):\nIf true prints the expected chisquare which is\ncorrected by effects caused by correlated input data.\nThis can take a while as the full correlation matrix\nhas to be calculated (default False).
      • \n
    • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
      • \n
    \n\n
    Notes
    \n\n

    Based on the orthogonal distance regression module of scipy.

    \n\n
    Returns
    \n\n
      \n
    • output (Fit_result):\nParameters and information on the fitted result.
      • \n
    \n", "signature": "(x, y, func, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.fit_lin": {"fullname": "pyerrors.fits.fit_lin", "modulename": "pyerrors.fits", "qualname": "fit_lin", "kind": "function", "doc": "

    Performs a linear fit to y = n + m * x and returns two Obs n, m.

    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nCan either be a list of floats in which case no xerror is assumed, or\na list of Obs, where the dvalues of the Obs are used as xerror for the fit.
      • \n
    • y (list):\nList of Obs, the dvalues of the Obs are used as yerror for the fit.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • fit_parameters (list[Obs]):\nLIist of fitted observables.
      • \n
    \n", "signature": "(x, y, **kwargs):", "funcdef": "def"}, "pyerrors.fits.qqplot": {"fullname": "pyerrors.fits.qqplot", "modulename": "pyerrors.fits", "qualname": "qqplot", "kind": "function", "doc": "

    Generates a quantile-quantile plot of the fit result which can be used to\n check if the residuals of the fit are gaussian distributed.

    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(x, o_y, func, p, title=''):", "funcdef": "def"}, "pyerrors.fits.residual_plot": {"fullname": "pyerrors.fits.residual_plot", "modulename": "pyerrors.fits", "qualname": "residual_plot", "kind": "function", "doc": "

    Generates a plot which compares the fit to the data and displays the corresponding residuals

    \n\n

    For uncorrelated data the residuals are expected to be distributed ~N(0,1).

    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(x, y, func, fit_res, title=''):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "

    Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.

    \n\n
    Returns
    \n\n
      \n
    • err (np.array(Obs)):\nError band for an array of sample values x
      • \n
    \n", "signature": "(x, func, beta):", "funcdef": "def"}, "pyerrors.fits.ks_test": {"fullname": "pyerrors.fits.ks_test", "modulename": "pyerrors.fits", "qualname": "ks_test", "kind": "function", "doc": "

    Performs a Kolmogorov\u2013Smirnov test for the p-values of all fit object.

    \n\n
    Parameters
    \n\n
      \n
    • objects (list):\nList of fit results to include in the analysis (optional).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(objects=None):", "funcdef": "def"}, "pyerrors.input": {"fullname": "pyerrors.input", "modulename": "pyerrors.input", "kind": "module", "doc": "

    pyerrors includes an input submodule in which input routines and parsers for the output of various numerical programs are contained.

    \n\n

    Jackknife samples

    \n\n

    For comparison with other analysis workflows pyerrors can also generate jackknife samples from an Obs object or import jackknife samples into an Obs object.\nSee pyerrors.obs.Obs.export_jackknife and pyerrors.obs.import_jackknife for details.

    \n"}, "pyerrors.input.bdio": {"fullname": "pyerrors.input.bdio", "modulename": "pyerrors.input.bdio", "kind": "module", "doc": "

    \n"}, "pyerrors.input.bdio.read_ADerrors": {"fullname": "pyerrors.input.bdio.read_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "read_ADerrors", "kind": "function", "doc": "

    Extract generic MCMC data from a bdio file

    \n\n

    read_ADerrors requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to

    \n\n

    all: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/

    \n\n
    Parameters
    \n\n
      \n
    • file_path -- path to the bdio file
      • \n
    • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (List[Obs]):\nExtracted data
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.write_ADerrors": {"fullname": "pyerrors.input.bdio.write_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "write_ADerrors", "kind": "function", "doc": "

    Write Obs to a bdio file according to ADerrors conventions

    \n\n

    read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to

    \n\n

    all: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/

    \n\n
    Parameters
    \n\n
      \n
    • file_path -- path to the bdio file
      • \n
    • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • success (int):\nreturns 0 is successful
      • \n
    \n", "signature": "(obs_list, file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_mesons": {"fullname": "pyerrors.input.bdio.read_mesons", "modulename": "pyerrors.input.bdio", "qualname": "read_mesons", "kind": "function", "doc": "

    Extract mesons data from a bdio file and return it as a dictionary

    \n\n

    The dictionary can be accessed with a tuple consisting of (type, source_position, kappa1, kappa2)

    \n\n

    read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to

    \n\n

    all: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/

    \n\n
    Parameters
    \n\n
      \n
    • file_path (str):\npath to the bdio file
      • \n
    • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    • start (int):\nThe first configuration to be read (default 1)
      • \n
    • stop (int):\nThe last configuration to be read (default None)
      • \n
    • step (int):\nFixed step size between two measurements (default 1)
      • \n
    • alternative_ensemble_name (str):\nManually overwrite ensemble name
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (dict):\nExtracted meson data
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_dSdm": {"fullname": "pyerrors.input.bdio.read_dSdm", "modulename": "pyerrors.input.bdio", "qualname": "read_dSdm", "kind": "function", "doc": "

    Extract dSdm data from a bdio file and return it as a dictionary

    \n\n

    The dictionary can be accessed with a tuple consisting of (type, kappa)

    \n\n

    read_dSdm requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to

    \n\n

    all: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/

    \n\n
    Parameters
    \n\n
      \n
    • file_path (str):\npath to the bdio file
      • \n
    • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    • start (int):\nThe first configuration to be read (default 1)
      • \n
    • stop (int):\nThe last configuration to be read (default None)
      • \n
    • step (int):\nFixed step size between two measurements (default 1)
      • \n
    • alternative_ensemble_name (str):\nManually overwrite ensemble name
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.dobs": {"fullname": "pyerrors.input.dobs", "modulename": "pyerrors.input.dobs", "kind": "module", "doc": "

    \n"}, "pyerrors.input.dobs.create_pobs_string": {"fullname": "pyerrors.input.dobs.create_pobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_pobs_string", "kind": "function", "doc": "

    Export a list of Obs or structures containing Obs to an xml string\naccording to the Zeuthen pobs format.

    \n\n

    Tags are not written or recovered automatically. The separator | is removed from the replica names.

    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • xml_str (str):\nXML formatted string of the input data
      • \n
    \n", "signature": "(obsl, name, spec='', origin='', symbol=[], enstag=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_pobs": {"fullname": "pyerrors.input.dobs.write_pobs", "modulename": "pyerrors.input.dobs", "qualname": "write_pobs", "kind": "function", "doc": "

    Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen pobs format.

    \n\n

    Tags are not written or recovered automatically. The separator | is removed from the replica names.

    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
      • \n
    • gz (bool):\nIf True, the output is a gzipped xml. If False, the output is an xml file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(\tobsl,\tfname,\tname,\tspec='',\torigin='',\tsymbol=[],\tenstag=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_pobs": {"fullname": "pyerrors.input.dobs.read_pobs", "modulename": "pyerrors.input.dobs", "qualname": "read_pobs", "kind": "function", "doc": "

    Import a list of Obs from an xml.gz file in the Zeuthen pobs format.

    \n\n

    Tags are not written or recovered automatically.

    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • separatior_insertion (str or int):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nNone (default): Replica names remain unchanged.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(fname, full_output=False, gz=True, separator_insertion=None):", "funcdef": "def"}, "pyerrors.input.dobs.import_dobs_string": {"fullname": "pyerrors.input.dobs.import_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "import_dobs_string", "kind": "function", "doc": "

    Import a list of Obs from a string in the Zeuthen dobs format.

    \n\n

    Tags are not written or recovered automatically.

    \n\n
    Parameters
    \n\n
      \n
    • content (str):\nXML string containing the data
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(content, full_output=False, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_dobs": {"fullname": "pyerrors.input.dobs.read_dobs", "modulename": "pyerrors.input.dobs", "qualname": "read_dobs", "kind": "function", "doc": "

    Import a list of Obs from an xml.gz file in the Zeuthen dobs format.

    \n\n

    Tags are not written or recovered automatically.

    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes XML file.
      • \n
    • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(fname, full_output=False, gz=True, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.create_dobs_string": {"fullname": "pyerrors.input.dobs.create_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_dobs_string", "kind": "function", "doc": "

    Generate the string for the export of a list of Obs or structures containing Obs\nto a .xml.gz file according to the Zeuthen dobs format.

    \n\n

    Tags are not written or recovered automatically. The separator |is removed from the replica names.

    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • who (str):\nProvide the name of the person that exports the data.
      • \n
    • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • xml_str (str):\nXML string generated from the data
      • \n
    \n", "signature": "(\tobsl,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_dobs": {"fullname": "pyerrors.input.dobs.write_dobs", "modulename": "pyerrors.input.dobs", "qualname": "write_dobs", "kind": "function", "doc": "

    Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen dobs format.

    \n\n

    Tags are not written or recovered automatically. The separator | is removed from the replica names.

    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • who (str):\nProvide the name of the person that exports the data.
      • \n
    • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
      • \n
    • gz (bool):\nIf True, the output is a gzipped XML. If False, the output is a XML file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(\tobsl,\tfname,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.hadrons": {"fullname": "pyerrors.input.hadrons", "modulename": "pyerrors.input.hadrons", "kind": "module", "doc": "

    \n"}, "pyerrors.input.hadrons.read_hd5": {"fullname": "pyerrors.input.hadrons.read_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_hd5", "kind": "function", "doc": "

    Read hadrons hdf5 file and extract entry based on attributes.

    \n\n
    Parameters
    \n\n
      \n
    • filestem (str):\nFull namestem of the files to read, including the full path.
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • group (str):\nlabel of the group to be extracted.
      • \n

    • attrs (dict or int):\nDictionary containing the attributes. For example

      \n\n
      \n
      attrs = {"gamma_snk": "Gamma5",\n        "gamma_src": "Gamma5"}\n
      \n
      \n\n

      Alternatively an integer can be specified to identify the sub group.\nThis is discouraged as the order in the file is not guaranteed.

      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • part (str):\nstring specifying whether to extract the real part ('real'),\nthe imaginary part ('imag') or a complex correlator ('complex').\nDefault 'real'.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr (Corr):\nCorrelator of the source sink combination in question.
      • \n
    \n", "signature": "(filestem, ens_id, group, attrs=None, idl=None, part='real'):", "funcdef": "def"}, "pyerrors.input.hadrons.read_meson_hd5": {"fullname": "pyerrors.input.hadrons.read_meson_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_meson_hd5", "kind": "function", "doc": "

    Read hadrons meson hdf5 file and extract the meson labeled 'meson'

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • meson (str):\nlabel of the meson to be extracted, standard value meson_0 which\ncorresponds to the pseudoscalar pseudoscalar two-point function.
      • \n
    • gammas (tuple of strings):\nInstrad of a meson label one can also provide a tuple of two strings\nindicating the gamma matrices at sink and source (gamma_snk, gamma_src).\n(\"Gamma5\", \"Gamma5\") corresponds to the pseudoscalar pseudoscalar\ntwo-point function. The gammas argument dominateds over meson.
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr (Corr):\nCorrelator of the source sink combination in question.
      • \n
    \n", "signature": "(path, filestem, ens_id, meson='meson_0', idl=None, gammas=None):", "funcdef": "def"}, "pyerrors.input.hadrons.extract_t0_hd5": {"fullname": "pyerrors.input.hadrons.extract_t0_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "extract_t0_hd5", "kind": "function", "doc": "

    Read hadrons FlowObservables hdf5 file and extract t0

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • obs (str):\nlabel of the observable from which t0 should be extracted.\nOptions: 'Clover energy density' and 'Plaquette energy density'
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
      • \n
    \n", "signature": "(\tpath,\tfilestem,\tens_id,\tobs='Clover energy density',\tfit_range=5,\tidl=None,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.hadrons.read_DistillationContraction_hd5": {"fullname": "pyerrors.input.hadrons.read_DistillationContraction_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_DistillationContraction_hd5", "kind": "function", "doc": "

    Read hadrons DistillationContraction hdf5 files in given directory structure

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the directories to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • diagrams (list):\nList of strings of the diagrams to extract, e.g. [\"direct\", \"box\", \"cross\"].
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (dict):\nextracted DistillationContration data
      • \n
    \n", "signature": "(path, ens_id, diagrams=['direct'], idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.Npr_matrix": {"fullname": "pyerrors.input.hadrons.Npr_matrix", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix", "kind": "class", "doc": "

    ndarray(shape, dtype=float, buffer=None, offset=0,\n strides=None, order=None)

    \n\n

    An array object represents a multidimensional, homogeneous array\nof fixed-size items. An associated data-type object describes the\nformat of each element in the array (its byte-order, how many bytes it\noccupies in memory, whether it is an integer, a floating point number,\nor something else, etc.)

    \n\n

    Arrays should be constructed using array, zeros or empty (refer\nto the See Also section below). The parameters given here refer to\na low-level method (ndarray(...)) for instantiating an array.

    \n\n

    For more information, refer to the numpy module and examine the\nmethods and attributes of an array.

    \n\n
    Parameters
    \n\n
      \n
    • (for the __new__ method; see Notes below)
      • \n
    • shape (tuple of ints):\nShape of created array.
      • \n
    • dtype (data-type, optional):\nAny object that can be interpreted as a numpy data type.
      • \n
    • buffer (object exposing buffer interface, optional):\nUsed to fill the array with data.
      • \n
    • offset (int, optional):\nOffset of array data in buffer.
      • \n
    • strides (tuple of ints, optional):\nStrides of data in memory.
      • \n
    • order ({'C', 'F'}, optional):\nRow-major (C-style) or column-major (Fortran-style) order.
      • \n
    \n\n
    Attributes
    \n\n
      \n
    • T (ndarray):\nTranspose of the array.
      • \n
    • data (buffer):\nThe array's elements, in memory.
      • \n
    • dtype (dtype object):\nDescribes the format of the elements in the array.
      • \n
    • flags (dict):\nDictionary containing information related to memory use, e.g.,\n'C_CONTIGUOUS', 'OWNDATA', 'WRITEABLE', etc.
      • \n
    • flat (numpy.flatiter object):\nFlattened version of the array as an iterator. The iterator\nallows assignments, e.g., x.flat = 3 (See ndarray.flat for\nassignment examples; TODO).
      • \n
    • imag (ndarray):\nImaginary part of the array.
      • \n
    • real (ndarray):\nReal part of the array.
      • \n
    • size (int):\nNumber of elements in the array.
      • \n
    • itemsize (int):\nThe memory use of each array element in bytes.
      • \n
    • nbytes (int):\nThe total number of bytes required to store the array data,\ni.e., itemsize * size.
      • \n
    • ndim (int):\nThe array's number of dimensions.
      • \n
    • shape (tuple of ints):\nShape of the array.
      • \n
    • strides (tuple of ints):\nThe step-size required to move from one element to the next in\nmemory. For example, a contiguous (3, 4) array of type\nint16 in C-order has strides (8, 2). This implies that\nto move from element to element in memory requires jumps of 2 bytes.\nTo move from row-to-row, one needs to jump 8 bytes at a time\n(2 * 4).
      • \n
    • ctypes (ctypes object):\nClass containing properties of the array needed for interaction\nwith ctypes.
      • \n
    • base (ndarray):\nIf the array is a view into another array, that array is its base\n(unless that array is also a view). The base array is where the\narray data is actually stored.
      • \n
    \n\n
    See Also
    \n\n

    array: Construct an array.
    \nzeros: Create an array, each element of which is zero.
    \nempty: Create an array, but leave its allocated memory unchanged (i.e.,\nit contains \"garbage\").
    \ndtype: Create a data-type.
    \nnumpy.typing.NDArray: An ndarray alias :term:generic <generic type>\nw.r.t. its dtype.type <numpy.dtype.type>.

    \n\n
    Notes
    \n\n

    There are two modes of creating an array using __new__:

    \n\n
      \n
    1. If buffer is None, then only shape, dtype, and order\nare used.
      2. \n
    2. If buffer is an object exposing the buffer interface, then\nall keywords are interpreted.
      3. \n
    \n\n

    No __init__ method is needed because the array is fully initialized\nafter the __new__ method.

    \n\n
    Examples
    \n\n

    These examples illustrate the low-level ndarray constructor. Refer\nto the See Also section above for easier ways of constructing an\nndarray.

    \n\n

    First mode, buffer is None:

    \n\n
    \n
    >>> np.ndarray(shape=(2,2), dtype=float, order='F')\narray([[0.0e+000, 0.0e+000], # random\n       [     nan, 2.5e-323]])\n
    \n
    \n\n

    Second mode:

    \n\n
    \n
    >>> np.ndarray((2,), buffer=np.array([1,2,3]),\n...            offset=np.int_().itemsize,\n...            dtype=int) # offset = 1*itemsize, i.e. skip first element\narray([2, 3])\n
    \n
    \n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "

    Gamma_5 hermitean conjugate

    \n\n

    Uses the fact that the propagator is gamma5 hermitean, so just the\nin and out momenta of the propagator are exchanged.

    \n"}, "pyerrors.input.hadrons.read_ExternalLeg_hd5": {"fullname": "pyerrors.input.hadrons.read_ExternalLeg_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_ExternalLeg_hd5", "kind": "function", "doc": "

    Read hadrons ExternalLeg hdf5 file and output an array of CObs

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (Npr_matrix):\nread Cobs-matrix
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Bilinear_hd5": {"fullname": "pyerrors.input.hadrons.read_Bilinear_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Bilinear_hd5", "kind": "function", "doc": "

    Read hadrons Bilinear hdf5 file and output an array of CObs

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result_dict (dict[Npr_matrix]):\nextracted Bilinears
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Fourquark_hd5": {"fullname": "pyerrors.input.hadrons.read_Fourquark_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Fourquark_hd5", "kind": "function", "doc": "

    Read hadrons FourquarkFullyConnected hdf5 file and output an array of CObs

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • vertices (list):\nVertex functions to be extracted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result_dict (dict):\nextracted fourquark matrizes
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None, vertices=['VA', 'AV']):", "funcdef": "def"}, "pyerrors.input.json": {"fullname": "pyerrors.input.json", "modulename": "pyerrors.input.json", "kind": "module", "doc": "

    \n"}, "pyerrors.input.json.create_json_string": {"fullname": "pyerrors.input.json.create_json_string", "modulename": "pyerrors.input.json", "qualname": "create_json_string", "kind": "function", "doc": "

    Generate the string for the export of a list of Obs or structures containing Obs\nto a .json(.gz) file

    \n\n
    Parameters
    \n\n
      \n
    • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • json_string (str):\nString for export to .json(.gz) file
      • \n
    \n", "signature": "(ol, description='', indent=1):", "funcdef": "def"}, "pyerrors.input.json.dump_to_json": {"fullname": "pyerrors.input.json.dump_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_to_json", "kind": "function", "doc": "

    Export a list of Obs or structures containing Obs to a .json(.gz) file.\nDict keys that are not JSON-serializable such as floats are converted to strings.

    \n\n
    Parameters
    \n\n
      \n
    • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Null
      • \n
    \n", "signature": "(ol, fname, description='', indent=1, gz=True):", "funcdef": "def"}, "pyerrors.input.json.import_json_string": {"fullname": "pyerrors.input.json.import_json_string", "modulename": "pyerrors.input.json", "qualname": "import_json_string", "kind": "function", "doc": "

    Reconstruct a list of Obs or structures containing Obs from a json string.

    \n\n

    The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.

    \n\n
    Parameters
    \n\n
      \n
    • json_string (str):\njson string containing the data.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nreconstructed list of observables from the json string
      • \n
    • or
      • \n
    • result (Obs):\nonly one observable if the list only has one entry
      • \n
    • or
      • \n
    • result (dict):\nif full_output=True
      • \n
    \n", "signature": "(json_string, verbose=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.load_json": {"fullname": "pyerrors.input.json.load_json", "modulename": "pyerrors.input.json", "qualname": "load_json", "kind": "function", "doc": "

    Import a list of Obs or structures containing Obs from a .json(.gz) file.

    \n\n

    The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.

    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nreconstructed list of observables from the json string
      • \n
    • or
      • \n
    • result (Obs):\nonly one observable if the list only has one entry
      • \n
    • or
      • \n
    • result (dict):\nif full_output=True
      • \n
    \n", "signature": "(fname, verbose=True, gz=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.dump_dict_to_json": {"fullname": "pyerrors.input.json.dump_dict_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_dict_to_json", "kind": "function", "doc": "

    Export a dict of Obs or structures containing Obs to a .json(.gz) file

    \n\n
    Parameters
    \n\n
      \n
    • od (dict):\nDict of JSON valid structures and objects that will be exported.\nAt the moment, these objects can be either of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    • reps (str):\nSpecify the structure of the placeholder in exported dict to be reps[0-9]+.
      • \n
    • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(od, fname, description='', indent=1, reps='DICTOBS', gz=True):", "funcdef": "def"}, "pyerrors.input.json.load_json_dict": {"fullname": "pyerrors.input.json.load_json_dict", "modulename": "pyerrors.input.json", "qualname": "load_json_dict", "kind": "function", "doc": "

    Import a dict of Obs or structures containing Obs from a .json(.gz) file.

    \n\n

    The following structures are supported: Obs, list, numpy.ndarray, Corr

    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    • reps (str):\nSpecify the structure of the placeholder in imported dict to be reps[0-9]+.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (Obs / list / Corr):\nRead data
      • \n
    • or
      • \n
    • data (dict):\nRead data and meta-data
      • \n
    \n", "signature": "(fname, verbose=True, gz=True, full_output=False, reps='DICTOBS'):", "funcdef": "def"}, "pyerrors.input.misc": {"fullname": "pyerrors.input.misc", "modulename": "pyerrors.input.misc", "kind": "module", "doc": "

    \n"}, "pyerrors.input.misc.fit_t0": {"fullname": "pyerrors.input.misc.fit_t0", "modulename": "pyerrors.input.misc", "qualname": "fit_t0", "kind": "function", "doc": "

    Compute the root of (flow-based) data based on a dictionary that contains\nthe necessary information in key-value pairs a la (flow time: observable at flow time).

    \n\n

    It is assumed that the data is monotonically increasing and passes zero from below.\nNo exception is thrown if this is not the case (several roots, no monotonic increase).\nAn exception is thrown if no root can be found in the data.

    \n\n

    A linear fit in the vicinity of the root is performed to exctract the root from the\ntwo fit parameters.

    \n\n
    Parameters
    \n\n
      \n
    • t2E_dict (dict):\nDictionary with pairs of (flow time: observable at flow time) where the flow times\nare of type float and the observables of type Obs.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data. (Default: False)
      • \n
    • observable (str):\nKeyword to identify the observable to print the correct ylabel (if plot_fit is True)\nfor the observables 't0' and 'w0'. No y label is printed otherwise. (Default: 't0')
      • \n
    \n\n
    Returns
    \n\n
      \n
    • root (Obs):\nThe root of the data series.
      • \n
    \n", "signature": "(t2E_dict, fit_range, plot_fit=False, observable='t0'):", "funcdef": "def"}, "pyerrors.input.misc.read_pbp": {"fullname": "pyerrors.input.misc.read_pbp", "modulename": "pyerrors.input.misc", "qualname": "read_pbp", "kind": "function", "doc": "

    Read pbp format from given folder structure.

    \n\n
    Parameters
    \n\n
      \n
    • r_start (list):\nlist which contains the first config to be read for each replicum
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nlist of observables read
      • \n
    \n", "signature": "(path, prefix, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD": {"fullname": "pyerrors.input.openQCD", "modulename": "pyerrors.input.openQCD", "kind": "module", "doc": "

    \n"}, "pyerrors.input.openQCD.read_rwms": {"fullname": "pyerrors.input.openQCD.read_rwms", "modulename": "pyerrors.input.openQCD", "qualname": "read_rwms", "kind": "function", "doc": "

    Read rwms format from given folder structure. Returns a list of length nrw

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath that contains the data files
      • \n
    • prefix (str):\nall files in path that start with prefix are considered as input files.\nMay be used together postfix to consider only special file endings.\nPrefix is ignored, if the keyword 'files' is used.
      • \n
    • version (str):\nversion of openQCD, default 2.0
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.ms1' for openQCD-files
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • print_err (bool):\nPrint additional information that is useful for debugging.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • rwms (Obs):\nReweighting factors read
      • \n
    \n", "signature": "(path, prefix, version='2.0', names=None, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_t0": {"fullname": "pyerrors.input.openQCD.extract_t0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_t0", "kind": "function", "doc": "

    Extract t0/a^2 from given .ms.dat files. Returns t0 as Obs.

    \n\n

    It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t^2 - c (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.

    \n\n

    It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to .ms.dat files
      • \n
    • prefix (str):\nEnsemble prefix
      • \n
    • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
      • \n
    • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
      • \n
    • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • postfix (str):\nPostfix of measurement file (Default: ms)
      • \n
    • c (float):\nConstant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1.
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • plaquette (bool):\nIf true extract the plaquette estimate of t0 instead.
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
      • \n
    • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
      • \n
    \n\n
    Returns
    \n\n
      \n
    • t0 (Obs):\nExtracted t0
      • \n
    \n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_w0": {"fullname": "pyerrors.input.openQCD.extract_w0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_w0", "kind": "function", "doc": "

    Extract w0/a from given .ms.dat files. Returns w0 as Obs.

    \n\n

    It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t d(t^2)/dt - (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.

    \n\n

    It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to .ms.dat files
      • \n
    • prefix (str):\nEnsemble prefix
      • \n
    • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
      • \n
    • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
      • \n
    • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • postfix (str):\nPostfix of measurement file (Default: ms)
      • \n
    • c (float):\nConstant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1.
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • plaquette (bool):\nIf true extract the plaquette estimate of w0 instead.
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of w0 is shown together with the data.
      • \n
    • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
      • \n
    \n\n
    Returns
    \n\n
      \n
    • w0 (Obs):\nExtracted w0
      • \n
    \n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop": {"fullname": "pyerrors.input.openQCD.read_qtop", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop", "kind": "function", "doc": "

    Read the topologial charge based on openQCD gradient flow measurements.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • version (str):\nEither openQCD or sfqcd, depending on the data.
      • \n
    • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
      • \n
    • integer_charge (bool):\nIf True, the charge is rounded towards the nearest integer on each config.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (Obs):\nRead topological charge
      • \n
    \n", "signature": "(path, prefix, c, dtr_cnfg=1, version='openQCD', **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_gf_coupling": {"fullname": "pyerrors.input.openQCD.read_gf_coupling", "modulename": "pyerrors.input.openQCD", "qualname": "read_gf_coupling", "kind": "function", "doc": "

    Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.

    \n\n

    Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used.
      • \n
    \n", "signature": "(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.qtop_projection": {"fullname": "pyerrors.input.openQCD.qtop_projection", "modulename": "pyerrors.input.openQCD", "qualname": "qtop_projection", "kind": "function", "doc": "

    Returns the projection to the topological charge sector defined by target.

    \n\n
    Parameters
    \n\n
      \n
    • path (Obs):\nTopological charge.
      • \n
    • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • reto (Obs):\nprojection to the topological charge sector defined by target
      • \n
    \n", "signature": "(qtop, target=0):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop_sector": {"fullname": "pyerrors.input.openQCD.read_qtop_sector", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop_sector", "kind": "function", "doc": "

    Constructs reweighting factors to a specified topological sector.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L
      • \n
    • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of trajectories\nbetween two configs.\nif it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • version (str):\nversion string of the openQCD (sfqcd) version used to create\nthe ensemble. Default is 2.0. May also be set to sfqcd.
      • \n
    • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
      • \n
    • r_start (list):\noffset of the first ensemble, making it easier to match\nlater on with other Obs
      • \n
    • r_stop (list):\nlast configurations that need to be read (per replicum)
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • reto (Obs):\nprojection to the topological charge sector defined by target
      • \n
    \n", "signature": "(path, prefix, c, target=0, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_ms5_xsf": {"fullname": "pyerrors.input.openQCD.read_ms5_xsf", "modulename": "pyerrors.input.openQCD", "qualname": "read_ms5_xsf", "kind": "function", "doc": "

    Read data from files in the specified directory with the specified prefix and quark combination extension, and return a Corr object containing the data.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nThe directory to search for the files in.
      • \n
    • prefix (str):\nThe prefix to match the files against.
      • \n
    • qc (str):\nThe quark combination extension to match the files against.
      • \n
    • corr (str):\nThe correlator to extract data for.
      • \n
    • sep (str, optional):\nThe separator to use when parsing the replika names.
      • \n

    • **kwargs: Additional keyword arguments. The following keyword arguments are recognized:

      \n\n
        \n
      • names (List[str]): A list of names to use for the replicas.
        • \n
      • files (List[str]): A list of files to read data from.
        • \n
      • idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given.
        • \n
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Corr: A complex valued Corr object containing the data read from the files. In case of boudary to bulk correlators.
      • \n
    • or
      • \n
    • CObs: A complex valued CObs object containing the data read from the files. In case of boudary to boundary correlators.
      • \n
    \n\n
    Raises
    \n\n
      \n
    • FileNotFoundError: If no files matching the specified prefix and quark combination extension are found in the specified directory.
      • \n
    • IOError: If there is an error reading a file.
      • \n
    • struct.error: If there is an error unpacking binary data.
      • \n
    \n", "signature": "(path, prefix, qc, corr, sep='r', **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas": {"fullname": "pyerrors.input.pandas", "modulename": "pyerrors.input.pandas", "kind": "module", "doc": "

    \n"}, "pyerrors.input.pandas.to_sql": {"fullname": "pyerrors.input.pandas.to_sql", "modulename": "pyerrors.input.pandas", "qualname": "to_sql", "kind": "function", "doc": "

    Write DataFrame including Obs or Corr valued columns to sqlite database.

    \n\n
    Parameters
    \n\n
      \n
    • df (pandas.DataFrame):\nDataframe to be written to the database.
      • \n
    • table_name (str):\nName of the table in the database.
      • \n
    • db (str):\nPath to the sqlite database.
      • \n
    • if exists (str):\nHow to behave if table already exists. Options 'fail', 'replace', 'append'.
      • \n
    • gz (bool):\nIf True the json strings are gzipped.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(df, table_name, db, if_exists='fail', gz=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.read_sql": {"fullname": "pyerrors.input.pandas.read_sql", "modulename": "pyerrors.input.pandas", "qualname": "read_sql", "kind": "function", "doc": "

    Execute SQL query on sqlite database and obtain DataFrame including Obs or Corr valued columns.

    \n\n
    Parameters
    \n\n
      \n
    • sql (str):\nSQL query to be executed.
      • \n
    • db (str):\nPath to the sqlite database.
      • \n
    • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
      • \n
    \n", "signature": "(sql, db, auto_gamma=False, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.dump_df": {"fullname": "pyerrors.input.pandas.dump_df", "modulename": "pyerrors.input.pandas", "qualname": "dump_df", "kind": "function", "doc": "

    Exports a pandas DataFrame containing Obs valued columns to a (gzipped) csv file.

    \n\n

    Before making use of pandas to_csv functionality Obs objects are serialized via the standardized\njson format of pyerrors.

    \n\n
    Parameters
    \n\n
      \n
    • df (pandas.DataFrame):\nDataframe to be dumped to a file.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • gz (bool):\nIf True, the output is a gzipped csv file. If False, the output is a csv file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(df, fname, gz=True):", "funcdef": "def"}, "pyerrors.input.pandas.load_df": {"fullname": "pyerrors.input.pandas.load_df", "modulename": "pyerrors.input.pandas", "qualname": "load_df", "kind": "function", "doc": "

    Imports a pandas DataFrame from a csv.(gz) file in which Obs objects are serialized as json strings.

    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
      • \n
    \n", "signature": "(fname, auto_gamma=False, gz=True):", "funcdef": "def"}, "pyerrors.input.sfcf": {"fullname": "pyerrors.input.sfcf", "modulename": "pyerrors.input.sfcf", "kind": "module", "doc": "

    \n"}, "pyerrors.input.sfcf.sep": {"fullname": "pyerrors.input.sfcf.sep", "modulename": "pyerrors.input.sfcf", "qualname": "sep", "kind": "variable", "doc": "

    \n", "default_value": "'/'"}, "pyerrors.input.sfcf.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "

    Read sfcf files from given folder structure.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to the sfcf files.
      • \n
    • prefix (str):\nPrefix of the sfcf files.
      • \n
    • name (str):\nName of the correlation function to read.
      • \n
    • quarks (str):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
      • \n
    • corr_type (str):\nType of correlation function to read. Can be\n
        \n
      • 'bi' for boundary-inner
        • \n
      • 'bb' for boundary-boundary
        • \n
      • 'bib' for boundary-inner-boundary
        • \n
      • \n
    • noffset (int):\nOffset of the source (only relevant when wavefunctions are used)
      • \n
    • wf (int):\nID of wave function
      • \n
    • wf2 (int):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
      • \n
    • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • ens_name (str):\nreplaces the name of the ensemble
      • \n
    • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
      • \n
    • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
      • \n
    • replica (list):\nlist of replica to be read, default is all
      • \n
    • files (list):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
      • \n
    • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
      • \n
    \n", "signature": "(\tpath,\tprefix,\tname,\tquarks='.*',\tcorr_type='bi',\tnoffset=0,\twf=0,\twf2=0,\tversion='1.0c',\tcfg_separator='n',\tsilent=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.sfcf.read_sfcf_multi": {"fullname": "pyerrors.input.sfcf.read_sfcf_multi", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf_multi", "kind": "function", "doc": "

    Read sfcf files from given folder structure.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to the sfcf files.
      • \n
    • prefix (str):\nPrefix of the sfcf files.
      • \n
    • name (str):\nName of the correlation function to read.
      • \n
    • quarks_list (list[str]):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
      • \n
    • corr_type_list (list[str]):\nType of correlation function to read. Can be\n
        \n
      • 'bi' for boundary-inner
        • \n
      • 'bb' for boundary-boundary
        • \n
      • 'bib' for boundary-inner-boundary
        • \n
      • \n
    • noffset_list (list[int]):\nOffset of the source (only relevant when wavefunctions are used)
      • \n
    • wf_list (int):\nID of wave function
      • \n
    • wf2_list (list[int]):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
      • \n
    • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • ens_name (str):\nreplaces the name of the ensemble
      • \n
    • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
      • \n
    • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
      • \n
    • replica (list):\nlist of replica to be read, default is all
      • \n
    • files (list):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
      • \n
    • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (dict[list[Obs]]):\ndict with one of the following properties:\nif keyed_out:\n dict[key] = list[Obs]\n where key has the form name/quarks/offset/wf/wf2\nif not keyed_out:\n dict[name][quarks][offset][wf][wf2] = list[Obs]
      • \n
    \n", "signature": "(\tpath,\tprefix,\tname_list,\tquarks_list=['.*'],\tcorr_type_list=['bi'],\tnoffset_list=[0],\twf_list=[0],\twf2_list=[0],\tversion='1.0c',\tcfg_separator='n',\tsilent=False,\tkeyed_out=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.utils": {"fullname": "pyerrors.input.utils", "modulename": "pyerrors.input.utils", "kind": "module", "doc": "

    Utilities for the input

    \n"}, "pyerrors.input.utils.sort_names": {"fullname": "pyerrors.input.utils.sort_names", "modulename": "pyerrors.input.utils", "qualname": "sort_names", "kind": "function", "doc": "

    Sorts a list of names of replika with searches for r and id in the replikum string.\nIf this search fails, a fallback method is used,\nwhere the strings are simply compared and the first diffeing numeral is used for differentiation.

    \n\n
    Parameters
    \n\n
      \n
    • ll (list):\nlist to sort
      • \n
    \n\n
    Returns
    \n\n
      \n
    • ll (list):\nsorted list
      • \n
    \n", "signature": "(ll):", "funcdef": "def"}, "pyerrors.input.utils.check_idl": {"fullname": "pyerrors.input.utils.check_idl", "modulename": "pyerrors.input.utils", "qualname": "check_idl", "kind": "function", "doc": "

    Checks if list of configurations is contained in an idl

    \n\n
    Parameters
    \n\n
      \n
    • idl (range or list):\nidl of the current replicum
      • \n
    • che (list):\nlist of configurations to be checked against
      • \n
    \n\n
    Returns
    \n\n
      \n
    • miss_str (str):\nstring with integers of which idls are missing
      • \n
    \n", "signature": "(idl, che):", "funcdef": "def"}, "pyerrors.input.utils.check_params": {"fullname": "pyerrors.input.utils.check_params", "modulename": "pyerrors.input.utils", "qualname": "check_params", "kind": "function", "doc": "

    Check if, for sfcf, the parameter hashes at the end of the parameter files are in fact the expected one.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nmeasurement path, same as for sfcf read method
      • \n
    • param_hash (str):\nexpected parameter hash
      • \n
    • prefix (str):\ndata prefix to find the appropriate replicum folders in path
      • \n
    • param_prefix (str):\nprefix of the parameter file. Defaults to 'parameters_'
      • \n
    \n\n
    Returns
    \n\n
      \n
    • nums (dict):\ndictionary of faulty parameter files sorted by the replica paths
      • \n
    \n", "signature": "(path, param_hash, prefix, param_prefix='parameters_'):", "funcdef": "def"}, "pyerrors.integrate": {"fullname": "pyerrors.integrate", "modulename": "pyerrors.integrate", "kind": "module", "doc": "

    \n"}, "pyerrors.integrate.quad": {"fullname": "pyerrors.integrate.quad", "modulename": "pyerrors.integrate", "qualname": "quad", "kind": "function", "doc": "

    Performs a (one-dimensional) numeric integration of f(p, x) from a to b.

    \n\n

    The integration is performed using scipy.integrate.quad().\nAll parameters that can be passed to scipy.integrate.quad may also be passed to this function.\nThe output is the same as for scipy.integrate.quad, the first element being an Obs.

    \n\n
    Parameters
    \n\n
      \n

    • func (object):\nfunction to integrate, has to be of the form

      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(p, x):\n   return p[0] + p[1] * x + p[2] * anp.sinh(x)\n
      \n
      \n\n

      where x is the integration variable.

      • \n
    • p (list of floats or Obs):\nparameters of the function func.
      • \n
    • a (float or Obs):\nLower limit of integration (use -numpy.inf for -infinity).
      • \n
    • b (float or Obs):\nUpper limit of integration (use -numpy.inf for -infinity).
      • \n
    • All parameters of scipy.integrate.quad
      • \n
    \n\n
    Returns
    \n\n
      \n
    • y (Obs):\nThe integral of func from a to b.
      • \n
    • abserr (float):\nAn estimate of the absolute error in the result.
      • \n
    • infodict (dict):\nA dictionary containing additional information.\nRun scipy.integrate.quad_explain() for more information.
      • \n
    • message: A convergence message.
      • \n
    • explain: Appended only with 'cos' or 'sin' weighting and infinite\nintegration limits, it contains an explanation of the codes in\ninfodict['ierlst']
      • \n
    \n", "signature": "(func, p, a, b, **kwargs):", "funcdef": "def"}, "pyerrors.linalg": {"fullname": "pyerrors.linalg", "modulename": "pyerrors.linalg", "kind": "module", "doc": "

    \n"}, "pyerrors.linalg.matmul": {"fullname": "pyerrors.linalg.matmul", "modulename": "pyerrors.linalg", "qualname": "matmul", "kind": "function", "doc": "

    Matrix multiply all operands.

    \n\n
    Parameters
    \n\n
      \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    • This implementation is faster compared to standard multiplication via the @ operator.
      • \n
    \n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.jack_matmul": {"fullname": "pyerrors.linalg.jack_matmul", "modulename": "pyerrors.linalg", "qualname": "jack_matmul", "kind": "function", "doc": "

    Matrix multiply both operands making use of the jackknife approximation.

    \n\n
    Parameters
    \n\n
      \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    • For large matrices this is considerably faster compared to matmul.
      • \n
    \n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.einsum": {"fullname": "pyerrors.linalg.einsum", "modulename": "pyerrors.linalg", "qualname": "einsum", "kind": "function", "doc": "

    Wrapper for numpy.einsum

    \n\n
    Parameters
    \n\n
      \n
    • subscripts (str):\nSubscripts for summation (see numpy documentation for details)
      • \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    \n", "signature": "(subscripts, *operands):", "funcdef": "def"}, "pyerrors.linalg.inv": {"fullname": "pyerrors.linalg.inv", "modulename": "pyerrors.linalg", "qualname": "inv", "kind": "function", "doc": "

    Inverse of Obs or CObs valued matrices.

    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.cholesky": {"fullname": "pyerrors.linalg.cholesky", "modulename": "pyerrors.linalg", "qualname": "cholesky", "kind": "function", "doc": "

    Cholesky decomposition of Obs valued matrices.

    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.det": {"fullname": "pyerrors.linalg.det", "modulename": "pyerrors.linalg", "qualname": "det", "kind": "function", "doc": "

    Determinant of Obs valued matrices.

    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.eigh": {"fullname": "pyerrors.linalg.eigh", "modulename": "pyerrors.linalg", "qualname": "eigh", "kind": "function", "doc": "

    Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.

    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eig": {"fullname": "pyerrors.linalg.eig", "modulename": "pyerrors.linalg", "qualname": "eig", "kind": "function", "doc": "

    Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig.

    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eigv": {"fullname": "pyerrors.linalg.eigv", "modulename": "pyerrors.linalg", "qualname": "eigv", "kind": "function", "doc": "

    Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.

    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.pinv": {"fullname": "pyerrors.linalg.pinv", "modulename": "pyerrors.linalg", "qualname": "pinv", "kind": "function", "doc": "

    Computes the Moore-Penrose pseudoinverse of a matrix of Obs.

    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.svd": {"fullname": "pyerrors.linalg.svd", "modulename": "pyerrors.linalg", "qualname": "svd", "kind": "function", "doc": "

    Computes the singular value decomposition of a matrix of Obs.

    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.misc": {"fullname": "pyerrors.misc", "modulename": "pyerrors.misc", "kind": "module", "doc": "

    \n"}, "pyerrors.misc.print_config": {"fullname": "pyerrors.misc.print_config", "modulename": "pyerrors.misc", "qualname": "print_config", "kind": "function", "doc": "

    Print information about version of python, pyerrors and dependencies.

    \n", "signature": "():", "funcdef": "def"}, "pyerrors.misc.errorbar": {"fullname": "pyerrors.misc.errorbar", "modulename": "pyerrors.misc", "qualname": "errorbar", "kind": "function", "doc": "

    pyerrors wrapper for the errorbars method of matplotlib

    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nA list of x-values which can be Obs.
      • \n
    • y (list):\nA list of y-values which can be Obs.
      • \n
    • axes ((matplotlib.pyplot.axes)):\nThe axes to plot on. default is plt.
      • \n
    \n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.10.13/x64/lib/python3.10/site-packages/matplotlib/pyplot.py'>,\t**kwargs):", "funcdef": "def"}, "pyerrors.misc.dump_object": {"fullname": "pyerrors.misc.dump_object", "modulename": "pyerrors.misc", "qualname": "dump_object", "kind": "function", "doc": "

    Dump object into pickle file.

    \n\n
    Parameters
    \n\n
      \n
    • obj (object):\nobject to be saved in the pickle file
      • \n
    • name (str):\nname of the file
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(obj, name, **kwargs):", "funcdef": "def"}, "pyerrors.misc.load_object": {"fullname": "pyerrors.misc.load_object", "modulename": "pyerrors.misc", "qualname": "load_object", "kind": "function", "doc": "

    Load object from pickle file.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the file
      • \n
    \n\n
    Returns
    \n\n
      \n
    • object (Obs):\nLoaded Object
      • \n
    \n", "signature": "(path):", "funcdef": "def"}, "pyerrors.misc.pseudo_Obs": {"fullname": "pyerrors.misc.pseudo_Obs", "modulename": "pyerrors.misc", "qualname": "pseudo_Obs", "kind": "function", "doc": "

    Generate an Obs object with given value, dvalue and name for test purposes

    \n\n
    Parameters
    \n\n
      \n
    • value (float):\ncentral value of the Obs to be generated.
      • \n
    • dvalue (float):\nerror of the Obs to be generated.
      • \n
    • name (str):\nname of the ensemble for which the Obs is to be generated.
      • \n
    • samples (int):\nnumber of samples for the Obs (default 1000).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (Obs):\nGenerated Observable
      • \n
    \n", "signature": "(value, dvalue, name, samples=1000):", "funcdef": "def"}, "pyerrors.misc.gen_correlated_data": {"fullname": "pyerrors.misc.gen_correlated_data", "modulename": "pyerrors.misc", "qualname": "gen_correlated_data", "kind": "function", "doc": "

    Generate observables with given covariance and autocorrelation times.

    \n\n
    Parameters
    \n\n
      \n
    • means (list):\nlist containing the mean value of each observable.
      • \n
    • cov (numpy.ndarray):\ncovariance matrix for the data to be generated.
      • \n
    • name (str):\nensemble name for the data to be geneated.
      • \n
    • tau (float or list):\ncan either be a real number or a list with an entry for\nevery dataset.
      • \n
    • samples (int):\nnumber of samples to be generated for each observable.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr_obs (list[Obs]):\nGenerated observable list
      • \n
    \n", "signature": "(means, cov, name, tau=0.5, samples=1000):", "funcdef": "def"}, "pyerrors.mpm": {"fullname": "pyerrors.mpm", "modulename": "pyerrors.mpm", "kind": "module", "doc": "

    \n"}, "pyerrors.mpm.matrix_pencil_method": {"fullname": "pyerrors.mpm.matrix_pencil_method", "modulename": "pyerrors.mpm", "qualname": "matrix_pencil_method", "kind": "function", "doc": "

    Matrix pencil method to extract k energy levels from data

    \n\n

    Implementation of the matrix pencil method based on\neq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)

    \n\n
    Parameters
    \n\n
      \n
    • data (list):\ncan be a list of Obs for the analysis of a single correlator, or a list of lists\nof Obs if several correlators are to analyzed at once.
      • \n
    • k (int):\nNumber of states to extract (default 1).
      • \n
    • p (int):\nmatrix pencil parameter which filters noise. The optimal value is expected between\nlen(data)/3 and 2*len(data)/3. The computation is more expensive the closer p is\nto len(data)/2 but could possibly suppress more noise (default len(data)//2).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • energy_levels (list[Obs]):\nExtracted energy levels
      • \n
    \n", "signature": "(corrs, k=1, p=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs": {"fullname": "pyerrors.obs", "modulename": "pyerrors.obs", "kind": "module", "doc": "

    \n"}, "pyerrors.obs.Obs": {"fullname": "pyerrors.obs.Obs", "modulename": "pyerrors.obs", "qualname": "Obs", "kind": "class", "doc": "

    Class for a general observable.

    \n\n

    Instances of Obs are the basic objects of a pyerrors error analysis.\nThey are initialized with a list which contains arrays of samples for\ndifferent ensembles/replica and another list of same length which contains\nthe names of the ensembles/replica. Mathematical operations can be\nperformed on instances. The result is another instance of Obs. The error of\nan instance can be computed with the gamma_method. Also contains additional\nmethods for output and visualization of the error calculation.

    \n\n
    Attributes
    \n\n
      \n
    • S_global (float):\nStandard value for S (default 2.0)
      • \n
    • S_dict (dict):\nDictionary for S values. If an entry for a given ensemble\nexists this overwrites the standard value for that ensemble.
      • \n
    • tau_exp_global (float):\nStandard value for tau_exp (default 0.0)
      • \n
    • tau_exp_dict (dict):\nDictionary for tau_exp values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
      • \n
    • N_sigma_global (float):\nStandard value for N_sigma (default 1.0)
      • \n
    • N_sigma_dict (dict):\nDictionary for N_sigma values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
      • \n
    \n"}, "pyerrors.obs.Obs.__init__": {"fullname": "pyerrors.obs.Obs.__init__", "modulename": "pyerrors.obs", "qualname": "Obs.__init__", "kind": "function", "doc": "

    Initialize Obs object.

    \n\n
    Parameters
    \n\n
      \n
    • samples (list):\nlist of numpy arrays containing the Monte Carlo samples
      • \n
    • names (list):\nlist of strings labeling the individual samples
      • \n
    • idl (list, optional):\nlist of ranges or lists on which the samples are defined
      • \n
    \n", "signature": "(samples, names, idl=None, **kwargs)"}, "pyerrors.obs.Obs.S_global": {"fullname": "pyerrors.obs.Obs.S_global", "modulename": "pyerrors.obs", "qualname": "Obs.S_global", "kind": "variable", "doc": "

    \n", "default_value": "2.0"}, "pyerrors.obs.Obs.S_dict": {"fullname": "pyerrors.obs.Obs.S_dict", "modulename": "pyerrors.obs", "qualname": "Obs.S_dict", "kind": "variable", "doc": "

    \n", "default_value": "{}"}, "pyerrors.obs.Obs.tau_exp_global": {"fullname": "pyerrors.obs.Obs.tau_exp_global", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_global", "kind": "variable", "doc": "

    \n", "default_value": "0.0"}, "pyerrors.obs.Obs.tau_exp_dict": {"fullname": "pyerrors.obs.Obs.tau_exp_dict", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_dict", "kind": "variable", "doc": "

    \n", "default_value": "{}"}, "pyerrors.obs.Obs.N_sigma_global": {"fullname": "pyerrors.obs.Obs.N_sigma_global", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_global", "kind": "variable", "doc": "

    \n", "default_value": "1.0"}, "pyerrors.obs.Obs.N_sigma_dict": {"fullname": "pyerrors.obs.Obs.N_sigma_dict", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_dict", "kind": "variable", "doc": "

    \n", "default_value": "{}"}, "pyerrors.obs.Obs.names": {"fullname": "pyerrors.obs.Obs.names", "modulename": "pyerrors.obs", "qualname": "Obs.names", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.shape": {"fullname": "pyerrors.obs.Obs.shape", "modulename": "pyerrors.obs", "qualname": "Obs.shape", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.r_values": {"fullname": "pyerrors.obs.Obs.r_values", "modulename": "pyerrors.obs", "qualname": "Obs.r_values", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.deltas": {"fullname": "pyerrors.obs.Obs.deltas", "modulename": "pyerrors.obs", "qualname": "Obs.deltas", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.N": {"fullname": "pyerrors.obs.Obs.N", "modulename": "pyerrors.obs", "qualname": "Obs.N", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.idl": {"fullname": "pyerrors.obs.Obs.idl", "modulename": "pyerrors.obs", "qualname": "Obs.idl", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.ddvalue": {"fullname": "pyerrors.obs.Obs.ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.ddvalue", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.reweighted": {"fullname": "pyerrors.obs.Obs.reweighted", "modulename": "pyerrors.obs", "qualname": "Obs.reweighted", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.tag": {"fullname": "pyerrors.obs.Obs.tag", "modulename": "pyerrors.obs", "qualname": "Obs.tag", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.value": {"fullname": "pyerrors.obs.Obs.value", "modulename": "pyerrors.obs", "qualname": "Obs.value", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.dvalue": {"fullname": "pyerrors.obs.Obs.dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.dvalue", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_names": {"fullname": "pyerrors.obs.Obs.e_names", "modulename": "pyerrors.obs", "qualname": "Obs.e_names", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.cov_names": {"fullname": "pyerrors.obs.Obs.cov_names", "modulename": "pyerrors.obs", "qualname": "Obs.cov_names", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.mc_names": {"fullname": "pyerrors.obs.Obs.mc_names", "modulename": "pyerrors.obs", "qualname": "Obs.mc_names", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_content": {"fullname": "pyerrors.obs.Obs.e_content", "modulename": "pyerrors.obs", "qualname": "Obs.e_content", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.covobs": {"fullname": "pyerrors.obs.Obs.covobs", "modulename": "pyerrors.obs", "qualname": "Obs.covobs", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.gamma_method": {"fullname": "pyerrors.obs.Obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "Obs.gamma_method", "kind": "function", "doc": "

    Estimate the error and related properties of the Obs.

    \n\n
    Parameters
    \n\n
      \n
    • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
      • \n
    • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
      • \n
    • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
      • \n
    • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
      • \n
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.gm": {"fullname": "pyerrors.obs.Obs.gm", "modulename": "pyerrors.obs", "qualname": "Obs.gm", "kind": "function", "doc": "

    Estimate the error and related properties of the Obs.

    \n\n
    Parameters
    \n\n
      \n
    • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
      • \n
    • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
      • \n
    • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
      • \n
    • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
      • \n
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.details": {"fullname": "pyerrors.obs.Obs.details", "modulename": "pyerrors.obs", "qualname": "Obs.details", "kind": "function", "doc": "

    Output detailed properties of the Obs.

    \n\n
    Parameters
    \n\n
      \n
    • ens_content (bool):\nprint details about the ensembles and replica if true.
      • \n
    \n", "signature": "(self, ens_content=True):", "funcdef": "def"}, "pyerrors.obs.Obs.reweight": {"fullname": "pyerrors.obs.Obs.reweight", "modulename": "pyerrors.obs", "qualname": "Obs.reweight", "kind": "function", "doc": "

    Reweight the obs with given rewighting factors.

    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
      • \n
    \n", "signature": "(self, weight):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero_within_error": {"fullname": "pyerrors.obs.Obs.is_zero_within_error", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero_within_error", "kind": "function", "doc": "

    Checks whether the observable is zero within 'sigma' standard errors.

    \n\n
    Parameters
    \n\n
      \n
    • sigma (int):\nNumber of standard errors used for the check.
      • \n
    • Works only properly when the gamma method was run.
      • \n
    \n", "signature": "(self, sigma=1):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero": {"fullname": "pyerrors.obs.Obs.is_zero", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero", "kind": "function", "doc": "

    Checks whether the observable is zero within a given tolerance.

    \n\n
    Parameters
    \n\n
      \n
    • atol (float):\nAbsolute tolerance (for details see numpy documentation).
      • \n
    \n", "signature": "(self, atol=1e-10):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_tauint": {"fullname": "pyerrors.obs.Obs.plot_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.plot_tauint", "kind": "function", "doc": "

    Plot integrated autocorrelation time for each ensemble.

    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rho": {"fullname": "pyerrors.obs.Obs.plot_rho", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rho", "kind": "function", "doc": "

    Plot normalized autocorrelation function time for each ensemble.

    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rep_dist": {"fullname": "pyerrors.obs.Obs.plot_rep_dist", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rep_dist", "kind": "function", "doc": "

    Plot replica distribution for each ensemble with more than one replicum.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_history": {"fullname": "pyerrors.obs.Obs.plot_history", "modulename": "pyerrors.obs", "qualname": "Obs.plot_history", "kind": "function", "doc": "

    Plot derived Monte Carlo history for each ensemble

    \n\n
    Parameters
    \n\n
      \n
    • expand (bool):\nshow expanded history for irregular Monte Carlo chains (default: True).
      • \n
    \n", "signature": "(self, expand=True):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_piechart": {"fullname": "pyerrors.obs.Obs.plot_piechart", "modulename": "pyerrors.obs", "qualname": "Obs.plot_piechart", "kind": "function", "doc": "

    Plot piechart which shows the fractional contribution of each\nensemble to the error and returns a dictionary containing the fractions.

    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.dump": {"fullname": "pyerrors.obs.Obs.dump", "modulename": "pyerrors.obs", "qualname": "Obs.dump", "kind": "function", "doc": "

    Dump the Obs to a file 'name' of chosen format.

    \n\n
    Parameters
    \n\n
      \n
    • filename (str):\nname of the file to be saved.
      • \n
    • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
      • \n
    • description (str):\nDescription for output file, only relevant for json.gz format.
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n", "signature": "(self, filename, datatype='json.gz', description='', **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.export_jackknife": {"fullname": "pyerrors.obs.Obs.export_jackknife", "modulename": "pyerrors.obs", "qualname": "Obs.export_jackknife", "kind": "function", "doc": "

    Export jackknife samples from the Obs

    \n\n
    Returns
    \n\n
      \n
    • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N jackknife samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived jackknife samples\nshould agree with samples from a full jackknife analysis up to O(1/N).
      • \n
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.export_bootstrap": {"fullname": "pyerrors.obs.Obs.export_bootstrap", "modulename": "pyerrors.obs", "qualname": "Obs.export_bootstrap", "kind": "function", "doc": "

    Export bootstrap samples from the Obs

    \n\n
    Parameters
    \n\n
      \n
    • samples (int):\nNumber of bootstrap samples to generate.
      • \n
    • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples.\nIf not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
      • \n
    • save_rng (str):\nSave the random numbers to a file if a path is specified.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N import_bootstrap samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived bootstrap samples\nshould agree with samples from a full bootstrap analysis up to O(1/N).
      • \n
    \n", "signature": "(self, samples=500, random_numbers=None, save_rng=None):", "funcdef": "def"}, "pyerrors.obs.Obs.sqrt": {"fullname": "pyerrors.obs.Obs.sqrt", "modulename": "pyerrors.obs", "qualname": "Obs.sqrt", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.log": {"fullname": "pyerrors.obs.Obs.log", "modulename": "pyerrors.obs", "qualname": "Obs.log", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.exp": {"fullname": "pyerrors.obs.Obs.exp", "modulename": "pyerrors.obs", "qualname": "Obs.exp", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sin": {"fullname": "pyerrors.obs.Obs.sin", "modulename": "pyerrors.obs", "qualname": "Obs.sin", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cos": {"fullname": "pyerrors.obs.Obs.cos", "modulename": "pyerrors.obs", "qualname": "Obs.cos", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tan": {"fullname": "pyerrors.obs.Obs.tan", "modulename": "pyerrors.obs", "qualname": "Obs.tan", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsin": {"fullname": "pyerrors.obs.Obs.arcsin", "modulename": "pyerrors.obs", "qualname": "Obs.arcsin", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccos": {"fullname": "pyerrors.obs.Obs.arccos", "modulename": "pyerrors.obs", "qualname": "Obs.arccos", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctan": {"fullname": "pyerrors.obs.Obs.arctan", "modulename": "pyerrors.obs", "qualname": "Obs.arctan", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sinh": {"fullname": "pyerrors.obs.Obs.sinh", "modulename": "pyerrors.obs", "qualname": "Obs.sinh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cosh": {"fullname": "pyerrors.obs.Obs.cosh", "modulename": "pyerrors.obs", "qualname": "Obs.cosh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tanh": {"fullname": "pyerrors.obs.Obs.tanh", "modulename": "pyerrors.obs", "qualname": "Obs.tanh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsinh": {"fullname": "pyerrors.obs.Obs.arcsinh", "modulename": "pyerrors.obs", "qualname": "Obs.arcsinh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccosh": {"fullname": "pyerrors.obs.Obs.arccosh", "modulename": "pyerrors.obs", "qualname": "Obs.arccosh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctanh": {"fullname": "pyerrors.obs.Obs.arctanh", "modulename": "pyerrors.obs", "qualname": "Obs.arctanh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.N_sigma": {"fullname": "pyerrors.obs.Obs.N_sigma", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.S": {"fullname": "pyerrors.obs.Obs.S", "modulename": "pyerrors.obs", "qualname": "Obs.S", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_ddvalue": {"fullname": "pyerrors.obs.Obs.e_ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_ddvalue", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_drho": {"fullname": "pyerrors.obs.Obs.e_drho", "modulename": "pyerrors.obs", "qualname": "Obs.e_drho", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_dtauint": {"fullname": "pyerrors.obs.Obs.e_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_dtauint", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_dvalue": {"fullname": "pyerrors.obs.Obs.e_dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_dvalue", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_n_dtauint": {"fullname": "pyerrors.obs.Obs.e_n_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_dtauint", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_n_tauint": {"fullname": "pyerrors.obs.Obs.e_n_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_tauint", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_rho": {"fullname": "pyerrors.obs.Obs.e_rho", "modulename": "pyerrors.obs", "qualname": "Obs.e_rho", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_tauint": {"fullname": "pyerrors.obs.Obs.e_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_tauint", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_windowsize": {"fullname": "pyerrors.obs.Obs.e_windowsize", "modulename": "pyerrors.obs", "qualname": "Obs.e_windowsize", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.tau_exp": {"fullname": "pyerrors.obs.Obs.tau_exp", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.CObs": {"fullname": "pyerrors.obs.CObs", "modulename": "pyerrors.obs", "qualname": "CObs", "kind": "class", "doc": "

    Class for a complex valued observable.

    \n"}, "pyerrors.obs.CObs.__init__": {"fullname": "pyerrors.obs.CObs.__init__", "modulename": "pyerrors.obs", "qualname": "CObs.__init__", "kind": "function", "doc": "

    \n", "signature": "(real, imag=0.0)"}, "pyerrors.obs.CObs.tag": {"fullname": "pyerrors.obs.CObs.tag", "modulename": "pyerrors.obs", "qualname": "CObs.tag", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.CObs.real": {"fullname": "pyerrors.obs.CObs.real", "modulename": "pyerrors.obs", "qualname": "CObs.real", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.CObs.imag": {"fullname": "pyerrors.obs.CObs.imag", "modulename": "pyerrors.obs", "qualname": "CObs.imag", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.CObs.gamma_method": {"fullname": "pyerrors.obs.CObs.gamma_method", "modulename": "pyerrors.obs", "qualname": "CObs.gamma_method", "kind": "function", "doc": "

    Executes the gamma_method for the real and the imaginary part.

    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.CObs.is_zero": {"fullname": "pyerrors.obs.CObs.is_zero", "modulename": "pyerrors.obs", "qualname": "CObs.is_zero", "kind": "function", "doc": "

    Checks whether both real and imaginary part are zero within machine precision.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs.conjugate": {"fullname": "pyerrors.obs.CObs.conjugate", "modulename": "pyerrors.obs", "qualname": "CObs.conjugate", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.gamma_method": {"fullname": "pyerrors.obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "gamma_method", "kind": "function", "doc": "

    Vectorized version of the gamma_method applicable to lists or arrays of Obs.

    \n\n

    See docstring of pe.Obs.gamma_method for details.

    \n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.gm": {"fullname": "pyerrors.obs.gm", "modulename": "pyerrors.obs", "qualname": "gm", "kind": "function", "doc": "

    Vectorized version of the gamma_method applicable to lists or arrays of Obs.

    \n\n

    See docstring of pe.Obs.gamma_method for details.

    \n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.derived_observable": {"fullname": "pyerrors.obs.derived_observable", "modulename": "pyerrors.obs", "qualname": "derived_observable", "kind": "function", "doc": "

    Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.

    \n\n
    Parameters
    \n\n
      \n
    • func (object):\narbitrary function of the form func(data, **kwargs). For the\nautomatic differentiation to work, all numpy functions have to have\nthe autograd wrapper (use 'import autograd.numpy as anp').
      • \n
    • data (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
      • \n
    • num_grad (bool):\nif True, numerical derivatives are used instead of autograd\n(default False). To control the numerical differentiation the\nkwargs of numdifftools.step_generators.MaxStepGenerator\ncan be used.
      • \n
    • man_grad (list):\nmanually supply a list or an array which contains the jacobian\nof func. Use cautiously, supplying the wrong derivative will\nnot be intercepted.
      • \n
    \n\n
    Notes
    \n\n

    For simple mathematical operations it can be practical to use anonymous\nfunctions. For the ratio of two observables one can e.g. use

    \n\n

    new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])

    \n", "signature": "(func, data, array_mode=False, **kwargs):", "funcdef": "def"}, "pyerrors.obs.reweight": {"fullname": "pyerrors.obs.reweight", "modulename": "pyerrors.obs", "qualname": "reweight", "kind": "function", "doc": "

    Reweight a list of observables.

    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • obs (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
      • \n
    \n", "signature": "(weight, obs, **kwargs):", "funcdef": "def"}, "pyerrors.obs.correlate": {"fullname": "pyerrors.obs.correlate", "modulename": "pyerrors.obs", "qualname": "correlate", "kind": "function", "doc": "

    Correlate two observables.

    \n\n
    Parameters
    \n\n
      \n
    • obs_a (Obs):\nFirst observable
      • \n
    • obs_b (Obs):\nSecond observable
      • \n
    \n\n
    Notes
    \n\n

    Keep in mind to only correlate primary observables which have not been reweighted\nyet. The reweighting has to be applied after correlating the observables.\nCurrently only works if ensembles are identical (this is not strictly necessary).

    \n", "signature": "(obs_a, obs_b):", "funcdef": "def"}, "pyerrors.obs.covariance": {"fullname": "pyerrors.obs.covariance", "modulename": "pyerrors.obs", "qualname": "covariance", "kind": "function", "doc": "

    Calculates the error covariance matrix of a set of observables.

    \n\n

    WARNING: This function should be used with care, especially for observables with support on multiple\n ensembles with differing autocorrelations. See the notes below for details.

    \n\n

    The gamma method has to be applied first to all observables.

    \n\n
    Parameters
    \n\n
      \n
    • obs (list or numpy.ndarray):\nList or one dimensional array of Obs
      • \n
    • visualize (bool):\nIf True plots the corresponding normalized correlation matrix (default False).
      • \n
    • correlation (bool):\nIf True the correlation matrix instead of the error covariance matrix is returned (default False).
      • \n
    • smooth (None or int):\nIf smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue\nsmoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the\nlargest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely\nsmall ones.
      • \n
    \n\n
    Notes
    \n\n

    The error covariance is defined such that it agrees with the squared standard error for two identical observables\n$$\\operatorname{cov}(a,a)=\\sum_{s=1}^N\\delta_a^s\\delta_a^s/N^2=\\Gamma_{aa}(0)/N=\\operatorname{var}(a)/N=\\sigma_a^2$$\nin the absence of autocorrelation.\nThe error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite\n$$\\sum_{i,j}v_i\\Gamma_{ij}(0)v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i,j}v_i\\delta_i^s\\delta_j^s v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i}|v_i\\delta_i^s|^2\\geq 0\\,,$$ for every $v\\in\\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.\nFor observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.\n$$\\tau_{\\mathrm{int}, ij}=\\sqrt{\\tau_{\\mathrm{int}, i}\\times \\tau_{\\mathrm{int}, j}}$$\nThis construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).

    \n", "signature": "(obs, visualize=False, correlation=False, smooth=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs.import_jackknife": {"fullname": "pyerrors.obs.import_jackknife", "modulename": "pyerrors.obs", "qualname": "import_jackknife", "kind": "function", "doc": "

    Imports jackknife samples and returns an Obs

    \n\n
    Parameters
    \n\n
      \n
    • jacks (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N jackknife samples as first to Nth entry.
      • \n
    • name (str):\nname of the ensemble the samples are defined on.
      • \n
    \n", "signature": "(jacks, name, idl=None):", "funcdef": "def"}, "pyerrors.obs.import_bootstrap": {"fullname": "pyerrors.obs.import_bootstrap", "modulename": "pyerrors.obs", "qualname": "import_bootstrap", "kind": "function", "doc": "

    Imports bootstrap samples and returns an Obs

    \n\n
    Parameters
    \n\n
      \n
    • boots (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N bootstrap samples as first to Nth entry.
      • \n
    • name (str):\nname of the ensemble the samples are defined on.
      • \n
    • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples,\nwhere samples is the number of bootstrap samples and length is the length of the original Monte Carlo\nchain to be reconstructed.
      • \n
    \n", "signature": "(boots, name, random_numbers):", "funcdef": "def"}, "pyerrors.obs.merge_obs": {"fullname": "pyerrors.obs.merge_obs", "modulename": "pyerrors.obs", "qualname": "merge_obs", "kind": "function", "doc": "

    Combine all observables in list_of_obs into one new observable

    \n\n
    Parameters
    \n\n
      \n
    • list_of_obs (list):\nlist of the Obs object to be combined
      • \n
    \n\n
    Notes
    \n\n

    It is not possible to combine obs which are based on the same replicum

    \n", "signature": "(list_of_obs):", "funcdef": "def"}, "pyerrors.obs.cov_Obs": {"fullname": "pyerrors.obs.cov_Obs", "modulename": "pyerrors.obs", "qualname": "cov_Obs", "kind": "function", "doc": "

    Create an Obs based on mean(s) and a covariance matrix

    \n\n
    Parameters
    \n\n
      \n
    • mean (list of floats or float):\nN mean value(s) of the new Obs
      • \n
    • cov (list or array):\n2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
      • \n
    • name (str):\nidentifier for the covariance matrix
      • \n
    • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
      • \n
    \n", "signature": "(means, cov, name, grad=None):", "funcdef": "def"}, "pyerrors.roots": {"fullname": "pyerrors.roots", "modulename": "pyerrors.roots", "kind": "module", "doc": "

    \n"}, "pyerrors.roots.find_root": {"fullname": "pyerrors.roots.find_root", "modulename": "pyerrors.roots", "qualname": "find_root", "kind": "function", "doc": "

    Finds the root of the function func(x, d) where d is an Obs.

    \n\n
    Parameters
    \n\n
      \n
    • d (Obs):\nObs passed to the function.
      • \n

    • func (object):\nFunction to be minimized. Any numpy functions have to use the autograd.numpy wrapper.\nExample:

      \n\n
      \n
      import autograd.numpy as anp\ndef root_func(x, d):\n   return anp.exp(-x ** 2) - d\n
      \n
      • \n

    • guess (float):\nInitial guess for the minimization.

      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (Obs):\nObs valued root of the function.
      • \n
    \n", "signature": "(d, func, guess=1.0, **kwargs):", "funcdef": "def"}, "pyerrors.special": {"fullname": "pyerrors.special", "modulename": "pyerrors.special", "kind": "module", "doc": "

    \n"}, "pyerrors.special.beta": {"fullname": "pyerrors.special.beta", "modulename": "pyerrors.special", "qualname": "beta", "kind": "function", "doc": "

    beta(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    beta(a, b, out=None)

    \n\n

    Beta function.

    \n\n

    This function is defined in 1 as

    \n\n

    $$B(a, b) = \\int_0^1 t^{a-1}(1-t)^{b-1}dt\n = \\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)},$$

    \n\n

    where \\( \\Gamma \\) is the gamma function.

    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nReal-valued arguments
      • \n
    • out (ndarray, optional):\nOptional output array for the function result
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the beta function
      • \n
    \n\n
    See Also
    \n\n

    gamma: the gamma function
    \nbetainc: the regularized incomplete beta function
    \nbetaln: the natural logarithm of the absolute\nvalue of the beta function

    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n

    The beta function relates to the gamma function by the\ndefinition given above:

    \n\n
    \n
    >>> sc.beta(2, 3)\n0.08333333333333333\n>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)\n0.08333333333333333\n
    \n
    \n\n

    As this relationship demonstrates, the beta function\nis symmetric:

    \n\n
    \n
    >>> sc.beta(1.7, 2.4)\n0.16567527689031739\n>>> sc.beta(2.4, 1.7)\n0.16567527689031739\n
    \n
    \n\n

    This function satisfies \\( B(1, b) = 1/b \\):

    \n\n
    \n
    >>> sc.beta(1, 4)\n0.25\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      NIST Digital Library of Mathematical Functions,\nEq. 5.12.1. https://dlmf.nist.gov/5.12 ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betainc": {"fullname": "pyerrors.special.betainc", "modulename": "pyerrors.special", "qualname": "betainc", "kind": "function", "doc": "

    betainc(x1, x2, x3, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    betainc(a, b, x, out=None)

    \n\n

    Regularized incomplete beta function.

    \n\n

    Computes the regularized incomplete beta function, defined as 1:

    \n\n

    $$I_x(a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^x\nt^{a-1}(1-t)^{b-1}dt,$$

    \n\n

    for \\( 0 \\leq x \\leq 1 \\).

    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nPositive, real-valued parameters
      • \n
    • x (array_like):\nReal-valued such that \\( 0 \\leq x \\leq 1 \\),\nthe upper limit of integration
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the regularized incomplete beta function
      • \n
    \n\n
    See Also
    \n\n

    beta: beta function
    \nbetaincinv: inverse of the regularized incomplete beta function

    \n\n
    Notes
    \n\n

    The term regularized in the name of this function refers to the\nscaling of the function by the gamma function terms shown in the\nformula. When not qualified as regularized, the name incomplete\nbeta function often refers to just the integral expression,\nwithout the gamma terms. One can use the function beta from\nscipy.special to get this \"nonregularized\" incomplete beta\nfunction by multiplying the result of betainc(a, b, x) by\nbeta(a, b).

    \n\n
    References
    \n\n
    Examples
    \n\n

    Let \\( B(a, b) \\) be the beta function.

    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n

    The coefficient in terms of gamma is equal to\n\\( 1/B(a, b) \\). Also, when \\( x=1 \\)\nthe integral is equal to \\( B(a, b) \\).\nTherefore, \\( I_{x=1}(a, b) = 1 \\) for any \\( a, b \\).

    \n\n
    \n
    >>> sc.betainc(0.2, 3.5, 1.0)\n1.0\n
    \n
    \n\n

    It satisfies\n\\( I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b)) \\),\nwhere \\( F \\) is the hypergeometric function hyp2f1:

    \n\n
    \n
    >>> a, b, x = 1.4, 3.1, 0.5\n>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))\n0.8148904036225295\n>>> sc.betainc(a, b, x)\n0.8148904036225296\n
    \n
    \n\n

    This functions satisfies the relationship\n\\( I_x(a, b) = 1 - I_{1-x}(b, a) \\):

    \n\n
    \n
    >>> sc.betainc(2.2, 3.1, 0.4)\n0.49339638807619446\n>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)\n0.49339638807619446\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/8.17 ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betaln": {"fullname": "pyerrors.special.betaln", "modulename": "pyerrors.special", "qualname": "betaln", "kind": "function", "doc": "

    betaln(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    betaln(a, b, out=None)

    \n\n

    Natural logarithm of absolute value of beta function.

    \n\n

    Computes ln(abs(beta(a, b))).

    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nPositive, real-valued parameters
      • \n
    • out (ndarray, optional):\nOptional output array for function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the betaln function
      • \n
    \n\n
    See Also
    \n\n

    gamma: the gamma function
    \nbetainc: the regularized incomplete beta function
    \nbeta: the beta function

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import betaln, beta\n
    \n
    \n\n

    Verify that, for moderate values of a and b, betaln(a, b)\nis the same as log(beta(a, b)):

    \n\n
    \n
    >>> betaln(3, 4)\n-4.0943445622221\n
    \n
    \n\n
    \n
    >>> np.log(beta(3, 4))\n-4.0943445622221\n
    \n
    \n\n

    In the following beta(a, b) underflows to 0, so we can't compute\nthe logarithm of the actual value.

    \n\n
    \n
    >>> a = 400\n>>> b = 900\n>>> beta(a, b)\n0.0\n
    \n
    \n\n

    We can compute the logarithm of beta(a, b) by using betaln:

    \n\n
    \n
    >>> betaln(a, b)\n-804.3069951764146\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.polygamma": {"fullname": "pyerrors.special.polygamma", "modulename": "pyerrors.special", "qualname": "polygamma", "kind": "function", "doc": "

    Polygamma functions.

    \n\n

    Defined as \\( \\psi^{(n)}(x) \\) where \\( \\psi \\) is the\ndigamma function. See [dlmf]_ for details.

    \n\n
    Parameters
    \n\n
      \n
    • n (array_like):\nThe order of the derivative of the digamma function; must be\nintegral
      • \n
    • x (array_like):\nReal valued input
      • \n
    \n\n
    Returns
    \n\n
      \n
    • ndarray: Function results
      • \n
    \n\n
    See Also
    \n\n

    digamma

    \n\n
    References
    \n\n

    .. [dlmf] NIST, Digital Library of Mathematical Functions,\n https://dlmf.nist.gov/5.15

    \n\n
    Examples
    \n\n
    \n
    >>> from scipy import special\n>>> x = [2, 3, 25.5]\n>>> special.polygamma(1, x)\narray([ 0.64493407,  0.39493407,  0.03999467])\n>>> special.polygamma(0, x) == special.psi(x)\narray([ True,  True,  True], dtype=bool)\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.psi": {"fullname": "pyerrors.special.psi", "modulename": "pyerrors.special", "qualname": "psi", "kind": "function", "doc": "

    psi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    psi(z, out=None)

    \n\n

    The digamma function.

    \n\n

    The logarithmic derivative of the gamma function evaluated at z.

    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex argument.
      • \n
    • out (ndarray, optional):\nArray for the computed values of psi.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • digamma (scalar or ndarray):\nComputed values of psi.
      • \n
    \n\n
    Notes
    \n\n

    For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.

    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n

    Verify psi(z) = psi(z + 1) - 1/z:

    \n\n
    \n
    >>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      2. \n\n
    2. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      3. \n\n
    3. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      4. \n\n
    4. \n

      Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩

      \n
      5. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.digamma": {"fullname": "pyerrors.special.digamma", "modulename": "pyerrors.special", "qualname": "digamma", "kind": "function", "doc": "

    psi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    psi(z, out=None)

    \n\n

    The digamma function.

    \n\n

    The logarithmic derivative of the gamma function evaluated at z.

    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex argument.
      • \n
    • out (ndarray, optional):\nArray for the computed values of psi.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • digamma (scalar or ndarray):\nComputed values of psi.
      • \n
    \n\n
    Notes
    \n\n

    For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.

    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n

    Verify psi(z) = psi(z + 1) - 1/z:

    \n\n
    \n
    >>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      2. \n\n
    2. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      3. \n\n
    3. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      4. \n\n
    4. \n

      Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩

      \n
      5. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gamma": {"fullname": "pyerrors.special.gamma", "modulename": "pyerrors.special", "qualname": "gamma", "kind": "function", "doc": "

    gamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    gamma(z, out=None)

    \n\n

    gamma function.

    \n\n

    The gamma function is defined as

    \n\n

    $$\\Gamma(z) = \\int_0^\\infty t^{z-1} e^{-t} dt$$

    \n\n

    for \\( \\Re(z) > 0 \\) and is extended to the rest of the complex\nplane by analytic continuation. See [dlmf]_ for more details.

    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex valued argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the gamma function
      • \n
    \n\n
    Notes
    \n\n

    The gamma function is often referred to as the generalized\nfactorial since \\( \\Gamma(n + 1) = n! \\) for natural numbers\n\\( n \\). More generally it satisfies the recurrence relation\n\\( \\Gamma(z + 1) = z \\cdot \\Gamma(z) \\) for complex \\( z \\),\nwhich, combined with the fact that \\( \\Gamma(1) = 1 \\), implies\nthe above identity for \\( z = n \\).

    \n\n
    References
    \n\n

    .. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5.2#E1

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import gamma, factorial\n
    \n
    \n\n
    \n
    >>> gamma([0, 0.5, 1, 5])\narray([         inf,   1.77245385,   1.        ,  24.        ])\n
    \n
    \n\n
    \n
    >>> z = 2.5 + 1j\n>>> gamma(z)\n(0.77476210455108352+0.70763120437959293j)\n>>> gamma(z+1), z*gamma(z)  # Recurrence property\n((1.2292740569981171+2.5438401155000685j),\n (1.2292740569981158+2.5438401155000658j))\n
    \n
    \n\n
    \n
    >>> gamma(0.5)**2  # gamma(0.5) = sqrt(pi)\n3.1415926535897927\n
    \n
    \n\n

    Plot gamma(x) for real x

    \n\n
    \n
    >>> x = np.linspace(-3.5, 5.5, 2251)\n>>> y = gamma(x)\n
    \n
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')\n>>> k = np.arange(1, 7)\n>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,\n...          label='(x-1)!, x = 1, 2, ...')\n>>> plt.xlim(-3.5, 5.5)\n>>> plt.ylim(-10, 25)\n>>> plt.grid()\n>>> plt.xlabel('x')\n>>> plt.legend(loc='lower right')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaln": {"fullname": "pyerrors.special.gammaln", "modulename": "pyerrors.special", "qualname": "gammaln", "kind": "function", "doc": "

    gammaln(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    gammaln(x, out=None)

    \n\n

    Logarithm of the absolute value of the gamma function.

    \n\n

    Defined as

    \n\n

    $$\\ln(\\lvert\\Gamma(x)\\rvert)$$

    \n\n

    where \\( \\Gamma \\) is the gamma function. For more details on\nthe gamma function, see [dlmf]_.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the log of the absolute value of gamma
      • \n
    \n\n
    See Also
    \n\n

    gammasgn: sign of the gamma function
    \nloggamma: principal branch of the logarithm of the gamma function

    \n\n
    Notes
    \n\n

    It is the same function as the Python standard library function\nmath.lgamma().

    \n\n

    When used in conjunction with gammasgn, this function is useful\nfor working in logspace on the real axis without having to deal\nwith complex numbers via the relation exp(gammaln(x)) =\ngammasgn(x) * gamma(x).

    \n\n

    For complex-valued log-gamma, use loggamma instead of gammaln.

    \n\n
    References
    \n\n

    .. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import scipy.special as sc\n
    \n
    \n\n

    It has two positive zeros.

    \n\n
    \n
    >>> sc.gammaln([1, 2])\narray([0., 0.])\n
    \n
    \n\n

    It has poles at nonpositive integers.

    \n\n
    \n
    >>> sc.gammaln([0, -1, -2, -3, -4])\narray([inf, inf, inf, inf, inf])\n
    \n
    \n\n

    It asymptotically approaches x * log(x) (Stirling's formula).

    \n\n
    \n
    >>> x = np.array([1e10, 1e20, 1e40, 1e80])\n>>> sc.gammaln(x)\narray([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])\n>>> x * np.log(x)\narray([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammainc": {"fullname": "pyerrors.special.gammainc", "modulename": "pyerrors.special", "qualname": "gammainc", "kind": "function", "doc": "

    gammainc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    gammainc(a, x, out=None)

    \n\n

    Regularized lower incomplete gamma function.

    \n\n

    It is defined as

    \n\n

    $$P(a, x) = \\frac{1}{\\Gamma(a)} \\int_0^x t^{a - 1}e^{-t} dt$$

    \n\n

    for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.

    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nPositive parameter
      • \n
    • x (array_like):\nNonnegative argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the lower incomplete gamma function
      • \n
    \n\n
    Notes
    \n\n

    The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammaincc is the regularized upper\nincomplete gamma function.

    \n\n

    The implementation largely follows that of [boost]_.

    \n\n
    See also
    \n\n

    gammaincc : regularized upper incomplete gamma function\ngammaincinv : inverse of the regularized lower incomplete gamma function\ngammainccinv : inverse of the regularized upper incomplete gamma function

    \n\n
    References
    \n\n

    .. [dlmf] NIST Digital Library of Mathematical functions\n https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html

    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n

    It is the CDF of the gamma distribution, so it starts at 0 and\nmonotonically increases to 1.

    \n\n
    \n
    >>> sc.gammainc(0.5, [0, 1, 10, 100])\narray([0.        , 0.84270079, 0.99999226, 1.        ])\n
    \n
    \n\n

    It is equal to one minus the upper incomplete gamma function.

    \n\n
    \n
    >>> a, x = 0.5, 0.4\n>>> sc.gammainc(a, x)\n0.6289066304773024\n>>> 1 - sc.gammaincc(a, x)\n0.6289066304773024\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaincc": {"fullname": "pyerrors.special.gammaincc", "modulename": "pyerrors.special", "qualname": "gammaincc", "kind": "function", "doc": "

    gammaincc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    gammaincc(a, x, out=None)

    \n\n

    Regularized upper incomplete gamma function.

    \n\n

    It is defined as

    \n\n

    $$Q(a, x) = \\frac{1}{\\Gamma(a)} \\int_x^\\infty t^{a - 1}e^{-t} dt$$

    \n\n

    for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.

    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nPositive parameter
      • \n
    • x (array_like):\nNonnegative argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the upper incomplete gamma function
      • \n
    \n\n
    Notes
    \n\n

    The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammainc is the regularized lower\nincomplete gamma function.

    \n\n

    The implementation largely follows that of [boost]_.

    \n\n
    See also
    \n\n

    gammainc : regularized lower incomplete gamma function\ngammaincinv : inverse of the regularized lower incomplete gamma function\ngammainccinv : inverse of the regularized upper incomplete gamma function

    \n\n
    References
    \n\n

    .. [dlmf] NIST Digital Library of Mathematical functions\n https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html

    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n

    It is the survival function of the gamma distribution, so it\nstarts at 1 and monotonically decreases to 0.

    \n\n
    \n
    >>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000])\narray([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45,\n       0.00000000e+00])\n
    \n
    \n\n

    It is equal to one minus the lower incomplete gamma function.

    \n\n
    \n
    >>> a, x = 0.5, 0.4\n>>> sc.gammaincc(a, x)\n0.37109336952269756\n>>> 1 - sc.gammainc(a, x)\n0.37109336952269756\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammasgn": {"fullname": "pyerrors.special.gammasgn", "modulename": "pyerrors.special", "qualname": "gammasgn", "kind": "function", "doc": "

    gammasgn(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    gammasgn(x, out=None)

    \n\n

    Sign of the gamma function.

    \n\n

    It is defined as

    \n\n

    $$\\text{gammasgn}(x) =\n\\begin{cases}\n +1 & \\Gamma(x) > 0 \\\n -1 & \\Gamma(x) < 0\n\\end{cases}$$

    \n\n

    where \\( \\Gamma \\) is the gamma function; see gamma. This\ndefinition is complete since the gamma function is never zero;\nsee the discussion after [dlmf]_.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Sign of the gamma function
      • \n
    \n\n
    Notes
    \n\n

    The gamma function can be computed as gammasgn(x) *\nnp.exp(gammaln(x)).

    \n\n
    See Also
    \n\n

    gamma: the gamma function
    \ngammaln: log of the absolute value of the gamma function
    \nloggamma: analytic continuation of the log of the gamma function

    \n\n
    References
    \n\n

    .. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5.2#E1

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import scipy.special as sc\n
    \n
    \n\n

    It is 1 for x > 0.

    \n\n
    \n
    >>> sc.gammasgn([1, 2, 3, 4])\narray([1., 1., 1., 1.])\n
    \n
    \n\n

    It alternates between -1 and 1 for negative integers.

    \n\n
    \n
    >>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5])\narray([-1.,  1., -1.,  1.])\n
    \n
    \n\n

    It can be used to compute the gamma function.

    \n\n
    \n
    >>> x = [1.5, 0.5, -0.5, -1.5]\n>>> sc.gammasgn(x) * np.exp(sc.gammaln(x))\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n>>> sc.gamma(x)\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.rgamma": {"fullname": "pyerrors.special.rgamma", "modulename": "pyerrors.special", "qualname": "rgamma", "kind": "function", "doc": "

    rgamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    rgamma(z, out=None)

    \n\n

    Reciprocal of the gamma function.

    \n\n

    Defined as \\( 1 / \\Gamma(z) \\), where \\( \\Gamma \\) is the\ngamma function. For more on the gamma function see gamma.

    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex valued input
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Function results
      • \n
    \n\n
    Notes
    \n\n

    The gamma function has no zeros and has simple poles at\nnonpositive integers, so rgamma is an entire function with zeros\nat the nonpositive integers. See the discussion in [dlmf]_ for\nmore details.

    \n\n
    See Also
    \n\n

    gamma,, gammaln,, loggamma

    \n\n
    References
    \n\n

    .. [dlmf] Nist, Digital Library of Mathematical functions,\n https://dlmf.nist.gov/5.2#i

    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n

    It is the reciprocal of the gamma function.

    \n\n
    \n
    >>> sc.rgamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n>>> 1 / sc.gamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n
    \n
    \n\n

    It is zero at nonpositive integers.

    \n\n
    \n
    >>> sc.rgamma([0, -1, -2, -3])\narray([0., 0., 0., 0.])\n
    \n
    \n\n

    It rapidly underflows to zero along the positive real axis.

    \n\n
    \n
    >>> sc.rgamma([10, 100, 179])\narray([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.multigammaln": {"fullname": "pyerrors.special.multigammaln", "modulename": "pyerrors.special", "qualname": "multigammaln", "kind": "function", "doc": "

    Returns the log of multivariate gamma, also sometimes called the\ngeneralized gamma.

    \n\n
    Parameters
    \n\n
      \n
    • a (ndarray):\nThe multivariate gamma is computed for each item of a.
      • \n
    • d (int):\nThe dimension of the space of integration.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (ndarray):\nThe values of the log multivariate gamma at the given points a.
      • \n
    \n\n
    Notes
    \n\n

    The formal definition of the multivariate gamma of dimension d for a real\na is

    \n\n

    $$\\Gamma_d(a) = \\int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA$$

    \n\n

    with the condition \\( a > (d-1)/2 \\), and \\( A > 0 \\) being the set of\nall the positive definite matrices of dimension d. Note that a is a\nscalar: the integrand only is multivariate, the argument is not (the\nfunction is defined over a subset of the real set).

    \n\n

    This can be proven to be equal to the much friendlier equation

    \n\n

    $$\\Gamma_d(a) = \\pi^{d(d-1)/4} \\prod_{i=1}^{d} \\Gamma(a - (i-1)/2).$$

    \n\n
    References
    \n\n

    R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in\nprobability and mathematical statistics).

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import multigammaln, gammaln\n>>> a = 23.5\n>>> d = 10\n>>> multigammaln(a, d)\n454.1488605074416\n
    \n
    \n\n

    Verify that the result agrees with the logarithm of the equation\nshown above:

    \n\n
    \n
    >>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()\n454.1488605074416\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.kn": {"fullname": "pyerrors.special.kn", "modulename": "pyerrors.special", "qualname": "kn", "kind": "function", "doc": "

    Modified Bessel function of the second kind of integer order n

    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j0": {"fullname": "pyerrors.special.j0", "modulename": "pyerrors.special", "qualname": "j0", "kind": "function", "doc": "

    j0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    j0(x, out=None)

    \n\n

    Bessel function of the first kind of order 0.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 0 at x.
      • \n
    \n\n
    Notes
    \n\n

    The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval the following rational approximation is used:

    \n\n

    $$J_0(x) \\approx (w - r_1^2)(w - r_2^2) \\frac{P_3(w)}{Q_8(w)},$$

    \n\n

    where \\( w = x^2 \\) and \\( r_1 \\), \\( r_2 \\) are the zeros of\n\\( J_0 \\), and \\( P_3 \\) and \\( Q_8 \\) are polynomials of degrees 3\nand 8, respectively.

    \n\n

    In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.

    \n\n

    This function is a wrapper for the Cephes 1 routine j0.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).

    \n\n
    See also
    \n\n

    jv : Bessel function of real order and complex argument.\nspherical_jn : spherical Bessel functions.

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import j0\n>>> j0(1.)\n0.7651976865579665\n
    \n
    \n\n

    Calculate the function at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> j0(np.array([-2., 0., 4.]))\narray([ 0.22389078,  1.        , -0.39714981])\n
    \n
    \n\n

    Plot the function from -20 to 20.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y0": {"fullname": "pyerrors.special.y0", "modulename": "pyerrors.special", "qualname": "y0", "kind": "function", "doc": "

    y0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    y0(x, out=None)

    \n\n

    Bessel function of the second kind of order 0.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 0 at x.
      • \n
    \n\n
    Notes
    \n\n

    The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval a rational approximation \\( R(x) \\) is employed to\ncompute,

    \n\n

    $$Y_0(x) = R(x) + \\frac{2 \\log(x) J_0(x)}{\\pi},$$

    \n\n

    where \\( J_0 \\) is the Bessel function of the first kind of order 0.

    \n\n

    In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.

    \n\n

    This function is a wrapper for the Cephes 1 routine y0.

    \n\n
    See also
    \n\n

    j0: Bessel function of the first kind of order 0\nyv: Bessel function of the first kind

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import y0\n>>> y0(1.)\n0.08825696421567697\n
    \n
    \n\n

    Calculate at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> y0(np.array([0.5, 2., 3.]))\narray([-0.44451873,  0.51037567,  0.37685001])\n
    \n
    \n\n

    Plot the function from 0 to 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j1": {"fullname": "pyerrors.special.j1", "modulename": "pyerrors.special", "qualname": "j1", "kind": "function", "doc": "

    j1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    j1(x, out=None)

    \n\n

    Bessel function of the first kind of order 1.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 1 at x.
      • \n
    \n\n
    Notes
    \n\n

    The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 24 term Chebyshev expansion is used. In the second, the\nasymptotic trigonometric representation is employed using two rational\nfunctions of degree 5/5.

    \n\n

    This function is a wrapper for the Cephes 1 routine j1.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).

    \n\n
    See also
    \n\n

    jv: Bessel function of the first kind\nspherical_jn: spherical Bessel functions.

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import j1\n>>> j1(1.)\n0.44005058574493355\n
    \n
    \n\n

    Calculate the function at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> j1(np.array([-2., 0., 4.]))\narray([-0.57672481,  0.        , -0.06604333])\n
    \n
    \n\n

    Plot the function from -20 to 20.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y1": {"fullname": "pyerrors.special.y1", "modulename": "pyerrors.special", "qualname": "y1", "kind": "function", "doc": "

    y1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    y1(x, out=None)

    \n\n

    Bessel function of the second kind of order 1.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 1 at x.
      • \n
    \n\n
    Notes
    \n\n

    The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 25 term Chebyshev expansion is used, and computing\n\\( J_1 \\) (the Bessel function of the first kind) is required. In the\nsecond, the asymptotic trigonometric representation is employed using two\nrational functions of degree 5/5.

    \n\n

    This function is a wrapper for the Cephes 1 routine y1.

    \n\n
    See also
    \n\n

    j1: Bessel function of the first kind of order 1\nyn: Bessel function of the second kind\nyv: Bessel function of the second kind

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import y1\n>>> y1(1.)\n-0.7812128213002888\n
    \n
    \n\n

    Calculate at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> y1(np.array([0.5, 2., 3.]))\narray([-1.47147239, -0.10703243,  0.32467442])\n
    \n
    \n\n

    Plot the function from 0 to 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.jn": {"fullname": "pyerrors.special.jn", "modulename": "pyerrors.special", "qualname": "jn", "kind": "function", "doc": "

    jv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    jv(v, z, out=None)

    \n\n

    Bessel function of the first kind of real order and complex argument.

    \n\n
    Parameters
    \n\n
      \n
    • v (array_like):\nOrder (float).
      • \n
    • z (array_like):\nArgument (float or complex).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function, \\( J_v(z) \\).
      • \n
    \n\n
    See also
    \n\n

    jve : \\( J_v \\) with leading exponential behavior stripped off.\nspherical_jn : spherical Bessel functions.\nj0 : faster version of this function for order 0.\nj1 : faster version of this function for order 1.

    \n\n
    Notes
    \n\n

    For positive v values, the computation is carried out using the AMOS\n1 zbesj routine, which exploits the connection to the modified\nBessel function \\( I_v \\),

    \n\n

    $$J_v(z) = \\exp(v\\pi\\imath/2) I_v(-\\imath z)\\qquad (\\Im z > 0)

    \n\n

    J_v(z) = \\exp(-v\\pi\\imath/2) I_v(\\imath z)\\qquad (\\Im z < 0)$$

    \n\n

    For negative v values the formula,

    \n\n

    $$J_{-v}(z) = J_v(z) \\cos(\\pi v) - Y_v(z) \\sin(\\pi v)$$

    \n\n

    is used, where \\( Y_v(z) \\) is the Bessel function of the second\nkind, computed using the AMOS routine zbesy. Note that the second\nterm is exactly zero for integer v; to improve accuracy the second\nterm is explicitly omitted for v values such that v = floor(v).

    \n\n

    Not to be confused with the spherical Bessel functions (see spherical_jn).

    \n\n
    References
    \n\n
    Examples
    \n\n

    Evaluate the function of order 0 at one point.

    \n\n
    \n
    >>> from scipy.special import jv\n>>> jv(0, 1.)\n0.7651976865579666\n
    \n
    \n\n

    Evaluate the function at one point for different orders.

    \n\n
    \n
    >>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)\n(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)\n
    \n
    \n\n

    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:

    \n\n
    \n
    >>> jv([0, 1, 1.5], 1.)\narray([0.76519769, 0.44005059, 0.24029784])\n
    \n
    \n\n

    Evaluate the function at several points for order 0 by providing an\narray for z.

    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> jv(0, points)\narray([ 0.22389078,  1.        , -0.26005195])\n
    \n
    \n\n

    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:

    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> jv(orders, points)\narray([[ 0.22389078,  1.        , -0.26005195],\n       [-0.57672481,  0.        ,  0.33905896]])\n
    \n
    \n\n

    Plot the functions of order 0 to 3 from -10 to 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, jv(i, x), label=f'$J_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.yn": {"fullname": "pyerrors.special.yn", "modulename": "pyerrors.special", "qualname": "yn", "kind": "function", "doc": "

    yn(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    yn(n, x, out=None)

    \n\n

    Bessel function of the second kind of integer order and real argument.

    \n\n
    Parameters
    \n\n
      \n
    • n (array_like):\nOrder (integer).
      • \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function, \\( Y_n(x) \\).
      • \n
    \n\n
    Notes
    \n\n

    Wrapper for the Cephes 1 routine yn.

    \n\n

    The function is evaluated by forward recurrence on n, starting with\nvalues computed by the Cephes routines y0 and y1. If n = 0 or 1,\nthe routine for y0 or y1 is called directly.

    \n\n
    See also
    \n\n

    yv : For real order and real or complex argument.\ny0: faster implementation of this function for order 0\ny1: faster implementation of this function for order 1

    \n\n
    References
    \n\n
    Examples
    \n\n

    Evaluate the function of order 0 at one point.

    \n\n
    \n
    >>> from scipy.special import yn\n>>> yn(0, 1.)\n0.08825696421567697\n
    \n
    \n\n

    Evaluate the function at one point for different orders.

    \n\n
    \n
    >>> yn(0, 1.), yn(1, 1.), yn(2, 1.)\n(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)\n
    \n
    \n\n

    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:

    \n\n
    \n
    >>> yn([0, 1, 2], 1.)\narray([ 0.08825696, -0.78121282, -1.65068261])\n
    \n
    \n\n

    Evaluate the function at several points for order 0 by providing an\narray for z.

    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([0.5, 3., 8.])\n>>> yn(0, points)\narray([-0.44451873,  0.37685001,  0.22352149])\n
    \n
    \n\n

    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:

    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> yn(orders, points)\narray([[-0.44451873,  0.37685001,  0.22352149],\n       [-1.47147239,  0.32467442, -0.15806046]])\n
    \n
    \n\n

    Plot the functions of order 0 to 3 from 0 to 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')\n>>> ax.set_ylim(-3, 1)\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i0": {"fullname": "pyerrors.special.i0", "modulename": "pyerrors.special", "qualname": "i0", "kind": "function", "doc": "

    i0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    i0(x, out=None)

    \n\n

    Modified Bessel function of order 0.

    \n\n

    Defined as,

    \n\n

    $$I_0(x) = \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{(k!)^2} = J_0(\\imath x),$$

    \n\n

    where \\( J_0 \\) is the Bessel function of the first kind of order 0.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float)
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • I (scalar or ndarray):\nValue of the modified Bessel function of order 0 at x.
      • \n
    \n\n
    Notes
    \n\n

    The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.

    \n\n

    This function is a wrapper for the Cephes 1 routine i0.

    \n\n
    See also
    \n\n

    iv: Modified Bessel function of any order\ni0e: Exponentially scaled modified Bessel function of order 0

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import i0\n>>> i0(1.)\n1.2660658777520082\n
    \n
    \n\n

    Calculate at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> i0(np.array([-2., 0., 3.5]))\narray([2.2795853 , 1.        , 7.37820343])\n
    \n
    \n\n

    Plot the function from -10 to 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i1": {"fullname": "pyerrors.special.i1", "modulename": "pyerrors.special", "qualname": "i1", "kind": "function", "doc": "

    i1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    i1(x, out=None)

    \n\n

    Modified Bessel function of order 1.

    \n\n

    Defined as,

    \n\n

    $$I_1(x) = \\frac{1}{2}x \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{k! (k + 1)!}\n = -\\imath J_1(\\imath x),$$

    \n\n

    where \\( J_1 \\) is the Bessel function of the first kind of order 1.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float)
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • I (scalar or ndarray):\nValue of the modified Bessel function of order 1 at x.
      • \n
    \n\n
    Notes
    \n\n

    The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.

    \n\n

    This function is a wrapper for the Cephes 1 routine i1.

    \n\n
    See also
    \n\n

    iv: Modified Bessel function of the first kind\ni1e: Exponentially scaled modified Bessel function of order 1

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import i1\n>>> i1(1.)\n0.5651591039924851\n
    \n
    \n\n

    Calculate the function at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> i1(np.array([-2., 0., 6.]))\narray([-1.59063685,  0.        , 61.34193678])\n
    \n
    \n\n

    Plot the function between -10 and 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.iv": {"fullname": "pyerrors.special.iv", "modulename": "pyerrors.special", "qualname": "iv", "kind": "function", "doc": "

    iv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    iv(v, z, out=None)

    \n\n

    Modified Bessel function of the first kind of real order.

    \n\n
    Parameters
    \n\n
      \n
    • v (array_like):\nOrder. If z is of real type and negative, v must be integer\nvalued.
      • \n
    • z (array_like of float or complex):\nArgument.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the modified Bessel function.
      • \n
    \n\n
    Notes
    \n\n

    For real z and \\( v \\in [-50, 50] \\), the evaluation is carried out\nusing Temme's method 1. For larger orders, uniform asymptotic\nexpansions are applied.

    \n\n

    For complex z and positive v, the AMOS 2 zbesi routine is\ncalled. It uses a power series for small z, the asymptotic expansion\nfor large abs(z), the Miller algorithm normalized by the Wronskian\nand a Neumann series for intermediate magnitudes, and the uniform\nasymptotic expansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large\norders. Backward recurrence is used to generate sequences or reduce\norders when necessary.

    \n\n

    The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,

    \n\n

    $$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$

    \n\n

    (valid when the real part of z is positive). For negative v, the\nformula

    \n\n

    $$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$

    \n\n

    is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.

    \n\n
    See also
    \n\n

    ive : This function with leading exponential behavior stripped off.\ni0 : Faster version of this function for order 0.\ni1 : Faster version of this function for order 1.

    \n\n
    References
    \n\n
    Examples
    \n\n

    Evaluate the function of order 0 at one point.

    \n\n
    \n
    >>> from scipy.special import iv\n>>> iv(0, 1.)\n1.2660658777520084\n
    \n
    \n\n

    Evaluate the function at one point for different orders.

    \n\n
    \n
    >>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)\n(1.2660658777520084, 0.565159103992485, 0.2935253263474798)\n
    \n
    \n\n

    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:

    \n\n
    \n
    >>> iv([0, 1, 1.5], 1.)\narray([1.26606588, 0.5651591 , 0.29352533])\n
    \n
    \n\n

    Evaluate the function at several points for order 0 by providing an\narray for z.

    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> iv(0, points)\narray([2.2795853 , 1.        , 4.88079259])\n
    \n
    \n\n

    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:

    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> iv(orders, points)\narray([[ 2.2795853 ,  1.        ,  4.88079259],\n       [-1.59063685,  0.        ,  3.95337022]])\n
    \n
    \n\n

    Plot the functions of order 0 to 3 from -5 to 5.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, iv(i, x), label=f'$I_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Temme, Journal of Computational Physics, vol 21, 343 (1976) ↩

      \n
      2. \n\n
    2. \n

      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩

      \n
      3. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.ive": {"fullname": "pyerrors.special.ive", "modulename": "pyerrors.special", "qualname": "ive", "kind": "function", "doc": "

    ive(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    ive(v, z, out=None)

    \n\n

    Exponentially scaled modified Bessel function of the first kind.

    \n\n

    Defined as::

    \n\n
    ive(v, z) = iv(v, z) * exp(-abs(z.real))\n
    \n\n

    For imaginary numbers without a real part, returns the unscaled\nBessel function of the first kind iv.

    \n\n
    Parameters
    \n\n
      \n
    • v (array_like of float):\nOrder.
      • \n
    • z (array_like of float or complex):\nArgument.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the exponentially scaled modified Bessel function.
      • \n
    \n\n
    Notes
    \n\n

    For positive v, the AMOS 1 zbesi routine is called. It uses a\npower series for small z, the asymptotic expansion for large\nabs(z), the Miller algorithm normalized by the Wronskian and a\nNeumann series for intermediate magnitudes, and the uniform asymptotic\nexpansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large orders.\nBackward recurrence is used to generate sequences or reduce orders when\nnecessary.

    \n\n

    The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,

    \n\n

    $$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$

    \n\n

    (valid when the real part of z is positive). For negative v, the\nformula

    \n\n

    $$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$

    \n\n

    is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.

    \n\n

    ive is useful for large arguments z: for these, iv easily overflows,\nwhile ive does not due to the exponential scaling.

    \n\n
    See also
    \n\n

    iv: Modified Bessel function of the first kind\ni0e: Faster implementation of this function for order 0\ni1e: Faster implementation of this function for order 1

    \n\n
    References
    \n\n
    Examples
    \n\n

    In the following example iv returns infinity whereas ive still returns\na finite number.

    \n\n
    \n
    >>> from scipy.special import iv, ive\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> iv(3, 1000.), ive(3, 1000.)\n(inf, 0.01256056218254712)\n
    \n
    \n\n

    Evaluate the function at one point for different orders by\nproviding a list or NumPy array as argument for the v parameter:

    \n\n
    \n
    >>> ive([0, 1, 1.5], 1.)\narray([0.46575961, 0.20791042, 0.10798193])\n
    \n
    \n\n

    Evaluate the function at several points for order 0 by providing an\narray for z.

    \n\n
    \n
    >>> points = np.array([-2., 0., 3.])\n>>> ive(0, points)\narray([0.30850832, 1.        , 0.24300035])\n
    \n
    \n\n

    Evaluate the function at several points for different orders by\nproviding arrays for both v for z. Both arrays have to be\nbroadcastable to the correct shape. To calculate the orders 0, 1\nand 2 for a 1D array of points:

    \n\n
    \n
    >>> ive([[0], [1], [2]], points)\narray([[ 0.30850832,  1.        ,  0.24300035],\n       [-0.21526929,  0.        ,  0.19682671],\n       [ 0.09323903,  0.        ,  0.11178255]])\n
    \n
    \n\n

    Plot the functions of order 0 to 3 from -5 to 5.

    \n\n
    \n
    >>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, ive(i, x), label=f'$I_{i!r}(z)\\cdot e^{{-|z|}}$')\n>>> ax.legend()\n>>> ax.set_xlabel(r"$z$")\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erf": {"fullname": "pyerrors.special.erf", "modulename": "pyerrors.special", "qualname": "erf", "kind": "function", "doc": "

    erf(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    erf(z, out=None)

    \n\n

    Returns the error function of complex argument.

    \n\n

    It is defined as 2/sqrt(pi)*integral(exp(-t**2), t=0..z).

    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nInput array.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (scalar or ndarray):\nThe values of the error function at the given points x.
      • \n
    \n\n
    See Also
    \n\n

    erfc,, erfinv,, erfcinv,, wofz,, erfcx,, erfi

    \n\n
    Notes
    \n\n

    The cumulative of the unit normal distribution is given by\nPhi(z) = 1/2[1 + erf(z/sqrt(2))].

    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erf(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erf(x)$')\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfc": {"fullname": "pyerrors.special.erfc", "modulename": "pyerrors.special", "qualname": "erfc", "kind": "function", "doc": "

    erfc(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    erfc(x, out=None)

    \n\n

    Complementary error function, 1 - erf(x).

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal or complex valued argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the complementary error function
      • \n
    \n\n
    See Also
    \n\n

    erf,, erfi,, erfcx,, dawsn,, wofz

    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erfc(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erfc(x)$')\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfinv": {"fullname": "pyerrors.special.erfinv", "modulename": "pyerrors.special", "qualname": "erfinv", "kind": "function", "doc": "

    erfinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    erfinv(y, out=None)

    \n\n

    Inverse of the error function.

    \n\n

    Computes the inverse of the error function.

    \n\n

    In the complex domain, there is no unique complex number w satisfying\nerf(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, -1 < x < 1, there is a unique real\nnumber satisfying erf(erfinv(x)) = x.

    \n\n
    Parameters
    \n\n
      \n
    • y (ndarray):\nArgument at which to evaluate. Domain: [-1, 1]
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • erfinv (scalar or ndarray):\nThe inverse of erf of y, element-wise
      • \n
    \n\n
    See Also
    \n\n

    erf: Error function of a complex argument
    \nerfc: Complementary error function, 1 - erf(x)
    \nerfcinv: Inverse of the complementary error function

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfinv, erf\n
    \n
    \n\n
    \n
    >>> erfinv(0.5)\n0.4769362762044699\n
    \n
    \n\n
    \n
    >>> y = np.linspace(-1.0, 1.0, num=9)\n>>> x = erfinv(y)\n>>> x\narray([       -inf, -0.81341985, -0.47693628, -0.22531206,  0.        ,\n        0.22531206,  0.47693628,  0.81341985,         inf])\n
    \n
    \n\n

    Verify that erf(erfinv(y)) is y.

    \n\n
    \n
    >>> erf(x)\narray([-1.  , -0.75, -0.5 , -0.25,  0.  ,  0.25,  0.5 ,  0.75,  1.  ])\n
    \n
    \n\n

    Plot the function:

    \n\n
    \n
    >>> y = np.linspace(-1, 1, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfinv(y)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfcinv": {"fullname": "pyerrors.special.erfcinv", "modulename": "pyerrors.special", "qualname": "erfcinv", "kind": "function", "doc": "

    erfcinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    erfcinv(y, out=None)

    \n\n

    Inverse of the complementary error function.

    \n\n

    Computes the inverse of the complementary error function.

    \n\n

    In the complex domain, there is no unique complex number w satisfying\nerfc(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, 0 < x < 2, there is a unique real\nnumber satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).

    \n\n

    It is related to inverse of the error function by erfcinv(1-x) = erfinv(x)

    \n\n
    Parameters
    \n\n
      \n
    • y (ndarray):\nArgument at which to evaluate. Domain: [0, 2]
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • erfcinv (scalar or ndarray):\nThe inverse of erfc of y, element-wise
      • \n
    \n\n
    See Also
    \n\n

    erf: Error function of a complex argument
    \nerfc: Complementary error function, 1 - erf(x)
    \nerfinv: Inverse of the error function

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfcinv\n
    \n
    \n\n
    \n
    >>> erfcinv(0.5)\n0.4769362762044699\n
    \n
    \n\n
    \n
    >>> y = np.linspace(0.0, 2.0, num=11)\n>>> erfcinv(y)\narray([        inf,  0.9061938 ,  0.59511608,  0.37080716,  0.17914345,\n       -0.        , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,\n              -inf])\n
    \n
    \n\n

    Plot the function:

    \n\n
    \n
    >>> y = np.linspace(0, 2, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfcinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfcinv(y)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logit": {"fullname": "pyerrors.special.logit", "modulename": "pyerrors.special", "qualname": "logit", "kind": "function", "doc": "

    logit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    logit(x, out=None)

    \n\n

    Logit ufunc for ndarrays.

    \n\n

    The logit function is defined as logit(p) = log(p/(1-p)).\nNote that logit(0) = -inf, logit(1) = inf, and logit(p)\nfor p<0 or p>1 yields nan.

    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nThe ndarray to apply logit to element-wise.
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare logit of the corresponding entry of x.
      • \n
    \n\n
    See Also
    \n\n

    expit

    \n\n
    Notes
    \n\n

    As a ufunc logit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs

    \n\n

    New in version 0.10.0.

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import logit, expit\n
    \n
    \n\n
    \n
    >>> logit([0, 0.25, 0.5, 0.75, 1])\narray([       -inf, -1.09861229,  0.        ,  1.09861229,         inf])\n
    \n
    \n\n

    expit is the inverse of logit:

    \n\n
    \n
    >>> expit(logit([0.1, 0.75, 0.999]))\narray([ 0.1  ,  0.75 ,  0.999])\n
    \n
    \n\n

    Plot logit(x) for x in [0, 1]:

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> x = np.linspace(0, 1, 501)\n>>> y = logit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.ylim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('logit(x)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.expit": {"fullname": "pyerrors.special.expit", "modulename": "pyerrors.special", "qualname": "expit", "kind": "function", "doc": "

    expit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    expit(x, out=None)

    \n\n

    Expit (a.k.a. logistic sigmoid) ufunc for ndarrays.

    \n\n

    The expit function, also known as the logistic sigmoid function, is\ndefined as expit(x) = 1/(1+exp(-x)). It is the inverse of the\nlogit function.

    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nThe ndarray to apply expit to element-wise.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare expit of the corresponding entry of x.
      • \n
    \n\n
    See Also
    \n\n

    logit

    \n\n
    Notes
    \n\n

    As a ufunc expit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs

    \n\n

    New in version 0.10.0.

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import expit, logit\n
    \n
    \n\n
    \n
    >>> expit([-np.inf, -1.5, 0, 1.5, np.inf])\narray([ 0.        ,  0.18242552,  0.5       ,  0.81757448,  1.        ])\n
    \n
    \n\n

    logit is the inverse of expit:

    \n\n
    \n
    >>> logit(expit([-2.5, 0, 3.1, 5.0]))\narray([-2.5,  0. ,  3.1,  5. ])\n
    \n
    \n\n

    Plot expit(x) for x in [-6, 6]:

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-6, 6, 121)\n>>> y = expit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.xlim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('expit(x)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logsumexp": {"fullname": "pyerrors.special.logsumexp", "modulename": "pyerrors.special", "qualname": "logsumexp", "kind": "function", "doc": "

    Compute the log of the sum of exponentials of input elements.

    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nInput array.
      • \n

    • axis (None or int or tuple of ints, optional):\nAxis or axes over which the sum is taken. By default axis is None,\nand all elements are summed.

      \n\n

      New in version 0.11.0.

      • \n

    • b (array-like, optional):\nScaling factor for exp(a) must be of the same shape as a or\nbroadcastable to a. These values may be negative in order to\nimplement subtraction.

      \n\n

      New in version 0.12.0.

      • \n

    • keepdims (bool, optional):\nIf this is set to True, the axes which are reduced are left in the\nresult as dimensions with size one. With this option, the result\nwill broadcast correctly against the original array.

      \n\n

      New in version 0.15.0.

      • \n

    • return_sign (bool, optional):\nIf this is set to True, the result will be a pair containing sign\ninformation; if False, results that are negative will be returned\nas NaN. Default is False (no sign information).

      \n\n

      New in version 0.16.0.

      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (ndarray):\nThe result, np.log(np.sum(np.exp(a))) calculated in a numerically\nmore stable way. If b is given then np.log(np.sum(b*np.exp(a)))\nis returned.
      • \n
    • sgn (ndarray):\nIf return_sign is True, this will be an array of floating-point\nnumbers matching res and +1, 0, or -1 depending on the sign\nof the result. If False, only one result is returned.
      • \n
    \n\n
    See Also
    \n\n

    numpy.logaddexp,, numpy.logaddexp2

    \n\n
    Notes
    \n\n

    NumPy has a logaddexp function which is very similar to logsumexp, but\nonly handles two arguments. logaddexp.reduce is similar to this\nfunction, but may be less stable.

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import logsumexp\n>>> a = np.arange(10)\n>>> logsumexp(a)\n9.4586297444267107\n>>> np.log(np.sum(np.exp(a)))\n9.4586297444267107\n
    \n
    \n\n

    With weights

    \n\n
    \n
    >>> a = np.arange(10)\n>>> b = np.arange(10, 0, -1)\n>>> logsumexp(a, b=b)\n9.9170178533034665\n>>> np.log(np.sum(b*np.exp(a)))\n9.9170178533034647\n
    \n
    \n\n

    Returning a sign flag

    \n\n
    \n
    >>> logsumexp([1,2],b=[1,-1],return_sign=True)\n(1.5413248546129181, -1.0)\n
    \n
    \n\n

    Notice that logsumexp does not directly support masked arrays. To use it\non a masked array, convert the mask into zero weights:

    \n\n
    \n
    >>> a = np.ma.array([np.log(2), 2, np.log(3)],\n...                  mask=[False, True, False])\n>>> b = (~a.mask).astype(int)\n>>> logsumexp(a.data, b=b), np.log(5)\n1.6094379124341005, 1.6094379124341005\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.version": {"fullname": "pyerrors.version", "modulename": "pyerrors.version", "kind": "module", "doc": "

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    What is pyerrors?

    \n\n

    pyerrors is a python package for error computation and propagation of Markov chain Monte Carlo data.\nIt is based on the gamma method arXiv:hep-lat/0306017. Some of its features are:

    \n\n
      \n
    • automatic differentiation for exact linear error propagation as suggested in arXiv:1809.01289 (partly based on the autograd package).
      • \n
    • treatment of slow modes in the simulation as suggested in arXiv:1009.5228.
      • \n
    • coherent error propagation for data from different Markov chains.
      • \n
    • non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in arXiv:1809.01289.
      • \n
    • real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).
      • \n
    \n\n

    More detailed examples can found in the GitHub repository \"badge\".

    \n\n

    If you use pyerrors for research that leads to a publication please consider citing:

    \n\n
      \n
    • Fabian Joswig, Simon Kuberski, Justus T. Kuhlmann, Jan Neuendorf, pyerrors: a python framework for error analysis of Monte Carlo data. Comput.Phys.Commun. 288 (2023) 108750.
      • \n
    • Ulli Wolff, Monte Carlo errors with less errors. Comput.Phys.Commun. 156 (2004) 143-153, Comput.Phys.Commun. 176 (2007) 383 (erratum).
      • \n
    • Alberto Ramos, Automatic differentiation for error analysis of Monte Carlo data. Comput.Phys.Commun. 238 (2019) 19-35.
      • \n
    \n\n

    and

    \n\n
      \n
    • Stefan Schaefer, Rainer Sommer, Francesco Virotta, Critical slowing down and error analysis in lattice QCD simulations. Nucl.Phys.B 845 (2011) 93-119.
      • \n
    \n\n

    where applicable.

    \n\n

    There exist similar publicly available implementations of gamma method error analysis suites in Fortran, Julia and Python.

    \n\n

    Installation

    \n\n

    Install the most recent release using pip and pypi:

    \n\n
    \n
    python -m pip install pyerrors     # Fresh install\npython -m pip install -U pyerrors  # Update\n
    \n
    \n\n

    Install the most recent release using conda and conda-forge:

    \n\n
    \n
    conda install -c conda-forge pyerrors  # Fresh install\nconda update -c conda-forge pyerrors   # Update\n
    \n
    \n\n

    Install the current develop version:

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    \n
    python -m pip install -U --no-deps --force-reinstall git+https://github.com/fjosw/pyerrors.git@develop\n
    \n
    \n\n

    (Also works for any feature branch).

    \n\n

    Basic example

    \n\n
    \n
    import numpy as np\nimport pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object\nmy_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object\nmy_new_obs.gamma_method()                     # Estimate the statistical error\nprint(my_new_obs)                             # Print the result to stdout\n> 0.31498(72)\n
    \n
    \n\n

    The Obs class

    \n\n

    pyerrors introduces a new datatype, Obs, which simplifies error propagation and estimation for auto- and cross-correlated data.\nAn Obs object can be initialized with two arguments, the first is a list containing the samples for an observable from a Monte Carlo chain.\nThe samples can either be provided as python list or as numpy array.\nThe second argument is a list containing the names of the respective Monte Carlo chains as strings. These strings uniquely identify a Monte Carlo chain/ensemble. It is crucial for the correct error propagation that observations from the same Monte Carlo history are labeled with the same name. See Multiple ensembles/replica for details.

    \n\n
    \n
    import pyerrors as pe\n\nmy_obs = pe.Obs([samples], ['ensemble_name'])\n
    \n
    \n\n

    Error propagation

    \n\n

    When performing mathematical operations on Obs objects the correct error propagation is intrinsically taken care of using a first order Taylor expansion\n$$\\delta_f^i=\\sum_\\alpha \\bar{f}_\\alpha \\delta_\\alpha^i\\,,\\quad \\delta_\\alpha^i=a_\\alpha^i-\\bar{a}_\\alpha\\,,$$\nas introduced in arXiv:hep-lat/0306017.\nThe required derivatives $\\bar{f}_\\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in arXiv:1809.01289.

    \n\n

    The Obs class is designed such that mathematical numpy functions can be used on Obs just as for regular floats.

    \n\n
    \n
    import numpy as np\nimport pyerrors as pe\n\nmy_obs1 = pe.Obs([samples1], ['ensemble_name'])\nmy_obs2 = pe.Obs([samples2], ['ensemble_name'])\n\nmy_sum = my_obs1 + my_obs2\n\nmy_m_eff = np.log(my_obs1 / my_obs2)\n\niamzero = my_m_eff - my_m_eff\n# Check that value and fluctuations are zero within machine precision\nprint(iamzero == 0.0)\n> True\n
    \n
    \n\n

    Error estimation

    \n\n

    The error estimation within pyerrors is based on the gamma method introduced in arXiv:hep-lat/0306017.\nAfter having arrived at the derived quantity of interest the gamma_method can be called as detailed in the following example.

    \n\n
    \n
    my_sum.gamma_method()\nprint(my_sum)\n> 1.70(57)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 5.72046658e-01 +/- 7.56746598e-02 (33.650%)\n>  t_int         2.71422900e+00 +/- 6.40320983e-01 S = 2.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n

    The gamma_method is not automatically called after every intermediate step in order to prevent computational overhead.

    \n\n

    We use the following definition of the integrated autocorrelation time established in Madras & Sokal 1988\n$$\\tau_\\mathrm{int}=\\frac{1}{2}+\\sum_{t=1}^{W}\\rho(t)\\geq \\frac{1}{2}\\,.$$\nThe window $W$ is determined via the automatic windowing procedure described in arXiv:hep-lat/0306017.\nThe standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the gamma_method as parameter.

    \n\n
    \n
    my_sum.gamma_method(S=3.0)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.30675201e-01 +/- 1.04585650e-01 (37.099%)\n>  t_int         3.29909703e+00 +/- 9.77310102e-01 S = 3.00\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n

    The integrated autocorrelation time $\\tau_\\mathrm{int}$ and the autocorrelation function $\\rho(W)$ can be monitored via the methods pyerrors.obs.Obs.plot_tauint and pyerrors.obs.Obs.plot_rho.

    \n\n

    If the parameter $S$ is set to zero it is assumed that the dataset does not exhibit any autocorrelation and the window size is chosen to be zero.\nIn this case the error estimate is identical to the sample standard error.

    \n\n

    Exponential tails

    \n\n

    Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\\rho$ as suggested in arXiv:1009.5228. The longest autocorrelation time in the history, $\\tau_\\mathrm{exp}$, can be passed to the gamma_method as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.

    \n\n
    \n
    my_sum.gamma_method(tau_exp=7.2)\nmy_sum.details()\n> Result         1.70000000e+00 +/- 6.28097762e-01 +/- 5.79077524e-02 (36.947%)\n>  t_int         3.27218667e+00 +/- 7.99583654e-01 tau_exp = 7.20,  N_sigma = 1\n> 1000 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble_name' : 1000 configurations (from 1 to 1000)\n
    \n
    \n\n

    For the full API see pyerrors.obs.Obs.gamma_method.

    \n\n

    Multiple ensembles/replica

    \n\n

    Error propagation for multiple ensembles (Markov chains with different simulation parameters) is handled automatically. Ensembles are uniquely identified by their name.

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    obs1 = pe.Obs([samples1], ['ensemble1'])\nobs2 = pe.Obs([samples2], ['ensemble2'])\n\nmy_sum = obs1 + obs2\nmy_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 2 ensembles:\n>   \u00b7 Ensemble 'ensemble1' : 1000 configurations (from 1 to 1000)\n>   \u00b7 Ensemble 'ensemble2' : 500 configurations (from 1 to 500)\n
    \n
    \n\n

    Observables from the same Monte Carlo chain have to be initialized with the same name for correct error propagation. If different names were used in this case the data would be treated as statistically independent resulting in loss of relevant information and a potential over or under estimate of the statistical error.

    \n\n

    pyerrors identifies multiple replica (independent Markov chains with identical simulation parameters) by the vertical bar | in the name of the data set.

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    \n
    obs1 = pe.Obs([samples1], ['ensemble1|r01'])\nobs2 = pe.Obs([samples2], ['ensemble1|r02'])\n\n> my_sum = obs1 + obs2\n> my_sum.details()\n> Result   2.00697958e+00\n> 1500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1'\n>     \u00b7 Replicum 'r01' : 1000 configurations (from 1 to 1000)\n>     \u00b7 Replicum 'r02' : 500 configurations (from 1 to 500)\n
    \n
    \n\n

    Error estimation for multiple ensembles

    \n\n

    In order to keep track of different error analysis parameters for different ensembles one can make use of global dictionaries as detailed in the following example.

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    \n
    pe.Obs.S_dict['ensemble1'] = 2.5\npe.Obs.tau_exp_dict['ensemble2'] = 8.0\npe.Obs.tau_exp_dict['ensemble3'] = 2.0\n
    \n
    \n\n

    In case the gamma_method is called without any parameters it will use the values specified in the dictionaries for the respective ensembles.\nPassing arguments to the gamma_method still dominates over the dictionaries.

    \n\n

    Irregular Monte Carlo chains

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    Obs objects defined on irregular Monte Carlo chains can be initialized with the parameter idl.

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    \n
    # Observable defined on configurations 20 to 519\nobs1 = pe.Obs([samples1], ['ensemble1'], idl=[range(20, 520)])\nobs1.details()\n> Result         9.98319881e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 20 to 519)\n\n# Observable defined on every second configuration between 5 and 1003\nobs2 = pe.Obs([samples2], ['ensemble1'], idl=[range(5, 1005, 2)])\nobs2.details()\n> Result         9.99100712e-01\n> 500 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 500 configurations (from 5 to 1003 in steps of 2)\n\n# Observable defined on configurations 2, 9, 28, 29 and 501\nobs3 = pe.Obs([samples3], ['ensemble1'], idl=[[2, 9, 28, 29, 501]])\nobs3.details()\n> Result         1.01718064e+00\n> 5 samples in 1 ensemble:\n>   \u00b7 Ensemble 'ensemble1' : 5 configurations (irregular range)\n
    \n
    \n\n

    Obs objects defined on regular and irregular histories of the same ensemble can be combined with each other and the correct error propagation and estimation is automatically taken care of.

    \n\n

    Warning: Irregular Monte Carlo chains can result in odd patterns in the autocorrelation functions.\nMake sure to check the autocorrelation time with e.g. pyerrors.obs.Obs.plot_rho or pyerrors.obs.Obs.plot_tauint.

    \n\n

    For the full API see pyerrors.obs.Obs.

    \n\n

    Correlators

    \n\n

    When one is not interested in single observables but correlation functions, pyerrors offers the Corr class which simplifies the corresponding error propagation and provides the user with a set of standard methods. In order to initialize a Corr objects one needs to arrange the data as a list of Obs

    \n\n
    \n
    my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0      0.7957(80)\n> 1      0.5156(51)\n> 2      0.3227(33)\n> 3      0.2041(21)\n
    \n
    \n\n

    In case the correlation functions are not defined on the outermost timeslices, for example because of fixed boundary conditions, a padding can be introduced.

    \n\n
    \n
    my_corr = pe.Corr([obs_0, obs_1, obs_2, obs_3], padding=[1, 1])\nprint(my_corr)\n> x0/a  Corr(x0/a)\n> ------------------\n> 0\n> 1      0.7957(80)\n> 2      0.5156(51)\n> 3      0.3227(33)\n> 4      0.2041(21)\n> 5\n
    \n
    \n\n

    The individual entries of a correlator can be accessed via slicing

    \n\n
    \n
    print(my_corr[3])\n> 0.3227(33)\n
    \n
    \n\n

    Error propagation with the Corr class works very similar to Obs objects. Mathematical operations are overloaded and Corr objects can be computed together with other Corr objects, Obs objects or real numbers and integers.

    \n\n
    \n
    my_new_corr = 0.3 * my_corr[2] * my_corr * my_corr + 12 / my_corr\n
    \n
    \n\n

    pyerrors provides the user with a set of regularly used methods for the manipulation of correlator objects:

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      \n
    • Corr.gamma_method applies the gamma method to all entries of the correlator.
      • \n
    • Corr.m_eff to construct effective masses. Various variants for periodic and fixed temporal boundary conditions are available.
      • \n
    • Corr.deriv returns the first derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
      • \n
    • Corr.second_deriv returns the second derivative of the correlator as Corr. Different discretizations of the numerical derivative are available.
      • \n
    • Corr.symmetric symmetrizes parity even correlations functions, assuming periodic boundary conditions.
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    • Corr.anti_symmetric anti-symmetrizes parity odd correlations functions, assuming periodic boundary conditions.
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    • Corr.T_symmetry averages a correlator with its time symmetry partner, assuming fixed boundary conditions.
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    • Corr.plateau extracts a plateau value from the correlator in a given range.
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    • Corr.roll periodically shifts the correlator.
      • \n
    • Corr.reverse reverses the time ordering of the correlator.
      • \n
    • Corr.correlate constructs a disconnected correlation function from the correlator and another Corr or Obs object.
      • \n
    • Corr.reweight reweights the correlator.
      • \n
    \n\n

    pyerrors can also handle matrices of correlation functions and extract energy states from these matrices via a generalized eigenvalue problem (see pyerrors.correlators.Corr.GEVP).

    \n\n

    For the full API see pyerrors.correlators.Corr.

    \n\n

    Complex valued observables

    \n\n

    pyerrors can handle complex valued observables via the class pyerrors.obs.CObs.\nCObs are initialized with a real and an imaginary part which both can be Obs valued.

    \n\n
    \n
    my_real_part = pe.Obs([samples1], ['ensemble1'])\nmy_imag_part = pe.Obs([samples2], ['ensemble1'])\n\nmy_cobs = pe.CObs(my_real_part, my_imag_part)\nmy_cobs.gamma_method()\nprint(my_cobs)\n> (0.9959(91)+0.659(28)j)\n
    \n
    \n\n

    Elementary mathematical operations are overloaded and samples are properly propagated as for the Obs class.

    \n\n
    \n
    my_derived_cobs = (my_cobs + my_cobs.conjugate()) / np.abs(my_cobs)\nmy_derived_cobs.gamma_method()\nprint(my_derived_cobs)\n> (1.668(23)+0.0j)\n
    \n
    \n\n

    The Covobs class

    \n\n

    In many projects, auxiliary data that is not based on Monte Carlo chains enters. Examples are experimentally determined mesons masses which are used to set the scale or renormalization constants. These numbers come with an error that has to be propagated through the analysis. The Covobs class allows to define such quantities in pyerrors. Furthermore, external input might consist of correlated quantities. An example are the parameters of an interpolation formula, which are defined via mean values and a covariance matrix between all parameters. The contribution of the interpolation formula to the error of a derived quantity therefore might depend on the complete covariance matrix.

    \n\n

    This concept is built into the definition of Covobs. In pyerrors, external input is defined by $M$ mean values, a $M\\times M$ covariance matrix, where $M=1$ is permissible, and a name that uniquely identifies the covariance matrix. Below, we define the pion mass, based on its mean value and error, 134.9768(5). Note, that the square of the error enters cov_Obs, since the second argument of this function is the covariance matrix of the Covobs.

    \n\n
    \n
    import pyerrors.obs as pe\n\nmpi = pe.cov_Obs(134.9768, 0.0005**2, 'pi^0 mass')\nmpi.gamma_method()\nmpi.details()\n> Result         1.34976800e+02 +/- 5.00000000e-04 +/- 0.00000000e+00 (0.000%)\n>  pi^0 mass     5.00000000e-04\n> 0 samples in 1 ensemble:\n>   \u00b7 Covobs   'pi^0 mass'\n
    \n
    \n\n

    The resulting object mpi is an Obs that contains a Covobs. In the following, it may be handled as any other Obs. The contribution of the covariance matrix to the error of an Obs is determined from the $M \\times M$ covariance matrix $\\Sigma$ and the gradient of the Obs with respect to the external quantities, which is the $1\\times M$ Jacobian matrix $J$, via\n$$s = \\sqrt{J^T \\Sigma J}\\,,$$\nwhere the Jacobian is computed for each derived quantity via automatic differentiation.

    \n\n

    Correlated auxiliary data is defined similarly to above, e.g., via

    \n\n
    \n
    RAP = pe.cov_Obs([16.7457, -19.0475], [[3.49591, -6.07560], [-6.07560, 10.5834]], 'R_AP, 1906.03445, (5.3a)')\nprint(RAP)\n> [Obs[16.7(1.9)], Obs[-19.0(3.3)]]\n
    \n
    \n\n

    where RAP now is a list of two Obs that contains the two correlated parameters.

    \n\n

    Since the gradient of a derived observable with respect to an external covariance matrix is propagated through the entire analysis, the Covobs class allows to quote the derivative of a result with respect to the external quantities. If these derivatives are published together with the result, small shifts in the definition of external quantities, e.g., the definition of the physical point, can be performed a posteriori based on the published information. This may help to compare results of different groups. The gradient of an Obs o with respect to a covariance matrix with the identifying string k may be accessed via

    \n\n
    \n
    o.covobs[k].grad\n
    \n
    \n\n

    Error propagation in iterative algorithms

    \n\n

    pyerrors supports exact linear error propagation for iterative algorithms like various variants of non-linear least squares fits or root finding. The derivatives required for the error propagation are calculated as described in arXiv:1809.01289.

    \n\n

    Least squares fits

    \n\n

    Standard non-linear least square fits with errors on the dependent but not the independent variables can be performed with pyerrors.fits.least_squares. As default solver the Levenberg-Marquardt algorithm implemented in scipy is used.

    \n\n

    Fit functions have to be of the following form

    \n\n
    \n
    import autograd.numpy as anp\n\ndef func(a, x):\n    return a[1] * anp.exp(-a[0] * x)\n
    \n
    \n\n

    It is important that numerical functions refer to autograd.numpy instead of numpy for the automatic differentiation in iterative algorithms to work properly.

    \n\n

    Fits can then be performed via

    \n\n
    \n
    fit_result = pe.fits.least_squares(x, y, func)\nprint("\\n", fit_result)\n> Fit with 2 parameters\n> Method: Levenberg-Marquardt\n> `ftol` termination condition is satisfied.\n> chisquare/d.o.f.: 0.9593035785160936\n\n>  Goodness of fit:\n> \u03c7\u00b2/d.o.f. = 0.959304\n> p-value   = 0.5673\n> Fit parameters:\n> 0      0.0548(28)\n> 1      1.933(64)\n
    \n
    \n\n

    where x is a list or numpy.array of floats and y is a list or numpy.array of Obs.

    \n\n

    Data stored in Corr objects can be fitted directly using the Corr.fit method.

    \n\n
    \n
    my_corr = pe.Corr(y)\nfit_result = my_corr.fit(func, fitrange=[12, 25])\n
    \n
    \n\n

    this can simplify working with absolute fit ranges and takes care of gaps in the data automatically.

    \n\n

    For fit functions with multiple independent variables the fit function can be of the form

    \n\n
    \n
    def func(a, x):\n    (x1, x2) = x\n    return a[0] * x1 ** 2 + a[1] * x2\n
    \n
    \n\n

    pyerrors also supports correlated fits which can be triggered via the parameter correlated_fit=True.\nDetails about how the required covariance matrix is estimated can be found in pyerrors.obs.covariance.\nDirect visualizations of the performed fits can be triggered via resplot=True or qqplot=True.

    \n\n

    For all available options including combined fits to multiple datasets see pyerrors.fits.least_squares.

    \n\n

    Total least squares fits

    \n\n

    pyerrors can also fit data with errors on both the dependent and independent variables using the total least squares method also referred to as orthogonal distance regression as implemented in scipy, see pyerrors.fits.least_squares. The syntax is identical to the standard least squares case, the only difference being that x also has to be a list or numpy.array of Obs.

    \n\n

    For the full API see pyerrors.fits for fits and pyerrors.roots for finding roots of functions.

    \n\n

    Matrix operations

    \n\n

    pyerrors provides wrappers for Obs- and CObs-valued matrix operations based on numpy.linalg. The supported functions include:

    \n\n
      \n
    • inv for the matrix inverse.
      • \n
    • cholseky for the Cholesky decomposition.
      • \n
    • det for the matrix determinant.
      • \n
    • eigh for eigenvalues and eigenvectors of hermitean matrices.
      • \n
    • eig for eigenvalues of general matrices.
      • \n
    • pinv for the Moore-Penrose pseudoinverse.
      • \n
    • svd for the singular-value-decomposition.
      • \n
    \n\n

    For the full API see pyerrors.linalg.

    \n\n

    Export data

    \n\n

    \n\n

    The preferred exported file format within pyerrors is json.gz. Files written to this format are valid JSON files that have been compressed using gzip. The structure of the content is inspired by the dobs format of the ALPHA collaboration. The aim of the format is to facilitate the storage of data in a self-contained way such that, even years after the creation of the file, it is possible to extract all necessary information:

    \n\n
      \n
    • What observables are stored? Possibly: How exactly are they defined.
      • \n
    • How does each single ensemble or external quantity contribute to the error of the observable?
      • \n
    • Who did write the file when and on which machine?
      • \n
    \n\n

    This can be achieved by storing all information in one single file. The export routines of pyerrors are written such that as much information as possible is written automatically as described in the following example

    \n\n
    \n
    my_obs = pe.Obs([samples], ["test_ensemble"])\nmy_obs.tag = "My observable"\n\npe.input.json.dump_to_json(my_obs, "test_output_file", description="This file contains a test observable")\n# For a single observable one can equivalently use the class method dump\nmy_obs.dump("test_output_file", description="This file contains a test observable")\n\ncheck = pe.input.json.load_json("test_output_file")\n\nprint(my_obs == check)\n> True\n
    \n
    \n\n

    The format also allows to directly write out the content of Corr objects or lists and arrays of Obs objects by passing the desired data to pyerrors.input.json.dump_to_json.

    \n\n

    json.gz format specification

    \n\n

    The first entries of the file provide optional auxiliary information:

    \n\n
      \n
    • program is a string that indicates which program was used to write the file.
      • \n
    • version is a string that specifies the version of the format.
      • \n
    • who is a string that specifies the user name of the creator of the file.
      • \n
    • date is a string and contains the creation date of the file.
      • \n
    • host is a string and contains the hostname of the machine where the file has been written.
      • \n
    • description contains information on the content of the file. This field is not filled automatically in pyerrors. The user is advised to provide as detailed information as possible in this field. Examples are: Input files of measurements or simulations, LaTeX formulae or references to publications to specify how the observables have been computed, details on the analysis strategy, ... This field may be any valid JSON type. Strings, arrays or objects (equivalent to dicts in python) are well suited to provide information.
      • \n
    \n\n

    The only necessary entry of the file is the field\n-obsdata, an array that contains the actual data.

    \n\n

    Each entry of the array belongs to a single structure of observables. Currently, these structures can be either of Obs, list, numpy.ndarray, Corr. All Obs inside a structure (with dimension > 0) have to be defined on the same set of configurations. Different structures, that are represented by entries of the array obsdata, are treated independently. Each entry of the array obsdata has the following required entries:

    \n\n
      \n
    • type is a string that specifies the type of the structure. This allows to parse the content to the correct form after reading the file. It is always possible to interpret the content as list of Obs.
      • \n
    • value is an array that contains the mean values of the Obs inside the structure.\nThe following entries are optional:
      • \n
    • layout is a string that specifies the layout of multi-dimensional structures. Examples are \"2, 2\" for a 2x2 dimensional matrix or \"64, 4, 4\" for a Corr with $T=64$ and 4x4 matrices on each time slices. \"1\" denotes a single Obs. Multi-dimensional structures are stored in row-major format (see below).
      • \n
    • tag is any JSON type. It contains additional information concerning the structure. The tag of an Obs in pyerrors is written here.
      • \n
    • reweighted is a Bool that may be used to specify, whether the Obs in the structure have been reweighted.
      • \n
    • data is an array that contains the data from MC chains. We will define it below.
      • \n
    • cdata is an array that contains the data from external quantities with an error (Covobs in pyerrors). We will define it below.
      • \n
    \n\n

    The array data contains the data from MC chains. Each entry of the array corresponds to one ensemble and contains:

    \n\n
      \n
    • id, a string that contains the name of the ensemble
      • \n
    • replica, an array that contains an entry per replica of the ensemble.
      • \n
    \n\n

    Each entry of replica contains\nname, a string that contains the name of the replica\ndeltas, an array that contains the actual data.

    \n\n

    Each entry in deltas corresponds to one configuration of the replica and has $1+N$ many entries. The first entry is an integer that specifies the configuration number that, together with ensemble and replica name, may be used to uniquely identify the configuration on which the data has been obtained. The following N entries specify the deltas, i.e., the deviation of the observable from the mean value on this configuration, of each Obs inside the structure. Multi-dimensional structures are stored in a row-major format. For primary observables, such as correlation functions, $value + delta_i$ matches the primary data obtained on the configuration.

    \n\n

    The array cdata contains information about the contribution of auxiliary observables, represented by Covobs in pyerrors, to the total error of the observables. Each entry of the array belongs to one auxiliary covariance matrix and contains:

    \n\n
      \n
    • id, a string that identifies the covariance matrix
      • \n
    • layout, a string that defines the dimensions of the $M\\times M$ covariance matrix (has to be \"M, M\" or \"1\").
      • \n
    • cov, an array that contains the $M\\times M$ many entries of the covariance matrix, stored in row-major format.
      • \n
    • grad, an array that contains N entries, one for each Obs inside the structure. Each entry itself is an array, that contains the M gradients of the Nth observable with respect to the quantity that corresponds to the Mth diagonal entry of the covariance matrix.
      • \n
    \n\n

    A JSON schema that may be used to verify the correctness of a file with respect to the format definition is stored in ./examples/json_schema.json. The schema is a self-descriptive format definition and contains an exemplary file.

    \n\n

    Julia I/O routines for the json.gz format, compatible with ADerrors.jl, can be found here.

    \n"}, "pyerrors.correlators": {"fullname": "pyerrors.correlators", "modulename": "pyerrors.correlators", "kind": "module", "doc": "

    \n"}, "pyerrors.correlators.Corr": {"fullname": "pyerrors.correlators.Corr", "modulename": "pyerrors.correlators", "qualname": "Corr", "kind": "class", "doc": "

    The class for a correlator (time dependent sequence of pe.Obs).

    \n\n

    Everything, this class does, can be achieved using lists or arrays of Obs.\nBut it is simply more convenient to have a dedicated object for correlators.\nOne often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient\nto iterate over all timeslices for every operation. This is especially true, when dealing with matrices.

    \n\n

    The correlator can have two types of content: An Obs at every timeslice OR a matrix at every timeslice.\nOther dependency (eg. spatial) are not supported.

    \n\n

    The Corr class can also deal with missing measurements or paddings for fixed boundary conditions.\nThe missing entries are represented via the None object.

    \n\n
    Initialization
    \n\n

    A simple correlator can be initialized with a list or a one-dimensional array of Obs or Cobs

    \n\n
    \n
    corr11 = pe.Corr([obs1, obs2])\ncorr11 = pe.Corr(np.array([obs1, obs2]))\n
    \n
    \n\n

    A matrix-valued correlator can either be initialized via a two-dimensional array of Corr objects

    \n\n
    \n
    matrix_corr = pe.Corr(np.array([[corr11, corr12], [corr21, corr22]]))\n
    \n
    \n\n

    or alternatively via a three-dimensional array of Obs or CObs of shape (T, N, N) where T is\nthe temporal extent of the correlator and N is the dimension of the matrix.

    \n"}, "pyerrors.correlators.Corr.__init__": {"fullname": "pyerrors.correlators.Corr.__init__", "modulename": "pyerrors.correlators", "qualname": "Corr.__init__", "kind": "function", "doc": "

    Initialize a Corr object.

    \n\n
    Parameters
    \n\n
      \n
    • data_input (list or array):\nlist of Obs or list of arrays of Obs or array of Corrs (see class docstring for details).
      • \n
    • padding (list, optional):\nList with two entries where the first labels the padding\nat the front of the correlator and the second the padding\nat the back.
      • \n
    • prange (list, optional):\nList containing the first and last timeslice of the plateau\nregion identified for this correlator.
      • \n
    \n", "signature": "(data_input, padding=[0, 0], prange=None)"}, "pyerrors.correlators.Corr.tag": {"fullname": "pyerrors.correlators.Corr.tag", "modulename": "pyerrors.correlators", "qualname": "Corr.tag", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.content": {"fullname": "pyerrors.correlators.Corr.content", "modulename": "pyerrors.correlators", "qualname": "Corr.content", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.T": {"fullname": "pyerrors.correlators.Corr.T", "modulename": "pyerrors.correlators", "qualname": "Corr.T", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.prange": {"fullname": "pyerrors.correlators.Corr.prange", "modulename": "pyerrors.correlators", "qualname": "Corr.prange", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.reweighted": {"fullname": "pyerrors.correlators.Corr.reweighted", "modulename": "pyerrors.correlators", "qualname": "Corr.reweighted", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.gamma_method": {"fullname": "pyerrors.correlators.Corr.gamma_method", "modulename": "pyerrors.correlators", "qualname": "Corr.gamma_method", "kind": "function", "doc": "

    Apply the gamma method to the content of the Corr.

    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.gm": {"fullname": "pyerrors.correlators.Corr.gm", "modulename": "pyerrors.correlators", "qualname": "Corr.gm", "kind": "function", "doc": "

    Apply the gamma method to the content of the Corr.

    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.projected": {"fullname": "pyerrors.correlators.Corr.projected", "modulename": "pyerrors.correlators", "qualname": "Corr.projected", "kind": "function", "doc": "

    We need to project the Correlator with a Vector to get a single value at each timeslice.

    \n\n

    The method can use one or two vectors.\nIf two are specified it returns v1@G@v2 (the order might be very important.)\nBy default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to

    \n", "signature": "(self, vector_l=None, vector_r=None, normalize=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.item": {"fullname": "pyerrors.correlators.Corr.item", "modulename": "pyerrors.correlators", "qualname": "Corr.item", "kind": "function", "doc": "

    Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.

    \n\n
    Parameters
    \n\n
      \n
    • i (int):\nFirst index to be picked.
      • \n
    • j (int):\nSecond index to be picked.
      • \n
    \n", "signature": "(self, i, j):", "funcdef": "def"}, "pyerrors.correlators.Corr.plottable": {"fullname": "pyerrors.correlators.Corr.plottable", "modulename": "pyerrors.correlators", "qualname": "Corr.plottable", "kind": "function", "doc": "

    Outputs the correlator in a plotable format.

    \n\n

    Outputs three lists containing the timeslice index, the value on each\ntimeslice and the error on each timeslice.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.symmetric": {"fullname": "pyerrors.correlators.Corr.symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.symmetric", "kind": "function", "doc": "

    Symmetrize the correlator around x0=0.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.anti_symmetric": {"fullname": "pyerrors.correlators.Corr.anti_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.anti_symmetric", "kind": "function", "doc": "

    Anti-symmetrize the correlator around x0=0.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.is_matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.is_matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.is_matrix_symmetric", "kind": "function", "doc": "

    Checks whether a correlator matrices is symmetric on every timeslice.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.trace": {"fullname": "pyerrors.correlators.Corr.trace", "modulename": "pyerrors.correlators", "qualname": "Corr.trace", "kind": "function", "doc": "

    Calculates the per-timeslice trace of a correlator matrix.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.matrix_symmetric": {"fullname": "pyerrors.correlators.Corr.matrix_symmetric", "modulename": "pyerrors.correlators", "qualname": "Corr.matrix_symmetric", "kind": "function", "doc": "

    Symmetrizes the correlator matrices on every timeslice.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.GEVP": {"fullname": "pyerrors.correlators.Corr.GEVP", "modulename": "pyerrors.correlators", "qualname": "Corr.GEVP", "kind": "function", "doc": "

    Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.

    \n\n

    The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the\nlargest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing

    \n\n
    \n
    C.GEVP(t0=2)[0]  # Ground state vector(s)\nC.GEVP(t0=2)[:3]  # Vectors for the lowest three states\n
    \n
    \n\n
    Parameters
    \n\n
      \n
    • t0 (int):\nThe time t0 for the right hand side of the GEVP according to $G(t)v_i=\\lambda_i G(t_0)v_i$
      • \n
    • ts (int):\nfixed time $G(t_s)v_i=\\lambda_i G(t_0)v_i$ if sort=None.\nIf sort=\"Eigenvector\" it gives a reference point for the sorting method.
      • \n
    • sort (string):\nIf this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.\n
        \n
      • \"Eigenvalue\": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice. (default)
        • \n
      • \"Eigenvector\": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.\nThe reference state is identified by its eigenvalue at $t=t_s$.
        • \n
      • None: The GEVP is solved only at ts, no sorting is necessary
        • \n
      • \n
    • vector_obs (bool):\nIf True, uncertainties are propagated in the eigenvector computation (default False).
      • \n
    \n\n
    Other Parameters
    \n\n
      \n
    • state (int):\nReturns only the vector(s) for a specified state. The lowest state is zero.
      • \n
    • method (str):\nMethod used to solve the GEVP.\n
        \n
      • \"eigh\": Use scipy.linalg.eigh to solve the GEVP. (default for vector_obs=False)
        • \n
      • \"cholesky\": Use manually implemented solution via the Cholesky decomposition. Automatically chosen if vector_obs==True.
        • \n
      • \n
    \n", "signature": "(self, t0, ts=None, sort='Eigenvalue', vector_obs=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Eigenvalue": {"fullname": "pyerrors.correlators.Corr.Eigenvalue", "modulename": "pyerrors.correlators", "qualname": "Corr.Eigenvalue", "kind": "function", "doc": "

    Determines the eigenvalue of the GEVP by solving and projecting the correlator

    \n\n
    Parameters
    \n\n
      \n
    • state (int):\nThe state one is interested in ordered by energy. The lowest state is zero.
      • \n
    • All other parameters are identical to the ones of Corr.GEVP.
      • \n
    \n", "signature": "(self, t0, ts=None, state=0, sort='Eigenvalue', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.Hankel": {"fullname": "pyerrors.correlators.Corr.Hankel", "modulename": "pyerrors.correlators", "qualname": "Corr.Hankel", "kind": "function", "doc": "

    Constructs an NxN Hankel matrix

    \n\n

    C(t) c(t+1) ... c(t+n-1)\nC(t+1) c(t+2) ... c(t+n)\n.................\nC(t+(n-1)) c(t+n) ... c(t+2(n-1))

    \n\n
    Parameters
    \n\n
      \n
    • N (int):\nDimension of the Hankel matrix
      • \n
    • periodic (bool, optional):\ndetermines whether the matrix is extended periodically
      • \n
    \n", "signature": "(self, N, periodic=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.roll": {"fullname": "pyerrors.correlators.Corr.roll", "modulename": "pyerrors.correlators", "qualname": "Corr.roll", "kind": "function", "doc": "

    Periodically shift the correlator by dt timeslices

    \n\n
    Parameters
    \n\n
      \n
    • dt (int):\nnumber of timeslices
      • \n
    \n", "signature": "(self, dt):", "funcdef": "def"}, "pyerrors.correlators.Corr.reverse": {"fullname": "pyerrors.correlators.Corr.reverse", "modulename": "pyerrors.correlators", "qualname": "Corr.reverse", "kind": "function", "doc": "

    Reverse the time ordering of the Corr

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.thin": {"fullname": "pyerrors.correlators.Corr.thin", "modulename": "pyerrors.correlators", "qualname": "Corr.thin", "kind": "function", "doc": "

    Thin out a correlator to suppress correlations

    \n\n
    Parameters
    \n\n
      \n
    • spacing (int):\nKeep only every 'spacing'th entry of the correlator
      • \n
    • offset (int):\nOffset the equal spacing
      • \n
    \n", "signature": "(self, spacing=2, offset=0):", "funcdef": "def"}, "pyerrors.correlators.Corr.correlate": {"fullname": "pyerrors.correlators.Corr.correlate", "modulename": "pyerrors.correlators", "qualname": "Corr.correlate", "kind": "function", "doc": "

    Correlate the correlator with another correlator or Obs

    \n\n
    Parameters
    \n\n
      \n
    • partner (Obs or Corr):\npartner to correlate the correlator with.\nCan either be an Obs which is correlated with all entries of the\ncorrelator or a Corr of same length.
      • \n
    \n", "signature": "(self, partner):", "funcdef": "def"}, "pyerrors.correlators.Corr.reweight": {"fullname": "pyerrors.correlators.Corr.reweight", "modulename": "pyerrors.correlators", "qualname": "Corr.reweight", "kind": "function", "doc": "

    Reweight the correlator.

    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl.
      • \n
    \n", "signature": "(self, weight, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.T_symmetry": {"fullname": "pyerrors.correlators.Corr.T_symmetry", "modulename": "pyerrors.correlators", "qualname": "Corr.T_symmetry", "kind": "function", "doc": "

    Return the time symmetry average of the correlator and its partner

    \n\n
    Parameters
    \n\n
      \n
    • partner (Corr):\nTime symmetry partner of the Corr
      • \n
    • parity (int):\nParity quantum number of the correlator, can be +1 or -1
      • \n
    \n", "signature": "(self, partner, parity=1):", "funcdef": "def"}, "pyerrors.correlators.Corr.deriv": {"fullname": "pyerrors.correlators.Corr.deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.deriv", "kind": "function", "doc": "

    Return the first derivative of the correlator with respect to x0.

    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice: symmetric, forward, backward, improved, log, default: symmetric
      • \n
    \n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.second_deriv": {"fullname": "pyerrors.correlators.Corr.second_deriv", "modulename": "pyerrors.correlators", "qualname": "Corr.second_deriv", "kind": "function", "doc": "

    Return the second derivative of the correlator with respect to x0.

    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\ndecides which definition of the finite differences derivative is used.\nAvailable choice:\n - symmetric (default)\n $$\\tilde{\\partial}^2_0 f(x_0) = f(x_0+1)-2f(x_0)+f(x_0-1)$$\n - big_symmetric\n $$\\partial^2_0 f(x_0) = \\frac{f(x_0+2)-2f(x_0)+f(x_0-2)}{4}$$\n - improved\n $$\\partial^2_0 f(x_0) = \\frac{-f(x_0+2) + 16 * f(x_0+1) - 30 * f(x_0) + 16 * f(x_0-1) - f(x_0-2)}{12}$$\n - log\n $$f(x) = \\tilde{\\partial}^2_0 log(f(x_0))+(\\tilde{\\partial}_0 log(f(x_0)))^2$$
      • \n
    \n", "signature": "(self, variant='symmetric'):", "funcdef": "def"}, "pyerrors.correlators.Corr.m_eff": {"fullname": "pyerrors.correlators.Corr.m_eff", "modulename": "pyerrors.correlators", "qualname": "Corr.m_eff", "kind": "function", "doc": "

    Returns the effective mass of the correlator as correlator object

    \n\n
    Parameters
    \n\n
      \n
    • variant (str):\nlog : uses the standard effective mass log(C(t) / C(t+1))\ncosh, periodic : Use periodicity of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.\nsinh : Use anti-periodicity of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.\nSee, e.g., arXiv:1205.5380\narccosh : Uses the explicit form of the symmetrized correlator (not recommended)\nlogsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
      • \n
    • guess (float):\nguess for the root finder, only relevant for the root variant
      • \n
    \n", "signature": "(self, variant='log', guess=1.0):", "funcdef": "def"}, "pyerrors.correlators.Corr.fit": {"fullname": "pyerrors.correlators.Corr.fit", "modulename": "pyerrors.correlators", "qualname": "Corr.fit", "kind": "function", "doc": "

    Fits function to the data

    \n\n
    Parameters
    \n\n
      \n
    • function (obj):\nfunction to fit to the data. See fits.least_squares for details.
      • \n
    • fitrange (list):\nTwo element list containing the timeslices on which the fit is supposed to start and stop.\nCaution: This range is inclusive as opposed to standard python indexing.\nfitrange=[4, 6] corresponds to the three entries 4, 5 and 6.\nIf not specified, self.prange or all timeslices are used.
      • \n
    • silent (bool):\nDecides whether output is printed to the standard output.
      • \n
    \n", "signature": "(self, function, fitrange=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.plateau": {"fullname": "pyerrors.correlators.Corr.plateau", "modulename": "pyerrors.correlators", "qualname": "Corr.plateau", "kind": "function", "doc": "

    Extract a plateau value from a Corr object

    \n\n
    Parameters
    \n\n
      \n
    • plateau_range (list):\nlist with two entries, indicating the first and the last timeslice\nof the plateau region.
      • \n
    • method (str):\nmethod to extract the plateau.\n 'fit' fits a constant to the plateau region\n 'avg', 'average' or 'mean' just average over the given timeslices.
      • \n
    • auto_gamma (bool):\napply gamma_method with default parameters to the Corr. Defaults to None
      • \n
    \n", "signature": "(self, plateau_range=None, method='fit', auto_gamma=False):", "funcdef": "def"}, "pyerrors.correlators.Corr.set_prange": {"fullname": "pyerrors.correlators.Corr.set_prange", "modulename": "pyerrors.correlators", "qualname": "Corr.set_prange", "kind": "function", "doc": "

    Sets the attribute prange of the Corr object.

    \n", "signature": "(self, prange):", "funcdef": "def"}, "pyerrors.correlators.Corr.show": {"fullname": "pyerrors.correlators.Corr.show", "modulename": "pyerrors.correlators", "qualname": "Corr.show", "kind": "function", "doc": "

    Plots the correlator using the tag of the correlator as label if available.

    \n\n
    Parameters
    \n\n
      \n
    • x_range (list):\nlist of two values, determining the range of the x-axis e.g. [4, 8].
      • \n
    • comp (Corr or list of Corr):\nCorrelator or list of correlators which are plotted for comparison.\nThe tags of these correlators are used as labels if available.
      • \n
    • logscale (bool):\nSets y-axis to logscale.
      • \n
    • plateau (Obs):\nPlateau value to be visualized in the figure.
      • \n
    • fit_res (Fit_result):\nFit_result object to be visualized.
      • \n
    • fit_key (str):\nKey for the fit function in Fit_result.fit_function (for combined fits).
      • \n
    • ylabel (str):\nLabel for the y-axis.
      • \n
    • save (str):\npath to file in which the figure should be saved.
      • \n
    • auto_gamma (bool):\nApply the gamma method with standard parameters to all correlators and plateau values before plotting.
      • \n
    • hide_sigma (float):\nHides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
      • \n
    • references (list):\nList of floating point values that are displayed as horizontal lines for reference.
      • \n
    • title (string):\nOptional title of the figure.
      • \n
    \n", "signature": "(\tself,\tx_range=None,\tcomp=None,\ty_range=None,\tlogscale=False,\tplateau=None,\tfit_res=None,\tfit_key=None,\tylabel=None,\tsave=None,\tauto_gamma=False,\thide_sigma=None,\treferences=None,\ttitle=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.spaghetti_plot": {"fullname": "pyerrors.correlators.Corr.spaghetti_plot", "modulename": "pyerrors.correlators", "qualname": "Corr.spaghetti_plot", "kind": "function", "doc": "

    Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.

    \n\n
    Parameters
    \n\n
      \n
    • logscale (bool):\nDetermines whether the scale of the y-axis is logarithmic or standard.
      • \n
    \n", "signature": "(self, logscale=True):", "funcdef": "def"}, "pyerrors.correlators.Corr.dump": {"fullname": "pyerrors.correlators.Corr.dump", "modulename": "pyerrors.correlators", "qualname": "Corr.dump", "kind": "function", "doc": "

    Dumps the Corr into a file of chosen type

    \n\n
    Parameters
    \n\n
      \n
    • filename (str):\nName of the file to be saved.
      • \n
    • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n", "signature": "(self, filename, datatype='json.gz', **kwargs):", "funcdef": "def"}, "pyerrors.correlators.Corr.print": {"fullname": "pyerrors.correlators.Corr.print", "modulename": "pyerrors.correlators", "qualname": "Corr.print", "kind": "function", "doc": "

    \n", "signature": "(self, print_range=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.sqrt": {"fullname": "pyerrors.correlators.Corr.sqrt", "modulename": "pyerrors.correlators", "qualname": "Corr.sqrt", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.log": {"fullname": "pyerrors.correlators.Corr.log", "modulename": "pyerrors.correlators", "qualname": "Corr.log", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.exp": {"fullname": "pyerrors.correlators.Corr.exp", "modulename": "pyerrors.correlators", "qualname": "Corr.exp", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sin": {"fullname": "pyerrors.correlators.Corr.sin", "modulename": "pyerrors.correlators", "qualname": "Corr.sin", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cos": {"fullname": "pyerrors.correlators.Corr.cos", "modulename": "pyerrors.correlators", "qualname": "Corr.cos", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tan": {"fullname": "pyerrors.correlators.Corr.tan", "modulename": "pyerrors.correlators", "qualname": "Corr.tan", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.sinh": {"fullname": "pyerrors.correlators.Corr.sinh", "modulename": "pyerrors.correlators", "qualname": "Corr.sinh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.cosh": {"fullname": "pyerrors.correlators.Corr.cosh", "modulename": "pyerrors.correlators", "qualname": "Corr.cosh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.tanh": {"fullname": "pyerrors.correlators.Corr.tanh", "modulename": "pyerrors.correlators", "qualname": "Corr.tanh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsin": {"fullname": "pyerrors.correlators.Corr.arcsin", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsin", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccos": {"fullname": "pyerrors.correlators.Corr.arccos", "modulename": "pyerrors.correlators", "qualname": "Corr.arccos", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctan": {"fullname": "pyerrors.correlators.Corr.arctan", "modulename": "pyerrors.correlators", "qualname": "Corr.arctan", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arcsinh": {"fullname": "pyerrors.correlators.Corr.arcsinh", "modulename": "pyerrors.correlators", "qualname": "Corr.arcsinh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arccosh": {"fullname": "pyerrors.correlators.Corr.arccosh", "modulename": "pyerrors.correlators", "qualname": "Corr.arccosh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.arctanh": {"fullname": "pyerrors.correlators.Corr.arctanh", "modulename": "pyerrors.correlators", "qualname": "Corr.arctanh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.correlators.Corr.real": {"fullname": "pyerrors.correlators.Corr.real", "modulename": "pyerrors.correlators", "qualname": "Corr.real", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.imag": {"fullname": "pyerrors.correlators.Corr.imag", "modulename": "pyerrors.correlators", "qualname": "Corr.imag", "kind": "variable", "doc": "

    \n"}, "pyerrors.correlators.Corr.prune": {"fullname": "pyerrors.correlators.Corr.prune", "modulename": "pyerrors.correlators", "qualname": "Corr.prune", "kind": "function", "doc": "

    Project large correlation matrix to lowest states

    \n\n

    This method can be used to reduce the size of an (N x N) correlation matrix\nto (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise\nis still small.

    \n\n
    Parameters
    \n\n
      \n
    • Ntrunc (int):\nRank of the target matrix.
      • \n
    • tproj (int):\nTime where the eigenvectors are evaluated, corresponds to ts in the GEVP method.\nThe default value is 3.
      • \n
    • t0proj (int):\nTime where the correlation matrix is inverted. Choosing t0proj=1 is strongly\ndiscouraged for O(a) improved theories, since the correctness of the procedure\ncannot be granted in this case. The default value is 2.
      • \n
    • basematrix (Corr):\nCorrelation matrix that is used to determine the eigenvectors of the\nlowest states based on a GEVP. basematrix is taken to be the Corr itself if\nis is not specified.
      • \n
    \n\n
    Notes
    \n\n

    We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving\nthe GEVP $$C(t) v_n(t, t_0) = \\lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \\equiv t_\\mathrm{proj}$\nand $t_0 \\equiv t_{0, \\mathrm{proj}}$. The target matrix is projected onto the subspace of the\nresulting eigenvectors $v_n, n=1,\\dots,N_\\mathrm{trunc}$ via\n$$G^\\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large\ncorrelation matrix and to remove some noise that is added by irrelevant operators.\nThis may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated\nbound $t_0 \\leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.

    \n", "signature": "(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):", "funcdef": "def"}, "pyerrors.correlators.Corr.N": {"fullname": "pyerrors.correlators.Corr.N", "modulename": "pyerrors.correlators", "qualname": "Corr.N", "kind": "variable", "doc": "

    \n"}, "pyerrors.covobs": {"fullname": "pyerrors.covobs", "modulename": "pyerrors.covobs", "kind": "module", "doc": "

    \n"}, "pyerrors.covobs.Covobs": {"fullname": "pyerrors.covobs.Covobs", "modulename": "pyerrors.covobs", "qualname": "Covobs", "kind": "class", "doc": "

    \n"}, "pyerrors.covobs.Covobs.__init__": {"fullname": "pyerrors.covobs.Covobs.__init__", "modulename": "pyerrors.covobs", "qualname": "Covobs.__init__", "kind": "function", "doc": "

    Initialize Covobs object.

    \n\n
    Parameters
    \n\n
      \n
    • mean (float):\nMean value of the new Obs
      • \n
    • cov (list or array):\n2d Covariance matrix or 1d diagonal entries
      • \n
    • name (str):\nidentifier for the covariance matrix
      • \n
    • pos (int):\nPosition of the variance belonging to mean in cov.\nIs taken to be 1 if cov is 0-dimensional
      • \n
    • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
      • \n
    \n", "signature": "(mean, cov, name, pos=None, grad=None)"}, "pyerrors.covobs.Covobs.name": {"fullname": "pyerrors.covobs.Covobs.name", "modulename": "pyerrors.covobs", "qualname": "Covobs.name", "kind": "variable", "doc": "

    \n"}, "pyerrors.covobs.Covobs.value": {"fullname": "pyerrors.covobs.Covobs.value", "modulename": "pyerrors.covobs", "qualname": "Covobs.value", "kind": "variable", "doc": "

    \n"}, "pyerrors.covobs.Covobs.errsq": {"fullname": "pyerrors.covobs.Covobs.errsq", "modulename": "pyerrors.covobs", "qualname": "Covobs.errsq", "kind": "function", "doc": "

    Return the variance (= square of the error) of the Covobs

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.covobs.Covobs.cov": {"fullname": "pyerrors.covobs.Covobs.cov", "modulename": "pyerrors.covobs", "qualname": "Covobs.cov", "kind": "variable", "doc": "

    \n"}, "pyerrors.covobs.Covobs.grad": {"fullname": "pyerrors.covobs.Covobs.grad", "modulename": "pyerrors.covobs", "qualname": "Covobs.grad", "kind": "variable", "doc": "

    \n"}, "pyerrors.dirac": {"fullname": "pyerrors.dirac", "modulename": "pyerrors.dirac", "kind": "module", "doc": "

    \n"}, "pyerrors.dirac.gammaX": {"fullname": "pyerrors.dirac.gammaX", "modulename": "pyerrors.dirac", "qualname": "gammaX", "kind": "variable", "doc": "

    \n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+1.j],\n [ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, -0.-1.j, 0.+0.j, 0.+0.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaY": {"fullname": "pyerrors.dirac.gammaY", "modulename": "pyerrors.dirac", "qualname": "gammaY", "kind": "variable", "doc": "

    \n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j],\n [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [-1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaZ": {"fullname": "pyerrors.dirac.gammaZ", "modulename": "pyerrors.dirac", "qualname": "gammaZ", "kind": "variable", "doc": "

    \n", "default_value": "array([[ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -0.-1.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+1.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gammaT": {"fullname": "pyerrors.dirac.gammaT", "modulename": "pyerrors.dirac", "qualname": "gammaT", "kind": "variable", "doc": "

    \n", "default_value": "array([[0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n [1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])"}, "pyerrors.dirac.gamma": {"fullname": "pyerrors.dirac.gamma", "modulename": "pyerrors.dirac", "qualname": "gamma", "kind": "variable", "doc": "

    \n", "default_value": "array([[[ 0.+0.j, 0.+0.j, 0.+0.j, 0.+1.j],\n [ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, -0.-1.j, 0.+0.j, 0.+0.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j],\n [ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [-1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 0.+1.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -0.-1.j],\n [-0.-1.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+1.j, 0.+0.j, 0.+0.j]],\n\n [[ 0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n [ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]]])"}, "pyerrors.dirac.gamma5": {"fullname": "pyerrors.dirac.gamma5", "modulename": "pyerrors.dirac", "qualname": "gamma5", "kind": "variable", "doc": "

    \n", "default_value": "array([[ 1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, -1.+0.j, 0.+0.j],\n [ 0.+0.j, 0.+0.j, 0.+0.j, -1.+0.j]])"}, "pyerrors.dirac.identity": {"fullname": "pyerrors.dirac.identity", "modulename": "pyerrors.dirac", "qualname": "identity", "kind": "variable", "doc": "

    \n", "default_value": "array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j],\n [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]])"}, "pyerrors.dirac.epsilon_tensor": {"fullname": "pyerrors.dirac.epsilon_tensor", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor", "kind": "function", "doc": "

    Rank-3 epsilon tensor

    \n\n

    Based on https://codegolf.stackexchange.com/a/160375

    \n\n
    Returns
    \n\n
      \n
    • elem (int):\nElement (i,j,k) of the epsilon tensor of rank 3
      • \n
    \n", "signature": "(i, j, k):", "funcdef": "def"}, "pyerrors.dirac.epsilon_tensor_rank4": {"fullname": "pyerrors.dirac.epsilon_tensor_rank4", "modulename": "pyerrors.dirac", "qualname": "epsilon_tensor_rank4", "kind": "function", "doc": "

    Rank-4 epsilon tensor

    \n\n

    Extension of https://codegolf.stackexchange.com/a/160375

    \n\n
    Returns
    \n\n
      \n
    • elem (int):\nElement (i,j,k,o) of the epsilon tensor of rank 4
      • \n
    \n", "signature": "(i, j, k, o):", "funcdef": "def"}, "pyerrors.dirac.Grid_gamma": {"fullname": "pyerrors.dirac.Grid_gamma", "modulename": "pyerrors.dirac", "qualname": "Grid_gamma", "kind": "function", "doc": "

    Returns gamma matrix in Grid labeling.

    \n", "signature": "(gamma_tag):", "funcdef": "def"}, "pyerrors.fits": {"fullname": "pyerrors.fits", "modulename": "pyerrors.fits", "kind": "module", "doc": "

    \n"}, "pyerrors.fits.Fit_result": {"fullname": "pyerrors.fits.Fit_result", "modulename": "pyerrors.fits", "qualname": "Fit_result", "kind": "class", "doc": "

    Represents fit results.

    \n\n
    Attributes
    \n\n
      \n
    • fit_parameters (list):\nresults for the individual fit parameters,\nalso accessible via indices.
      • \n
    • chisquare_by_dof (float):\nreduced chisquare.
      • \n
    • p_value (float):\np-value of the fit
      • \n
    • t2_p_value (float):\nHotelling t-squared p-value for correlated fits.
      • \n
    \n", "bases": "collections.abc.Sequence"}, "pyerrors.fits.Fit_result.fit_parameters": {"fullname": "pyerrors.fits.Fit_result.fit_parameters", "modulename": "pyerrors.fits", "qualname": "Fit_result.fit_parameters", "kind": "variable", "doc": "

    \n"}, "pyerrors.fits.Fit_result.gamma_method": {"fullname": "pyerrors.fits.Fit_result.gamma_method", "modulename": "pyerrors.fits", "qualname": "Fit_result.gamma_method", "kind": "function", "doc": "

    Apply the gamma method to all fit parameters

    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.Fit_result.gm": {"fullname": "pyerrors.fits.Fit_result.gm", "modulename": "pyerrors.fits", "qualname": "Fit_result.gm", "kind": "function", "doc": "

    Apply the gamma method to all fit parameters

    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.fits.least_squares": {"fullname": "pyerrors.fits.least_squares", "modulename": "pyerrors.fits", "qualname": "least_squares", "kind": "function", "doc": "

    Performs a non-linear fit to y = func(x).\n ```

    \n\n
    Parameters
    \n\n
      \n
    • For an uncombined fit:
      • \n
    • x (list):\nlist of floats.
      • \n
    • y (list):\nlist of Obs.
      • \n

    • func (object):\nfit function, has to be of the form

      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(a, x):\n   return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
      \n
      \n\n

      For multiple x values func can be of the form

      \n\n
      \n
      def func(a, x):\n   (x1, x2) = x\n   return a[0] * x1 ** 2 + a[1] * x2\n
      \n
      \n\n

      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.

      • \n
    • OR For a combined fit:
      • \n
    • x (dict):\ndict of lists.
      • \n
    • y (dict):\ndict of lists of Obs.
      • \n

    • funcs (dict):\ndict of objects\nfit functions have to be of the form (here a[0] is the common fit parameter)\n```python\nimport autograd.numpy as anp\nfuncs = {\"a\": func_a,\n \"b\": func_b}

      \n\n

      def func_a(a, x):\n return a[1] * anp.exp(-a[0] * x)

      \n\n

      def func_b(a, x):\n return a[2] * anp.exp(-a[0] * x)

      \n\n

      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.

      • \n
    • priors (dict or list, optional):\npriors can either be a dictionary with integer keys and the corresponding priors as values or\na list with an entry for every parameter in the fit. The entries can either be\nObs (e.g. results from a previous fit) or strings containing a value and an error formatted like\n0.548(23), 500(40) or 0.5(0.4)
      • \n
    • silent (bool, optional):\nIf true all output to the console is omitted (default False).
      • \n
    • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for\nnon-linear fits with many parameters. In case of correlated fits the guess is used to perform\nan uncorrelated fit which then serves as guess for the correlated fit.
      • \n
    • method (str, optional):\ncan be used to choose an alternative method for the minimization of chisquare.\nThe possible methods are the ones which can be used for scipy.optimize.minimize and\nmigrad of iminuit. If no method is specified, Levenberg-Marquard is used.\nReliable alternatives are migrad, Powell and Nelder-Mead.
      • \n
    • tol (float, optional):\ncan be used (only for combined fits and methods other than Levenberg-Marquard) to set the tolerance for convergence\nto a different value to either speed up convergence at the cost of a larger error on the fitted parameters (and possibly\ninvalid estimates for parameter uncertainties) or smaller values to get more accurate parameter values\nThe stopping criterion depends on the method, e.g. migrad: edm_max = 0.002 * tol * errordef (EDM criterion: edm < edm_max)
      • \n
    • correlated_fit (bool):\nIf True, use the full inverse covariance matrix in the definition of the chisquare cost function.\nFor details about how the covariance matrix is estimated see pyerrors.obs.covariance.\nIn practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).\nThis procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
      • \n
    • expected_chisquare (bool):\nIf True estimates the expected chisquare which is\ncorrected by effects caused by correlated input data (default False).
      • \n
    • resplot (bool):\nIf True, a plot which displays fit, data and residuals is generated (default False).
      • \n
    • qqplot (bool):\nIf True, a quantile-quantile plot of the fit result is generated (default False).
      • \n
    • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • output (Fit_result):\nParameters and information on the fitted result.
      • \n
    \n", "signature": "(x, y, func, priors=None, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.total_least_squares": {"fullname": "pyerrors.fits.total_least_squares", "modulename": "pyerrors.fits", "qualname": "total_least_squares", "kind": "function", "doc": "

    Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.

    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nlist of Obs, or a tuple of lists of Obs
      • \n
    • y (list):\nlist of Obs. The dvalues of the Obs are used as x- and yerror for the fit.
      • \n

    • func (object):\nfunc has to be of the form

      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(a, x):\n   return a[0] + a[1] * x + a[2] * anp.sinh(x)\n
      \n
      \n\n

      For multiple x values func can be of the form

      \n\n
      \n
      def func(a, x):\n   (x1, x2) = x\n   return a[0] * x1 ** 2 + a[1] * x2\n
      \n
      \n\n

      It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation\nwill not work.

      • \n
    • silent (bool, optional):\nIf true all output to the console is omitted (default False).
      • \n
    • initial_guess (list):\ncan provide an initial guess for the input parameters. Relevant for non-linear\nfits with many parameters.
      • \n
    • expected_chisquare (bool):\nIf true prints the expected chisquare which is\ncorrected by effects caused by correlated input data.\nThis can take a while as the full correlation matrix\nhas to be calculated (default False).
      • \n
    • num_grad (bool):\nUse numerical differentation instead of automatic differentiation to perform the error propagation (default False).
      • \n
    \n\n
    Notes
    \n\n

    Based on the orthogonal distance regression module of scipy.

    \n\n
    Returns
    \n\n
      \n
    • output (Fit_result):\nParameters and information on the fitted result.
      • \n
    \n", "signature": "(x, y, func, silent=False, **kwargs):", "funcdef": "def"}, "pyerrors.fits.fit_lin": {"fullname": "pyerrors.fits.fit_lin", "modulename": "pyerrors.fits", "qualname": "fit_lin", "kind": "function", "doc": "

    Performs a linear fit to y = n + m * x and returns two Obs n, m.

    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nCan either be a list of floats in which case no xerror is assumed, or\na list of Obs, where the dvalues of the Obs are used as xerror for the fit.
      • \n
    • y (list):\nList of Obs, the dvalues of the Obs are used as yerror for the fit.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • fit_parameters (list[Obs]):\nLIist of fitted observables.
      • \n
    \n", "signature": "(x, y, **kwargs):", "funcdef": "def"}, "pyerrors.fits.qqplot": {"fullname": "pyerrors.fits.qqplot", "modulename": "pyerrors.fits", "qualname": "qqplot", "kind": "function", "doc": "

    Generates a quantile-quantile plot of the fit result which can be used to\n check if the residuals of the fit are gaussian distributed.

    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(x, o_y, func, p, title=''):", "funcdef": "def"}, "pyerrors.fits.residual_plot": {"fullname": "pyerrors.fits.residual_plot", "modulename": "pyerrors.fits", "qualname": "residual_plot", "kind": "function", "doc": "

    Generates a plot which compares the fit to the data and displays the corresponding residuals

    \n\n

    For uncorrelated data the residuals are expected to be distributed ~N(0,1).

    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(x, y, func, fit_res, title=''):", "funcdef": "def"}, "pyerrors.fits.error_band": {"fullname": "pyerrors.fits.error_band", "modulename": "pyerrors.fits", "qualname": "error_band", "kind": "function", "doc": "

    Calculate the error band for an array of sample values x, for given fit function func with optimized parameters beta.

    \n\n
    Returns
    \n\n
      \n
    • err (np.array(Obs)):\nError band for an array of sample values x
      • \n
    \n", "signature": "(x, func, beta):", "funcdef": "def"}, "pyerrors.fits.ks_test": {"fullname": "pyerrors.fits.ks_test", "modulename": "pyerrors.fits", "qualname": "ks_test", "kind": "function", "doc": "

    Performs a Kolmogorov\u2013Smirnov test for the p-values of all fit object.

    \n\n
    Parameters
    \n\n
      \n
    • objects (list):\nList of fit results to include in the analysis (optional).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(objects=None):", "funcdef": "def"}, "pyerrors.input": {"fullname": "pyerrors.input", "modulename": "pyerrors.input", "kind": "module", "doc": "

    pyerrors includes an input submodule in which input routines and parsers for the output of various numerical programs are contained.

    \n\n

    Jackknife samples

    \n\n

    For comparison with other analysis workflows pyerrors can also generate jackknife samples from an Obs object or import jackknife samples into an Obs object.\nSee pyerrors.obs.Obs.export_jackknife and pyerrors.obs.import_jackknife for details.

    \n"}, "pyerrors.input.bdio": {"fullname": "pyerrors.input.bdio", "modulename": "pyerrors.input.bdio", "kind": "module", "doc": "

    \n"}, "pyerrors.input.bdio.read_ADerrors": {"fullname": "pyerrors.input.bdio.read_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "read_ADerrors", "kind": "function", "doc": "

    Extract generic MCMC data from a bdio file

    \n\n

    read_ADerrors requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to

    \n\n

    all: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/

    \n\n
    Parameters
    \n\n
      \n
    • file_path -- path to the bdio file
      • \n
    • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (List[Obs]):\nExtracted data
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.write_ADerrors": {"fullname": "pyerrors.input.bdio.write_ADerrors", "modulename": "pyerrors.input.bdio", "qualname": "write_ADerrors", "kind": "function", "doc": "

    Write Obs to a bdio file according to ADerrors conventions

    \n\n

    read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to

    \n\n

    all: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/

    \n\n
    Parameters
    \n\n
      \n
    • file_path -- path to the bdio file
      • \n
    • bdio_path -- path to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • success (int):\nreturns 0 is successful
      • \n
    \n", "signature": "(obs_list, file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_mesons": {"fullname": "pyerrors.input.bdio.read_mesons", "modulename": "pyerrors.input.bdio", "qualname": "read_mesons", "kind": "function", "doc": "

    Extract mesons data from a bdio file and return it as a dictionary

    \n\n

    The dictionary can be accessed with a tuple consisting of (type, source_position, kappa1, kappa2)

    \n\n

    read_mesons requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to

    \n\n

    all: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/

    \n\n
    Parameters
    \n\n
      \n
    • file_path (str):\npath to the bdio file
      • \n
    • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    • start (int):\nThe first configuration to be read (default 1)
      • \n
    • stop (int):\nThe last configuration to be read (default None)
      • \n
    • step (int):\nFixed step size between two measurements (default 1)
      • \n
    • alternative_ensemble_name (str):\nManually overwrite ensemble name
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (dict):\nExtracted meson data
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.bdio.read_dSdm": {"fullname": "pyerrors.input.bdio.read_dSdm", "modulename": "pyerrors.input.bdio", "qualname": "read_dSdm", "kind": "function", "doc": "

    Extract dSdm data from a bdio file and return it as a dictionary

    \n\n

    The dictionary can be accessed with a tuple consisting of (type, kappa)

    \n\n

    read_dSdm requires bdio to be compiled into a shared library. This can be achieved by\nadding the flag -fPIC to CC and changing the all target to

    \n\n

    all: bdio.o $(LIBDIR)\n gcc -shared -Wl,-soname,libbdio.so -o $(BUILDDIR)/libbdio.so $(BUILDDIR)/bdio.o\n cp $(BUILDDIR)/libbdio.so $(LIBDIR)/

    \n\n
    Parameters
    \n\n
      \n
    • file_path (str):\npath to the bdio file
      • \n
    • bdio_path (str):\npath to the shared bdio library libbdio.so (default ./libbdio.so)
      • \n
    • start (int):\nThe first configuration to be read (default 1)
      • \n
    • stop (int):\nThe last configuration to be read (default None)
      • \n
    • step (int):\nFixed step size between two measurements (default 1)
      • \n
    • alternative_ensemble_name (str):\nManually overwrite ensemble name
      • \n
    \n", "signature": "(file_path, bdio_path='./libbdio.so', **kwargs):", "funcdef": "def"}, "pyerrors.input.dobs": {"fullname": "pyerrors.input.dobs", "modulename": "pyerrors.input.dobs", "kind": "module", "doc": "

    \n"}, "pyerrors.input.dobs.create_pobs_string": {"fullname": "pyerrors.input.dobs.create_pobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_pobs_string", "kind": "function", "doc": "

    Export a list of Obs or structures containing Obs to an xml string\naccording to the Zeuthen pobs format.

    \n\n

    Tags are not written or recovered automatically. The separator | is removed from the replica names.

    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • xml_str (str):\nXML formatted string of the input data
      • \n
    \n", "signature": "(obsl, name, spec='', origin='', symbol=[], enstag=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_pobs": {"fullname": "pyerrors.input.dobs.write_pobs", "modulename": "pyerrors.input.dobs", "qualname": "write_pobs", "kind": "function", "doc": "

    Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen pobs format.

    \n\n

    Tags are not written or recovered automatically. The separator | is removed from the replica names.

    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure have to be defined on the same ensemble.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • enstag (str):\nEnstag that is written to pobs. If None, the ensemble name is used.
      • \n
    • gz (bool):\nIf True, the output is a gzipped xml. If False, the output is an xml file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(\tobsl,\tfname,\tname,\tspec='',\torigin='',\tsymbol=[],\tenstag=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_pobs": {"fullname": "pyerrors.input.dobs.read_pobs", "modulename": "pyerrors.input.dobs", "qualname": "read_pobs", "kind": "function", "doc": "

    Import a list of Obs from an xml.gz file in the Zeuthen pobs format.

    \n\n

    Tags are not written or recovered automatically.

    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • separatior_insertion (str or int):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nNone (default): Replica names remain unchanged.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(fname, full_output=False, gz=True, separator_insertion=None):", "funcdef": "def"}, "pyerrors.input.dobs.import_dobs_string": {"fullname": "pyerrors.input.dobs.import_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "import_dobs_string", "kind": "function", "doc": "

    Import a list of Obs from a string in the Zeuthen dobs format.

    \n\n

    Tags are not written or recovered automatically.

    \n\n
    Parameters
    \n\n
      \n
    • content (str):\nXML string containing the data
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(content, full_output=False, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.read_dobs": {"fullname": "pyerrors.input.dobs.read_dobs", "modulename": "pyerrors.input.dobs", "qualname": "read_dobs", "kind": "function", "doc": "

    Import a list of Obs from an xml.gz file in the Zeuthen dobs format.

    \n\n

    Tags are not written or recovered automatically.

    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned as list.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes XML file.
      • \n
    • separatior_insertion (str, int or bool):\nstr: replace all occurences of \"separator_insertion\" within the replica names\nby \"|%s\" % (separator_insertion) when constructing the names of the replica.\nint: Insert the separator \"|\" at the position given by separator_insertion.\nTrue (default): separator \"|\" is inserted after len(ensname), assuming that the\nensemble name is a prefix to the replica name.\nNone or False: No separator is inserted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (list[Obs]):\nImported data
      • \n
    • or
      • \n
    • res (dict):\nImported data and meta-data
      • \n
    \n", "signature": "(fname, full_output=False, gz=True, separator_insertion=True):", "funcdef": "def"}, "pyerrors.input.dobs.create_dobs_string": {"fullname": "pyerrors.input.dobs.create_dobs_string", "modulename": "pyerrors.input.dobs", "qualname": "create_dobs_string", "kind": "function", "doc": "

    Generate the string for the export of a list of Obs or structures containing Obs\nto a .xml.gz file according to the Zeuthen dobs format.

    \n\n

    Tags are not written or recovered automatically. The separator |is removed from the replica names.

    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • who (str):\nProvide the name of the person that exports the data.
      • \n
    • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • xml_str (str):\nXML string generated from the data
      • \n
    \n", "signature": "(\tobsl,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None):", "funcdef": "def"}, "pyerrors.input.dobs.write_dobs": {"fullname": "pyerrors.input.dobs.write_dobs", "modulename": "pyerrors.input.dobs", "qualname": "write_dobs", "kind": "function", "doc": "

    Export a list of Obs or structures containing Obs to a .xml.gz file\naccording to the Zeuthen dobs format.

    \n\n

    Tags are not written or recovered automatically. The separator | is removed from the replica names.

    \n\n
    Parameters
    \n\n
      \n
    • obsl (list):\nList of Obs that will be exported.\nThe Obs inside a structure do not have to be defined on the same set of configurations,\nbut the storage requirement is increased, if this is not the case.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • name (str):\nThe name of the observable.
      • \n
    • spec (str):\nOptional string that describes the contents of the file.
      • \n
    • origin (str):\nSpecify where the data has its origin.
      • \n
    • symbol (list):\nA list of symbols that describe the observables to be written. May be empty.
      • \n
    • who (str):\nProvide the name of the person that exports the data.
      • \n
    • enstags (dict):\nProvide alternative enstag for ensembles in the form enstags = {ename: enstag}\nOtherwise, the ensemble name is used.
      • \n
    • gz (bool):\nIf True, the output is a gzipped XML. If False, the output is a XML file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(\tobsl,\tfname,\tname,\tspec='dobs v1.0',\torigin='',\tsymbol=[],\twho=None,\tenstags=None,\tgz=True):", "funcdef": "def"}, "pyerrors.input.hadrons": {"fullname": "pyerrors.input.hadrons", "modulename": "pyerrors.input.hadrons", "kind": "module", "doc": "

    \n"}, "pyerrors.input.hadrons.read_hd5": {"fullname": "pyerrors.input.hadrons.read_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_hd5", "kind": "function", "doc": "

    Read hadrons hdf5 file and extract entry based on attributes.

    \n\n
    Parameters
    \n\n
      \n
    • filestem (str):\nFull namestem of the files to read, including the full path.
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • group (str):\nlabel of the group to be extracted.
      • \n

    • attrs (dict or int):\nDictionary containing the attributes. For example

      \n\n
      \n
      attrs = {"gamma_snk": "Gamma5",\n        "gamma_src": "Gamma5"}\n
      \n
      \n\n

      Alternatively an integer can be specified to identify the sub group.\nThis is discouraged as the order in the file is not guaranteed.

      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • part (str):\nstring specifying whether to extract the real part ('real'),\nthe imaginary part ('imag') or a complex correlator ('complex').\nDefault 'real'.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr (Corr):\nCorrelator of the source sink combination in question.
      • \n
    \n", "signature": "(filestem, ens_id, group, attrs=None, idl=None, part='real'):", "funcdef": "def"}, "pyerrors.input.hadrons.read_meson_hd5": {"fullname": "pyerrors.input.hadrons.read_meson_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_meson_hd5", "kind": "function", "doc": "

    Read hadrons meson hdf5 file and extract the meson labeled 'meson'

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • meson (str):\nlabel of the meson to be extracted, standard value meson_0 which\ncorresponds to the pseudoscalar pseudoscalar two-point function.
      • \n
    • gammas (tuple of strings):\nInstrad of a meson label one can also provide a tuple of two strings\nindicating the gamma matrices at sink and source (gamma_snk, gamma_src).\n(\"Gamma5\", \"Gamma5\") corresponds to the pseudoscalar pseudoscalar\ntwo-point function. The gammas argument dominateds over meson.
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr (Corr):\nCorrelator of the source sink combination in question.
      • \n
    \n", "signature": "(path, filestem, ens_id, meson='meson_0', idl=None, gammas=None):", "funcdef": "def"}, "pyerrors.input.hadrons.extract_t0_hd5": {"fullname": "pyerrors.input.hadrons.extract_t0_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "extract_t0_hd5", "kind": "function", "doc": "

    Read hadrons FlowObservables hdf5 file and extract t0

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • obs (str):\nlabel of the observable from which t0 should be extracted.\nOptions: 'Clover energy density' and 'Plaquette energy density'
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
      • \n
    \n", "signature": "(\tpath,\tfilestem,\tens_id,\tobs='Clover energy density',\tfit_range=5,\tidl=None,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.hadrons.read_DistillationContraction_hd5": {"fullname": "pyerrors.input.hadrons.read_DistillationContraction_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_DistillationContraction_hd5", "kind": "function", "doc": "

    Read hadrons DistillationContraction hdf5 files in given directory structure

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the directories to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • diagrams (list):\nList of strings of the diagrams to extract, e.g. [\"direct\", \"box\", \"cross\"].
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (dict):\nextracted DistillationContration data
      • \n
    \n", "signature": "(path, ens_id, diagrams=['direct'], idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.Npr_matrix": {"fullname": "pyerrors.input.hadrons.Npr_matrix", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix", "kind": "class", "doc": "

    ndarray(shape, dtype=float, buffer=None, offset=0,\n strides=None, order=None)

    \n\n

    An array object represents a multidimensional, homogeneous array\nof fixed-size items. An associated data-type object describes the\nformat of each element in the array (its byte-order, how many bytes it\noccupies in memory, whether it is an integer, a floating point number,\nor something else, etc.)

    \n\n

    Arrays should be constructed using array, zeros or empty (refer\nto the See Also section below). The parameters given here refer to\na low-level method (ndarray(...)) for instantiating an array.

    \n\n

    For more information, refer to the numpy module and examine the\nmethods and attributes of an array.

    \n\n
    Parameters
    \n\n
      \n
    • (for the __new__ method; see Notes below)
      • \n
    • shape (tuple of ints):\nShape of created array.
      • \n
    • dtype (data-type, optional):\nAny object that can be interpreted as a numpy data type.
      • \n
    • buffer (object exposing buffer interface, optional):\nUsed to fill the array with data.
      • \n
    • offset (int, optional):\nOffset of array data in buffer.
      • \n
    • strides (tuple of ints, optional):\nStrides of data in memory.
      • \n
    • order ({'C', 'F'}, optional):\nRow-major (C-style) or column-major (Fortran-style) order.
      • \n
    \n\n
    Attributes
    \n\n
      \n
    • T (ndarray):\nTranspose of the array.
      • \n
    • data (buffer):\nThe array's elements, in memory.
      • \n
    • dtype (dtype object):\nDescribes the format of the elements in the array.
      • \n
    • flags (dict):\nDictionary containing information related to memory use, e.g.,\n'C_CONTIGUOUS', 'OWNDATA', 'WRITEABLE', etc.
      • \n
    • flat (numpy.flatiter object):\nFlattened version of the array as an iterator. The iterator\nallows assignments, e.g., x.flat = 3 (See ndarray.flat for\nassignment examples; TODO).
      • \n
    • imag (ndarray):\nImaginary part of the array.
      • \n
    • real (ndarray):\nReal part of the array.
      • \n
    • size (int):\nNumber of elements in the array.
      • \n
    • itemsize (int):\nThe memory use of each array element in bytes.
      • \n
    • nbytes (int):\nThe total number of bytes required to store the array data,\ni.e., itemsize * size.
      • \n
    • ndim (int):\nThe array's number of dimensions.
      • \n
    • shape (tuple of ints):\nShape of the array.
      • \n
    • strides (tuple of ints):\nThe step-size required to move from one element to the next in\nmemory. For example, a contiguous (3, 4) array of type\nint16 in C-order has strides (8, 2). This implies that\nto move from element to element in memory requires jumps of 2 bytes.\nTo move from row-to-row, one needs to jump 8 bytes at a time\n(2 * 4).
      • \n
    • ctypes (ctypes object):\nClass containing properties of the array needed for interaction\nwith ctypes.
      • \n
    • base (ndarray):\nIf the array is a view into another array, that array is its base\n(unless that array is also a view). The base array is where the\narray data is actually stored.
      • \n
    \n\n
    See Also
    \n\n

    array: Construct an array.
    \nzeros: Create an array, each element of which is zero.
    \nempty: Create an array, but leave its allocated memory unchanged (i.e.,\nit contains \"garbage\").
    \ndtype: Create a data-type.
    \nnumpy.typing.NDArray: An ndarray alias :term:generic <generic type>\nw.r.t. its dtype.type <numpy.dtype.type>.

    \n\n
    Notes
    \n\n

    There are two modes of creating an array using __new__:

    \n\n
      \n
    1. If buffer is None, then only shape, dtype, and order\nare used.
      2. \n
    2. If buffer is an object exposing the buffer interface, then\nall keywords are interpreted.
      3. \n
    \n\n

    No __init__ method is needed because the array is fully initialized\nafter the __new__ method.

    \n\n
    Examples
    \n\n

    These examples illustrate the low-level ndarray constructor. Refer\nto the See Also section above for easier ways of constructing an\nndarray.

    \n\n

    First mode, buffer is None:

    \n\n
    \n
    >>> np.ndarray(shape=(2,2), dtype=float, order='F')\narray([[0.0e+000, 0.0e+000], # random\n       [     nan, 2.5e-323]])\n
    \n
    \n\n

    Second mode:

    \n\n
    \n
    >>> np.ndarray((2,), buffer=np.array([1,2,3]),\n...            offset=np.int_().itemsize,\n...            dtype=int) # offset = 1*itemsize, i.e. skip first element\narray([2, 3])\n
    \n
    \n", "bases": "numpy.ndarray"}, "pyerrors.input.hadrons.Npr_matrix.g5H": {"fullname": "pyerrors.input.hadrons.Npr_matrix.g5H", "modulename": "pyerrors.input.hadrons", "qualname": "Npr_matrix.g5H", "kind": "variable", "doc": "

    Gamma_5 hermitean conjugate

    \n\n

    Uses the fact that the propagator is gamma5 hermitean, so just the\nin and out momenta of the propagator are exchanged.

    \n"}, "pyerrors.input.hadrons.read_ExternalLeg_hd5": {"fullname": "pyerrors.input.hadrons.read_ExternalLeg_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_ExternalLeg_hd5", "kind": "function", "doc": "

    Read hadrons ExternalLeg hdf5 file and output an array of CObs

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (Npr_matrix):\nread Cobs-matrix
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Bilinear_hd5": {"fullname": "pyerrors.input.hadrons.read_Bilinear_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Bilinear_hd5", "kind": "function", "doc": "

    Read hadrons Bilinear hdf5 file and output an array of CObs

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result_dict (dict[Npr_matrix]):\nextracted Bilinears
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None):", "funcdef": "def"}, "pyerrors.input.hadrons.read_Fourquark_hd5": {"fullname": "pyerrors.input.hadrons.read_Fourquark_hd5", "modulename": "pyerrors.input.hadrons", "qualname": "read_Fourquark_hd5", "kind": "function", "doc": "

    Read hadrons FourquarkFullyConnected hdf5 file and output an array of CObs

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the files to read
      • \n
    • filestem (str):\nnamestem of the files to read
      • \n
    • ens_id (str):\nname of the ensemble, required for internal bookkeeping
      • \n
    • idl (range):\nIf specified only configurations in the given range are read in.
      • \n
    • vertices (list):\nVertex functions to be extracted.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result_dict (dict):\nextracted fourquark matrizes
      • \n
    \n", "signature": "(path, filestem, ens_id, idl=None, vertices=['VA', 'AV']):", "funcdef": "def"}, "pyerrors.input.json": {"fullname": "pyerrors.input.json", "modulename": "pyerrors.input.json", "kind": "module", "doc": "

    \n"}, "pyerrors.input.json.create_json_string": {"fullname": "pyerrors.input.json.create_json_string", "modulename": "pyerrors.input.json", "qualname": "create_json_string", "kind": "function", "doc": "

    Generate the string for the export of a list of Obs or structures containing Obs\nto a .json(.gz) file

    \n\n
    Parameters
    \n\n
      \n
    • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • json_string (str):\nString for export to .json(.gz) file
      • \n
    \n", "signature": "(ol, description='', indent=1):", "funcdef": "def"}, "pyerrors.input.json.dump_to_json": {"fullname": "pyerrors.input.json.dump_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_to_json", "kind": "function", "doc": "

    Export a list of Obs or structures containing Obs to a .json(.gz) file.\nDict keys that are not JSON-serializable such as floats are converted to strings.

    \n\n
    Parameters
    \n\n
      \n
    • ol (list):\nList of objects that will be exported. At the moment, these objects can be\neither of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Null
      • \n
    \n", "signature": "(ol, fname, description='', indent=1, gz=True):", "funcdef": "def"}, "pyerrors.input.json.import_json_string": {"fullname": "pyerrors.input.json.import_json_string", "modulename": "pyerrors.input.json", "qualname": "import_json_string", "kind": "function", "doc": "

    Reconstruct a list of Obs or structures containing Obs from a json string.

    \n\n

    The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.

    \n\n
    Parameters
    \n\n
      \n
    • json_string (str):\njson string containing the data.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nreconstructed list of observables from the json string
      • \n
    • or
      • \n
    • result (Obs):\nonly one observable if the list only has one entry
      • \n
    • or
      • \n
    • result (dict):\nif full_output=True
      • \n
    \n", "signature": "(json_string, verbose=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.load_json": {"fullname": "pyerrors.input.json.load_json", "modulename": "pyerrors.input.json", "qualname": "load_json", "kind": "function", "doc": "

    Import a list of Obs or structures containing Obs from a .json(.gz) file.

    \n\n

    The following structures are supported: Obs, list, numpy.ndarray, Corr\nIf the list contains only one element, it is unpacked from the list.

    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nreconstructed list of observables from the json string
      • \n
    • or
      • \n
    • result (Obs):\nonly one observable if the list only has one entry
      • \n
    • or
      • \n
    • result (dict):\nif full_output=True
      • \n
    \n", "signature": "(fname, verbose=True, gz=True, full_output=False):", "funcdef": "def"}, "pyerrors.input.json.dump_dict_to_json": {"fullname": "pyerrors.input.json.dump_dict_to_json", "modulename": "pyerrors.input.json", "qualname": "dump_dict_to_json", "kind": "function", "doc": "

    Export a dict of Obs or structures containing Obs to a .json(.gz) file

    \n\n
    Parameters
    \n\n
      \n
    • od (dict):\nDict of JSON valid structures and objects that will be exported.\nAt the moment, these objects can be either of: Obs, list, numpy.ndarray, Corr.\nAll Obs inside a structure have to be defined on the same set of configurations.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • description (str):\nOptional string that describes the contents of the json file.
      • \n
    • indent (int):\nSpecify the indentation level of the json file. None or 0 is permissible and\nsaves disk space.
      • \n
    • reps (str):\nSpecify the structure of the placeholder in exported dict to be reps[0-9]+.
      • \n
    • gz (bool):\nIf True, the output is a gzipped json. If False, the output is a json file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(od, fname, description='', indent=1, reps='DICTOBS', gz=True):", "funcdef": "def"}, "pyerrors.input.json.load_json_dict": {"fullname": "pyerrors.input.json.load_json_dict", "modulename": "pyerrors.input.json", "qualname": "load_json_dict", "kind": "function", "doc": "

    Import a dict of Obs or structures containing Obs from a .json(.gz) file.

    \n\n

    The following structures are supported: Obs, list, numpy.ndarray, Corr

    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • verbose (bool):\nPrint additional information that was written to the file.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    • full_output (bool):\nIf True, a dict containing auxiliary information and the data is returned.\nIf False, only the data is returned.
      • \n
    • reps (str):\nSpecify the structure of the placeholder in imported dict to be reps[0-9]+.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (Obs / list / Corr):\nRead data
      • \n
    • or
      • \n
    • data (dict):\nRead data and meta-data
      • \n
    \n", "signature": "(fname, verbose=True, gz=True, full_output=False, reps='DICTOBS'):", "funcdef": "def"}, "pyerrors.input.misc": {"fullname": "pyerrors.input.misc", "modulename": "pyerrors.input.misc", "kind": "module", "doc": "

    \n"}, "pyerrors.input.misc.fit_t0": {"fullname": "pyerrors.input.misc.fit_t0", "modulename": "pyerrors.input.misc", "qualname": "fit_t0", "kind": "function", "doc": "

    Compute the root of (flow-based) data based on a dictionary that contains\nthe necessary information in key-value pairs a la (flow time: observable at flow time).

    \n\n

    It is assumed that the data is monotonically increasing and passes zero from below.\nNo exception is thrown if this is not the case (several roots, no monotonic increase).\nAn exception is thrown if no root can be found in the data.

    \n\n

    A linear fit in the vicinity of the root is performed to exctract the root from the\ntwo fit parameters.

    \n\n
    Parameters
    \n\n
      \n
    • t2E_dict (dict):\nDictionary with pairs of (flow time: observable at flow time) where the flow times\nare of type float and the observables of type Obs.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data. (Default: False)
      • \n
    • observable (str):\nKeyword to identify the observable to print the correct ylabel (if plot_fit is True)\nfor the observables 't0' and 'w0'. No y label is printed otherwise. (Default: 't0')
      • \n
    \n\n
    Returns
    \n\n
      \n
    • root (Obs):\nThe root of the data series.
      • \n
    \n", "signature": "(t2E_dict, fit_range, plot_fit=False, observable='t0'):", "funcdef": "def"}, "pyerrors.input.misc.read_pbp": {"fullname": "pyerrors.input.misc.read_pbp", "modulename": "pyerrors.input.misc", "qualname": "read_pbp", "kind": "function", "doc": "

    Read pbp format from given folder structure.

    \n\n
    Parameters
    \n\n
      \n
    • r_start (list):\nlist which contains the first config to be read for each replicum
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nlist of observables read
      • \n
    \n", "signature": "(path, prefix, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD": {"fullname": "pyerrors.input.openQCD", "modulename": "pyerrors.input.openQCD", "kind": "module", "doc": "

    \n"}, "pyerrors.input.openQCD.read_rwms": {"fullname": "pyerrors.input.openQCD.read_rwms", "modulename": "pyerrors.input.openQCD", "qualname": "read_rwms", "kind": "function", "doc": "

    Read rwms format from given folder structure. Returns a list of length nrw

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath that contains the data files
      • \n
    • prefix (str):\nall files in path that start with prefix are considered as input files.\nMay be used together postfix to consider only special file endings.\nPrefix is ignored, if the keyword 'files' is used.
      • \n
    • version (str):\nversion of openQCD, default 2.0
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.ms1' for openQCD-files
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • print_err (bool):\nPrint additional information that is useful for debugging.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • rwms (Obs):\nReweighting factors read
      • \n
    \n", "signature": "(path, prefix, version='2.0', names=None, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_t0": {"fullname": "pyerrors.input.openQCD.extract_t0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_t0", "kind": "function", "doc": "

    Extract t0/a^2 from given .ms.dat files. Returns t0 as Obs.

    \n\n

    It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t^2 - c (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.

    \n\n

    It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to .ms.dat files
      • \n
    • prefix (str):\nEnsemble prefix
      • \n
    • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
      • \n
    • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
      • \n
    • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • postfix (str):\nPostfix of measurement file (Default: ms)
      • \n
    • c (float):\nConstant that defines the flow scale. Default 0.3 for t_0, choose 2./3 for t_1.
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • plaquette (bool):\nIf true extract the plaquette estimate of t0 instead.
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of t0 is shown together with the data.
      • \n
    • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
      • \n
    \n\n
    Returns
    \n\n
      \n
    • t0 (Obs):\nExtracted t0
      • \n
    \n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.extract_w0": {"fullname": "pyerrors.input.openQCD.extract_w0", "modulename": "pyerrors.input.openQCD", "qualname": "extract_w0", "kind": "function", "doc": "

    Extract w0/a from given .ms.dat files. Returns w0 as Obs.

    \n\n

    It is assumed that all boundary effects have\nsufficiently decayed at x0=xmin.\nThe data around the zero crossing of t d(t^2)/dt - (where c=0.3 by default)\nis fitted with a linear function\nfrom which the exact root is extracted.

    \n\n

    It is assumed that one measurement is performed for each config.\nIf this is not the case, the resulting idl, as well as the handling\nof r_start, r_stop and r_step is wrong and the user has to correct\nthis in the resulting observable.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to .ms.dat files
      • \n
    • prefix (str):\nEnsemble prefix
      • \n
    • dtr_read (int):\nDetermines how many trajectories should be skipped\nwhen reading the ms.dat files.\nCorresponds to dtr_cnfg / dtr_ms in the openQCD input file.
      • \n
    • xmin (int):\nFirst timeslice where the boundary\neffects have sufficiently decayed.
      • \n
    • spatial_extent (int):\nspatial extent of the lattice, required for normalization.
      • \n
    • fit_range (int):\nNumber of data points left and right of the zero\ncrossing to be included in the linear fit. (Default: 5)
      • \n
    • postfix (str):\nPostfix of measurement file (Default: ms)
      • \n
    • c (float):\nConstant that defines the flow scale. Default 0.3 for w_0, choose 2./3 for w_1.
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • r_step (int):\ninteger that defines a fixed step size between two measurements (in units of configs)\nIf not given, r_step=1 is assumed.
      • \n
    • plaquette (bool):\nIf true extract the plaquette estimate of w0 instead.
      • \n
    • names (list):\nlist of names that is assigned to the data according according\nto the order in the file list. Use careful, if you do not provide file names!
      • \n
    • files (list):\nlist which contains the filenames to be read. No automatic detection of\nfiles performed if given.
      • \n
    • plot_fit (bool):\nIf true, the fit for the extraction of w0 is shown together with the data.
      • \n
    • assume_thermalization (bool):\nIf True: If the first record divided by the distance between two measurements is larger than\n1, it is assumed that this is due to thermalization and the first measurement belongs\nto the first config (default).\nIf False: The config numbers are assumed to be traj_number // difference
      • \n
    \n\n
    Returns
    \n\n
      \n
    • w0 (Obs):\nExtracted w0
      • \n
    \n", "signature": "(\tpath,\tprefix,\tdtr_read,\txmin,\tspatial_extent,\tfit_range=5,\tpostfix='ms',\tc=0.3,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop": {"fullname": "pyerrors.input.openQCD.read_qtop", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop", "kind": "function", "doc": "

    Read the topologial charge based on openQCD gradient flow measurements.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • version (str):\nEither openQCD or sfqcd, depending on the data.
      • \n
    • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
      • \n
    • integer_charge (bool):\nIf True, the charge is rounded towards the nearest integer on each config.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (Obs):\nRead topological charge
      • \n
    \n", "signature": "(path, prefix, c, dtr_cnfg=1, version='openQCD', **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_gf_coupling": {"fullname": "pyerrors.input.openQCD.read_gf_coupling", "modulename": "pyerrors.input.openQCD", "qualname": "read_gf_coupling", "kind": "function", "doc": "

    Read the gradient flow coupling based on sfqcd gradient flow measurements. See 1607.06423 for details.

    \n\n

    Note: The current implementation only works for c=0.3 and T=L. The definition of the coupling in 1607.06423 requires projection to topological charge zero which is not done within this function but has to be performed in a separate step.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat.\nIgnored if file names are passed explicitly via keyword files.
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L.
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of measurements\nbetween two configs.\nIf it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • r_start (list):\nlist which contains the first config to be read for each replicum.
      • \n
    • r_stop (list):\nlist which contains the last config to be read for each replicum.
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length.
      • \n
    • postfix (str):\npostfix of the file to read, e.g. '.gfms.dat' for openQCD-files
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for the coupling. If False, the Wilson flow is used.
      • \n
    \n", "signature": "(path, prefix, c, dtr_cnfg=1, Zeuthen_flow=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.qtop_projection": {"fullname": "pyerrors.input.openQCD.qtop_projection", "modulename": "pyerrors.input.openQCD", "qualname": "qtop_projection", "kind": "function", "doc": "

    Returns the projection to the topological charge sector defined by target.

    \n\n
    Parameters
    \n\n
      \n
    • path (Obs):\nTopological charge.
      • \n
    • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
      • \n
    \n\n
    Returns
    \n\n
      \n
    • reto (Obs):\nprojection to the topological charge sector defined by target
      • \n
    \n", "signature": "(qtop, target=0):", "funcdef": "def"}, "pyerrors.input.openQCD.read_qtop_sector": {"fullname": "pyerrors.input.openQCD.read_qtop_sector", "modulename": "pyerrors.input.openQCD", "qualname": "read_qtop_sector", "kind": "function", "doc": "

    Constructs reweighting factors to a specified topological sector.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath of the measurement files
      • \n
    • prefix (str):\nprefix of the measurement files, e.g. _id0_r0.ms.dat
      • \n
    • c (double):\nSmearing radius in units of the lattice extent, c = sqrt(8 t0) / L
      • \n
    • target (int):\nSpecifies the topological sector to be reweighted to (default 0)
      • \n
    • dtr_cnfg (int):\n(optional) parameter that specifies the number of trajectories\nbetween two configs.\nif it is not set, the distance between two measurements\nin the file is assumed to be the distance between two configurations.
      • \n
    • steps (int):\n(optional) Distance between two configurations in units of trajectories /\n cycles. Assumed to be the distance between two measurements * dtr_cnfg if not given
      • \n
    • version (str):\nversion string of the openQCD (sfqcd) version used to create\nthe ensemble. Default is 2.0. May also be set to sfqcd.
      • \n
    • L (int):\nspatial length of the lattice in L/a.\nHAS to be set if version != sfqcd, since openQCD does not provide\nthis in the header
      • \n
    • r_start (list):\noffset of the first ensemble, making it easier to match\nlater on with other Obs
      • \n
    • r_stop (list):\nlast configurations that need to be read (per replicum)
      • \n
    • files (list):\nspecify the exact files that need to be read\nfrom path, practical if e.g. only one replicum is needed
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • Zeuthen_flow (bool):\n(optional) If True, the Zeuthen flow is used for Qtop. Only possible\nfor version=='sfqcd' If False, the Wilson flow is used.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • reto (Obs):\nprojection to the topological charge sector defined by target
      • \n
    \n", "signature": "(path, prefix, c, target=0, **kwargs):", "funcdef": "def"}, "pyerrors.input.openQCD.read_ms5_xsf": {"fullname": "pyerrors.input.openQCD.read_ms5_xsf", "modulename": "pyerrors.input.openQCD", "qualname": "read_ms5_xsf", "kind": "function", "doc": "

    Read data from files in the specified directory with the specified prefix and quark combination extension, and return a Corr object containing the data.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nThe directory to search for the files in.
      • \n
    • prefix (str):\nThe prefix to match the files against.
      • \n
    • qc (str):\nThe quark combination extension to match the files against.
      • \n
    • corr (str):\nThe correlator to extract data for.
      • \n
    • sep (str, optional):\nThe separator to use when parsing the replika names.
      • \n

    • **kwargs: Additional keyword arguments. The following keyword arguments are recognized:

      \n\n
        \n
      • names (List[str]): A list of names to use for the replicas.
        • \n
      • files (List[str]): A list of files to read data from.
        • \n
      • idl (List[List[int]]): A list of idls per replicum, resticting data to the idls given.
        • \n
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Corr: A complex valued Corr object containing the data read from the files. In case of boudary to bulk correlators.
      • \n
    • or
      • \n
    • CObs: A complex valued CObs object containing the data read from the files. In case of boudary to boundary correlators.
      • \n
    \n\n
    Raises
    \n\n
      \n
    • FileNotFoundError: If no files matching the specified prefix and quark combination extension are found in the specified directory.
      • \n
    • IOError: If there is an error reading a file.
      • \n
    • struct.error: If there is an error unpacking binary data.
      • \n
    \n", "signature": "(path, prefix, qc, corr, sep='r', **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas": {"fullname": "pyerrors.input.pandas", "modulename": "pyerrors.input.pandas", "kind": "module", "doc": "

    \n"}, "pyerrors.input.pandas.to_sql": {"fullname": "pyerrors.input.pandas.to_sql", "modulename": "pyerrors.input.pandas", "qualname": "to_sql", "kind": "function", "doc": "

    Write DataFrame including Obs or Corr valued columns to sqlite database.

    \n\n
    Parameters
    \n\n
      \n
    • df (pandas.DataFrame):\nDataframe to be written to the database.
      • \n
    • table_name (str):\nName of the table in the database.
      • \n
    • db (str):\nPath to the sqlite database.
      • \n
    • if exists (str):\nHow to behave if table already exists. Options 'fail', 'replace', 'append'.
      • \n
    • gz (bool):\nIf True the json strings are gzipped.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(df, table_name, db, if_exists='fail', gz=True, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.read_sql": {"fullname": "pyerrors.input.pandas.read_sql", "modulename": "pyerrors.input.pandas", "qualname": "read_sql", "kind": "function", "doc": "

    Execute SQL query on sqlite database and obtain DataFrame including Obs or Corr valued columns.

    \n\n
    Parameters
    \n\n
      \n
    • sql (str):\nSQL query to be executed.
      • \n
    • db (str):\nPath to the sqlite database.
      • \n
    • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
      • \n
    \n", "signature": "(sql, db, auto_gamma=False, **kwargs):", "funcdef": "def"}, "pyerrors.input.pandas.dump_df": {"fullname": "pyerrors.input.pandas.dump_df", "modulename": "pyerrors.input.pandas", "qualname": "dump_df", "kind": "function", "doc": "

    Exports a pandas DataFrame containing Obs valued columns to a (gzipped) csv file.

    \n\n

    Before making use of pandas to_csv functionality Obs objects are serialized via the standardized\njson format of pyerrors.

    \n\n
    Parameters
    \n\n
      \n
    • df (pandas.DataFrame):\nDataframe to be dumped to a file.
      • \n
    • fname (str):\nFilename of the output file.
      • \n
    • gz (bool):\nIf True, the output is a gzipped csv file. If False, the output is a csv file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(df, fname, gz=True):", "funcdef": "def"}, "pyerrors.input.pandas.load_df": {"fullname": "pyerrors.input.pandas.load_df", "modulename": "pyerrors.input.pandas", "qualname": "load_df", "kind": "function", "doc": "

    Imports a pandas DataFrame from a csv.(gz) file in which Obs objects are serialized as json strings.

    \n\n
    Parameters
    \n\n
      \n
    • fname (str):\nFilename of the input file.
      • \n
    • auto_gamma (bool):\nIf True applies the gamma_method to all imported Obs objects with the default parameters for\nthe error analysis. Default False.
      • \n
    • gz (bool):\nIf True, assumes that data is gzipped. If False, assumes JSON file.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • data (pandas.DataFrame):\nDataframe with the content of the sqlite database.
      • \n
    \n", "signature": "(fname, auto_gamma=False, gz=True):", "funcdef": "def"}, "pyerrors.input.sfcf": {"fullname": "pyerrors.input.sfcf", "modulename": "pyerrors.input.sfcf", "kind": "module", "doc": "

    \n"}, "pyerrors.input.sfcf.sep": {"fullname": "pyerrors.input.sfcf.sep", "modulename": "pyerrors.input.sfcf", "qualname": "sep", "kind": "variable", "doc": "

    \n", "default_value": "'/'"}, "pyerrors.input.sfcf.read_sfcf": {"fullname": "pyerrors.input.sfcf.read_sfcf", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf", "kind": "function", "doc": "

    Read sfcf files from given folder structure.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to the sfcf files.
      • \n
    • prefix (str):\nPrefix of the sfcf files.
      • \n
    • name (str):\nName of the correlation function to read.
      • \n
    • quarks (str):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
      • \n
    • corr_type (str):\nType of correlation function to read. Can be\n
        \n
      • 'bi' for boundary-inner
        • \n
      • 'bb' for boundary-boundary
        • \n
      • 'bib' for boundary-inner-boundary
        • \n
      • \n
    • noffset (int):\nOffset of the source (only relevant when wavefunctions are used)
      • \n
    • wf (int):\nID of wave function
      • \n
    • wf2 (int):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
      • \n
    • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • ens_name (str):\nreplaces the name of the ensemble
      • \n
    • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
      • \n
    • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
      • \n
    • replica (list):\nlist of replica to be read, default is all
      • \n
    • files (list):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
      • \n
    • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (list[Obs]):\nlist of Observables with length T, observable per timeslice.\nbb-type correlators have length 1.
      • \n
    \n", "signature": "(\tpath,\tprefix,\tname,\tquarks='.*',\tcorr_type='bi',\tnoffset=0,\twf=0,\twf2=0,\tversion='1.0c',\tcfg_separator='n',\tsilent=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.sfcf.read_sfcf_multi": {"fullname": "pyerrors.input.sfcf.read_sfcf_multi", "modulename": "pyerrors.input.sfcf", "qualname": "read_sfcf_multi", "kind": "function", "doc": "

    Read sfcf files from given folder structure.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nPath to the sfcf files.
      • \n
    • prefix (str):\nPrefix of the sfcf files.
      • \n
    • name (str):\nName of the correlation function to read.
      • \n
    • quarks_list (list[str]):\nLabel of the quarks used in the sfcf input file. e.g. \"quark quark\"\nfor version 0.0 this does NOT need to be given with the typical \" - \"\nthat is present in the output file,\nthis is done automatically for this version
      • \n
    • corr_type_list (list[str]):\nType of correlation function to read. Can be\n
        \n
      • 'bi' for boundary-inner
        • \n
      • 'bb' for boundary-boundary
        • \n
      • 'bib' for boundary-inner-boundary
        • \n
      • \n
    • noffset_list (list[int]):\nOffset of the source (only relevant when wavefunctions are used)
      • \n
    • wf_list (int):\nID of wave function
      • \n
    • wf2_list (list[int]):\nID of the second wavefunction\n(only relevant for boundary-to-boundary correlation functions)
      • \n
    • im (bool):\nif True, read imaginary instead of real part\nof the correlation function.
      • \n
    • names (list):\nAlternative labeling for replicas/ensembles.\nHas to have the appropriate length
      • \n
    • ens_name (str):\nreplaces the name of the ensemble
      • \n
    • version (str):\nversion of SFCF, with which the measurement was done.\nif the compact output option (-c) was specified,\nappend a \"c\" to the version (e.g. \"1.0c\")\nif the append output option (-a) was specified,\nappend an \"a\" to the version
      • \n
    • cfg_separator (str):\nString that separates the ensemble identifier from the configuration number (default 'n').
      • \n
    • replica (list):\nlist of replica to be read, default is all
      • \n
    • files (list):\nlist of files to be read per replica, default is all.\nfor non-compact output format, hand the folders to be read here.
      • \n
    • check_configs (list[list[int]]):\nlist of list of supposed configs, eg. [range(1,1000)]\nfor one replicum with 1000 configs
      • \n
    \n\n
    Returns
    \n\n
      \n
    • result (dict[list[Obs]]):\ndict with one of the following properties:\nif keyed_out:\n dict[key] = list[Obs]\n where key has the form name/quarks/offset/wf/wf2\nif not keyed_out:\n dict[name][quarks][offset][wf][wf2] = list[Obs]
      • \n
    \n", "signature": "(\tpath,\tprefix,\tname_list,\tquarks_list=['.*'],\tcorr_type_list=['bi'],\tnoffset_list=[0],\twf_list=[0],\twf2_list=[0],\tversion='1.0c',\tcfg_separator='n',\tsilent=False,\tkeyed_out=False,\t**kwargs):", "funcdef": "def"}, "pyerrors.input.utils": {"fullname": "pyerrors.input.utils", "modulename": "pyerrors.input.utils", "kind": "module", "doc": "

    Utilities for the input

    \n"}, "pyerrors.input.utils.sort_names": {"fullname": "pyerrors.input.utils.sort_names", "modulename": "pyerrors.input.utils", "qualname": "sort_names", "kind": "function", "doc": "

    Sorts a list of names of replika with searches for r and id in the replikum string.\nIf this search fails, a fallback method is used,\nwhere the strings are simply compared and the first diffeing numeral is used for differentiation.

    \n\n
    Parameters
    \n\n
      \n
    • ll (list):\nlist to sort
      • \n
    \n\n
    Returns
    \n\n
      \n
    • ll (list):\nsorted list
      • \n
    \n", "signature": "(ll):", "funcdef": "def"}, "pyerrors.input.utils.check_idl": {"fullname": "pyerrors.input.utils.check_idl", "modulename": "pyerrors.input.utils", "qualname": "check_idl", "kind": "function", "doc": "

    Checks if list of configurations is contained in an idl

    \n\n
    Parameters
    \n\n
      \n
    • idl (range or list):\nidl of the current replicum
      • \n
    • che (list):\nlist of configurations to be checked against
      • \n
    \n\n
    Returns
    \n\n
      \n
    • miss_str (str):\nstring with integers of which idls are missing
      • \n
    \n", "signature": "(idl, che):", "funcdef": "def"}, "pyerrors.input.utils.check_params": {"fullname": "pyerrors.input.utils.check_params", "modulename": "pyerrors.input.utils", "qualname": "check_params", "kind": "function", "doc": "

    Check if, for sfcf, the parameter hashes at the end of the parameter files are in fact the expected one.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\nmeasurement path, same as for sfcf read method
      • \n
    • param_hash (str):\nexpected parameter hash
      • \n
    • prefix (str):\ndata prefix to find the appropriate replicum folders in path
      • \n
    • param_prefix (str):\nprefix of the parameter file. Defaults to 'parameters_'
      • \n
    \n\n
    Returns
    \n\n
      \n
    • nums (dict):\ndictionary of faulty parameter files sorted by the replica paths
      • \n
    \n", "signature": "(path, param_hash, prefix, param_prefix='parameters_'):", "funcdef": "def"}, "pyerrors.integrate": {"fullname": "pyerrors.integrate", "modulename": "pyerrors.integrate", "kind": "module", "doc": "

    \n"}, "pyerrors.integrate.quad": {"fullname": "pyerrors.integrate.quad", "modulename": "pyerrors.integrate", "qualname": "quad", "kind": "function", "doc": "

    Performs a (one-dimensional) numeric integration of f(p, x) from a to b.

    \n\n

    The integration is performed using scipy.integrate.quad().\nAll parameters that can be passed to scipy.integrate.quad may also be passed to this function.\nThe output is the same as for scipy.integrate.quad, the first element being an Obs.

    \n\n
    Parameters
    \n\n
      \n

    • func (object):\nfunction to integrate, has to be of the form

      \n\n
      \n
      import autograd.numpy as anp\n\ndef func(p, x):\n   return p[0] + p[1] * x + p[2] * anp.sinh(x)\n
      \n
      \n\n

      where x is the integration variable.

      • \n
    • p (list of floats or Obs):\nparameters of the function func.
      • \n
    • a (float or Obs):\nLower limit of integration (use -numpy.inf for -infinity).
      • \n
    • b (float or Obs):\nUpper limit of integration (use -numpy.inf for -infinity).
      • \n
    • All parameters of scipy.integrate.quad
      • \n
    \n\n
    Returns
    \n\n
      \n
    • y (Obs):\nThe integral of func from a to b.
      • \n
    • abserr (float):\nAn estimate of the absolute error in the result.
      • \n
    • infodict (dict):\nA dictionary containing additional information.\nRun scipy.integrate.quad_explain() for more information.
      • \n
    • message: A convergence message.
      • \n
    • explain: Appended only with 'cos' or 'sin' weighting and infinite\nintegration limits, it contains an explanation of the codes in\ninfodict['ierlst']
      • \n
    \n", "signature": "(func, p, a, b, **kwargs):", "funcdef": "def"}, "pyerrors.linalg": {"fullname": "pyerrors.linalg", "modulename": "pyerrors.linalg", "kind": "module", "doc": "

    \n"}, "pyerrors.linalg.matmul": {"fullname": "pyerrors.linalg.matmul", "modulename": "pyerrors.linalg", "qualname": "matmul", "kind": "function", "doc": "

    Matrix multiply all operands.

    \n\n
    Parameters
    \n\n
      \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    • This implementation is faster compared to standard multiplication via the @ operator.
      • \n
    \n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.jack_matmul": {"fullname": "pyerrors.linalg.jack_matmul", "modulename": "pyerrors.linalg", "qualname": "jack_matmul", "kind": "function", "doc": "

    Matrix multiply both operands making use of the jackknife approximation.

    \n\n
    Parameters
    \n\n
      \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    • For large matrices this is considerably faster compared to matmul.
      • \n
    \n", "signature": "(*operands):", "funcdef": "def"}, "pyerrors.linalg.einsum": {"fullname": "pyerrors.linalg.einsum", "modulename": "pyerrors.linalg", "qualname": "einsum", "kind": "function", "doc": "

    Wrapper for numpy.einsum

    \n\n
    Parameters
    \n\n
      \n
    • subscripts (str):\nSubscripts for summation (see numpy documentation for details)
      • \n
    • operands (numpy.ndarray):\nArbitrary number of 2d-numpy arrays which can be real or complex\nObs valued.
      • \n
    \n", "signature": "(subscripts, *operands):", "funcdef": "def"}, "pyerrors.linalg.inv": {"fullname": "pyerrors.linalg.inv", "modulename": "pyerrors.linalg", "qualname": "inv", "kind": "function", "doc": "

    Inverse of Obs or CObs valued matrices.

    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.cholesky": {"fullname": "pyerrors.linalg.cholesky", "modulename": "pyerrors.linalg", "qualname": "cholesky", "kind": "function", "doc": "

    Cholesky decomposition of Obs valued matrices.

    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.det": {"fullname": "pyerrors.linalg.det", "modulename": "pyerrors.linalg", "qualname": "det", "kind": "function", "doc": "

    Determinant of Obs valued matrices.

    \n", "signature": "(x):", "funcdef": "def"}, "pyerrors.linalg.eigh": {"fullname": "pyerrors.linalg.eigh", "modulename": "pyerrors.linalg", "qualname": "eigh", "kind": "function", "doc": "

    Computes the eigenvalues and eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.

    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eig": {"fullname": "pyerrors.linalg.eig", "modulename": "pyerrors.linalg", "qualname": "eig", "kind": "function", "doc": "

    Computes the eigenvalues of a given matrix of Obs according to np.linalg.eig.

    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.eigv": {"fullname": "pyerrors.linalg.eigv", "modulename": "pyerrors.linalg", "qualname": "eigv", "kind": "function", "doc": "

    Computes the eigenvectors of a given hermitian matrix of Obs according to np.linalg.eigh.

    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.pinv": {"fullname": "pyerrors.linalg.pinv", "modulename": "pyerrors.linalg", "qualname": "pinv", "kind": "function", "doc": "

    Computes the Moore-Penrose pseudoinverse of a matrix of Obs.

    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.linalg.svd": {"fullname": "pyerrors.linalg.svd", "modulename": "pyerrors.linalg", "qualname": "svd", "kind": "function", "doc": "

    Computes the singular value decomposition of a matrix of Obs.

    \n", "signature": "(obs, **kwargs):", "funcdef": "def"}, "pyerrors.misc": {"fullname": "pyerrors.misc", "modulename": "pyerrors.misc", "kind": "module", "doc": "

    \n"}, "pyerrors.misc.print_config": {"fullname": "pyerrors.misc.print_config", "modulename": "pyerrors.misc", "qualname": "print_config", "kind": "function", "doc": "

    Print information about version of python, pyerrors and dependencies.

    \n", "signature": "():", "funcdef": "def"}, "pyerrors.misc.errorbar": {"fullname": "pyerrors.misc.errorbar", "modulename": "pyerrors.misc", "qualname": "errorbar", "kind": "function", "doc": "

    pyerrors wrapper for the errorbars method of matplotlib

    \n\n
    Parameters
    \n\n
      \n
    • x (list):\nA list of x-values which can be Obs.
      • \n
    • y (list):\nA list of y-values which can be Obs.
      • \n
    • axes ((matplotlib.pyplot.axes)):\nThe axes to plot on. default is plt.
      • \n
    \n", "signature": "(\tx,\ty,\taxes=<module 'matplotlib.pyplot' from '/opt/hostedtoolcache/Python/3.10.13/x64/lib/python3.10/site-packages/matplotlib/pyplot.py'>,\t**kwargs):", "funcdef": "def"}, "pyerrors.misc.dump_object": {"fullname": "pyerrors.misc.dump_object", "modulename": "pyerrors.misc", "qualname": "dump_object", "kind": "function", "doc": "

    Dump object into pickle file.

    \n\n
    Parameters
    \n\n
      \n
    • obj (object):\nobject to be saved in the pickle file
      • \n
    • name (str):\nname of the file
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n\n
    Returns
    \n\n
      \n
    • None
      • \n
    \n", "signature": "(obj, name, **kwargs):", "funcdef": "def"}, "pyerrors.misc.load_object": {"fullname": "pyerrors.misc.load_object", "modulename": "pyerrors.misc", "qualname": "load_object", "kind": "function", "doc": "

    Load object from pickle file.

    \n\n
    Parameters
    \n\n
      \n
    • path (str):\npath to the file
      • \n
    \n\n
    Returns
    \n\n
      \n
    • object (Obs):\nLoaded Object
      • \n
    \n", "signature": "(path):", "funcdef": "def"}, "pyerrors.misc.pseudo_Obs": {"fullname": "pyerrors.misc.pseudo_Obs", "modulename": "pyerrors.misc", "qualname": "pseudo_Obs", "kind": "function", "doc": "

    Generate an Obs object with given value, dvalue and name for test purposes

    \n\n
    Parameters
    \n\n
      \n
    • value (float):\ncentral value of the Obs to be generated.
      • \n
    • dvalue (float):\nerror of the Obs to be generated.
      • \n
    • name (str):\nname of the ensemble for which the Obs is to be generated.
      • \n
    • samples (int):\nnumber of samples for the Obs (default 1000).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (Obs):\nGenerated Observable
      • \n
    \n", "signature": "(value, dvalue, name, samples=1000):", "funcdef": "def"}, "pyerrors.misc.gen_correlated_data": {"fullname": "pyerrors.misc.gen_correlated_data", "modulename": "pyerrors.misc", "qualname": "gen_correlated_data", "kind": "function", "doc": "

    Generate observables with given covariance and autocorrelation times.

    \n\n
    Parameters
    \n\n
      \n
    • means (list):\nlist containing the mean value of each observable.
      • \n
    • cov (numpy.ndarray):\ncovariance matrix for the data to be generated.
      • \n
    • name (str):\nensemble name for the data to be geneated.
      • \n
    • tau (float or list):\ncan either be a real number or a list with an entry for\nevery dataset.
      • \n
    • samples (int):\nnumber of samples to be generated for each observable.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • corr_obs (list[Obs]):\nGenerated observable list
      • \n
    \n", "signature": "(means, cov, name, tau=0.5, samples=1000):", "funcdef": "def"}, "pyerrors.mpm": {"fullname": "pyerrors.mpm", "modulename": "pyerrors.mpm", "kind": "module", "doc": "

    \n"}, "pyerrors.mpm.matrix_pencil_method": {"fullname": "pyerrors.mpm.matrix_pencil_method", "modulename": "pyerrors.mpm", "qualname": "matrix_pencil_method", "kind": "function", "doc": "

    Matrix pencil method to extract k energy levels from data

    \n\n

    Implementation of the matrix pencil method based on\neq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)

    \n\n
    Parameters
    \n\n
      \n
    • data (list):\ncan be a list of Obs for the analysis of a single correlator, or a list of lists\nof Obs if several correlators are to analyzed at once.
      • \n
    • k (int):\nNumber of states to extract (default 1).
      • \n
    • p (int):\nmatrix pencil parameter which filters noise. The optimal value is expected between\nlen(data)/3 and 2*len(data)/3. The computation is more expensive the closer p is\nto len(data)/2 but could possibly suppress more noise (default len(data)//2).
      • \n
    \n\n
    Returns
    \n\n
      \n
    • energy_levels (list[Obs]):\nExtracted energy levels
      • \n
    \n", "signature": "(corrs, k=1, p=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs": {"fullname": "pyerrors.obs", "modulename": "pyerrors.obs", "kind": "module", "doc": "

    \n"}, "pyerrors.obs.Obs": {"fullname": "pyerrors.obs.Obs", "modulename": "pyerrors.obs", "qualname": "Obs", "kind": "class", "doc": "

    Class for a general observable.

    \n\n

    Instances of Obs are the basic objects of a pyerrors error analysis.\nThey are initialized with a list which contains arrays of samples for\ndifferent ensembles/replica and another list of same length which contains\nthe names of the ensembles/replica. Mathematical operations can be\nperformed on instances. The result is another instance of Obs. The error of\nan instance can be computed with the gamma_method. Also contains additional\nmethods for output and visualization of the error calculation.

    \n\n
    Attributes
    \n\n
      \n
    • S_global (float):\nStandard value for S (default 2.0)
      • \n
    • S_dict (dict):\nDictionary for S values. If an entry for a given ensemble\nexists this overwrites the standard value for that ensemble.
      • \n
    • tau_exp_global (float):\nStandard value for tau_exp (default 0.0)
      • \n
    • tau_exp_dict (dict):\nDictionary for tau_exp values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
      • \n
    • N_sigma_global (float):\nStandard value for N_sigma (default 1.0)
      • \n
    • N_sigma_dict (dict):\nDictionary for N_sigma values. If an entry for a given ensemble exists\nthis overwrites the standard value for that ensemble.
      • \n
    \n"}, "pyerrors.obs.Obs.__init__": {"fullname": "pyerrors.obs.Obs.__init__", "modulename": "pyerrors.obs", "qualname": "Obs.__init__", "kind": "function", "doc": "

    Initialize Obs object.

    \n\n
    Parameters
    \n\n
      \n
    • samples (list):\nlist of numpy arrays containing the Monte Carlo samples
      • \n
    • names (list):\nlist of strings labeling the individual samples
      • \n
    • idl (list, optional):\nlist of ranges or lists on which the samples are defined
      • \n
    \n", "signature": "(samples, names, idl=None, **kwargs)"}, "pyerrors.obs.Obs.S_global": {"fullname": "pyerrors.obs.Obs.S_global", "modulename": "pyerrors.obs", "qualname": "Obs.S_global", "kind": "variable", "doc": "

    \n", "default_value": "2.0"}, "pyerrors.obs.Obs.S_dict": {"fullname": "pyerrors.obs.Obs.S_dict", "modulename": "pyerrors.obs", "qualname": "Obs.S_dict", "kind": "variable", "doc": "

    \n", "default_value": "{}"}, "pyerrors.obs.Obs.tau_exp_global": {"fullname": "pyerrors.obs.Obs.tau_exp_global", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_global", "kind": "variable", "doc": "

    \n", "default_value": "0.0"}, "pyerrors.obs.Obs.tau_exp_dict": {"fullname": "pyerrors.obs.Obs.tau_exp_dict", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp_dict", "kind": "variable", "doc": "

    \n", "default_value": "{}"}, "pyerrors.obs.Obs.N_sigma_global": {"fullname": "pyerrors.obs.Obs.N_sigma_global", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_global", "kind": "variable", "doc": "

    \n", "default_value": "1.0"}, "pyerrors.obs.Obs.N_sigma_dict": {"fullname": "pyerrors.obs.Obs.N_sigma_dict", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma_dict", "kind": "variable", "doc": "

    \n", "default_value": "{}"}, "pyerrors.obs.Obs.names": {"fullname": "pyerrors.obs.Obs.names", "modulename": "pyerrors.obs", "qualname": "Obs.names", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.shape": {"fullname": "pyerrors.obs.Obs.shape", "modulename": "pyerrors.obs", "qualname": "Obs.shape", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.r_values": {"fullname": "pyerrors.obs.Obs.r_values", "modulename": "pyerrors.obs", "qualname": "Obs.r_values", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.deltas": {"fullname": "pyerrors.obs.Obs.deltas", "modulename": "pyerrors.obs", "qualname": "Obs.deltas", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.N": {"fullname": "pyerrors.obs.Obs.N", "modulename": "pyerrors.obs", "qualname": "Obs.N", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.idl": {"fullname": "pyerrors.obs.Obs.idl", "modulename": "pyerrors.obs", "qualname": "Obs.idl", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.ddvalue": {"fullname": "pyerrors.obs.Obs.ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.ddvalue", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.reweighted": {"fullname": "pyerrors.obs.Obs.reweighted", "modulename": "pyerrors.obs", "qualname": "Obs.reweighted", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.tag": {"fullname": "pyerrors.obs.Obs.tag", "modulename": "pyerrors.obs", "qualname": "Obs.tag", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.value": {"fullname": "pyerrors.obs.Obs.value", "modulename": "pyerrors.obs", "qualname": "Obs.value", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.dvalue": {"fullname": "pyerrors.obs.Obs.dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.dvalue", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_names": {"fullname": "pyerrors.obs.Obs.e_names", "modulename": "pyerrors.obs", "qualname": "Obs.e_names", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.cov_names": {"fullname": "pyerrors.obs.Obs.cov_names", "modulename": "pyerrors.obs", "qualname": "Obs.cov_names", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.mc_names": {"fullname": "pyerrors.obs.Obs.mc_names", "modulename": "pyerrors.obs", "qualname": "Obs.mc_names", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_content": {"fullname": "pyerrors.obs.Obs.e_content", "modulename": "pyerrors.obs", "qualname": "Obs.e_content", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.covobs": {"fullname": "pyerrors.obs.Obs.covobs", "modulename": "pyerrors.obs", "qualname": "Obs.covobs", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.gamma_method": {"fullname": "pyerrors.obs.Obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "Obs.gamma_method", "kind": "function", "doc": "

    Estimate the error and related properties of the Obs.

    \n\n
    Parameters
    \n\n
      \n
    • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
      • \n
    • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
      • \n
    • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
      • \n
    • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
      • \n
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.gm": {"fullname": "pyerrors.obs.Obs.gm", "modulename": "pyerrors.obs", "qualname": "Obs.gm", "kind": "function", "doc": "

    Estimate the error and related properties of the Obs.

    \n\n
    Parameters
    \n\n
      \n
    • S (float):\nspecifies a custom value for the parameter S (default 2.0).\nIf set to 0 it is assumed that the data exhibits no\nautocorrelation. In this case the error estimates coincides\nwith the sample standard error.
      • \n
    • tau_exp (float):\npositive value triggers the critical slowing down analysis\n(default 0.0).
      • \n
    • N_sigma (float):\nnumber of standard deviations from zero until the tail is\nattached to the autocorrelation function (default 1).
      • \n
    • fft (bool):\ndetermines whether the fft algorithm is used for the computation\nof the autocorrelation function (default True)
      • \n
    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.details": {"fullname": "pyerrors.obs.Obs.details", "modulename": "pyerrors.obs", "qualname": "Obs.details", "kind": "function", "doc": "

    Output detailed properties of the Obs.

    \n\n
    Parameters
    \n\n
      \n
    • ens_content (bool):\nprint details about the ensembles and replica if true.
      • \n
    \n", "signature": "(self, ens_content=True):", "funcdef": "def"}, "pyerrors.obs.Obs.reweight": {"fullname": "pyerrors.obs.Obs.reweight", "modulename": "pyerrors.obs", "qualname": "Obs.reweight", "kind": "function", "doc": "

    Reweight the obs with given rewighting factors.

    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
      • \n
    \n", "signature": "(self, weight):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero_within_error": {"fullname": "pyerrors.obs.Obs.is_zero_within_error", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero_within_error", "kind": "function", "doc": "

    Checks whether the observable is zero within 'sigma' standard errors.

    \n\n
    Parameters
    \n\n
      \n
    • sigma (int):\nNumber of standard errors used for the check.
      • \n
    • Works only properly when the gamma method was run.
      • \n
    \n", "signature": "(self, sigma=1):", "funcdef": "def"}, "pyerrors.obs.Obs.is_zero": {"fullname": "pyerrors.obs.Obs.is_zero", "modulename": "pyerrors.obs", "qualname": "Obs.is_zero", "kind": "function", "doc": "

    Checks whether the observable is zero within a given tolerance.

    \n\n
    Parameters
    \n\n
      \n
    • atol (float):\nAbsolute tolerance (for details see numpy documentation).
      • \n
    \n", "signature": "(self, atol=1e-10):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_tauint": {"fullname": "pyerrors.obs.Obs.plot_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.plot_tauint", "kind": "function", "doc": "

    Plot integrated autocorrelation time for each ensemble.

    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rho": {"fullname": "pyerrors.obs.Obs.plot_rho", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rho", "kind": "function", "doc": "

    Plot normalized autocorrelation function time for each ensemble.

    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_rep_dist": {"fullname": "pyerrors.obs.Obs.plot_rep_dist", "modulename": "pyerrors.obs", "qualname": "Obs.plot_rep_dist", "kind": "function", "doc": "

    Plot replica distribution for each ensemble with more than one replicum.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_history": {"fullname": "pyerrors.obs.Obs.plot_history", "modulename": "pyerrors.obs", "qualname": "Obs.plot_history", "kind": "function", "doc": "

    Plot derived Monte Carlo history for each ensemble

    \n\n
    Parameters
    \n\n
      \n
    • expand (bool):\nshow expanded history for irregular Monte Carlo chains (default: True).
      • \n
    \n", "signature": "(self, expand=True):", "funcdef": "def"}, "pyerrors.obs.Obs.plot_piechart": {"fullname": "pyerrors.obs.Obs.plot_piechart", "modulename": "pyerrors.obs", "qualname": "Obs.plot_piechart", "kind": "function", "doc": "

    Plot piechart which shows the fractional contribution of each\nensemble to the error and returns a dictionary containing the fractions.

    \n\n
    Parameters
    \n\n
      \n
    • save (str):\nsaves the figure to a file named 'save' if.
      • \n
    \n", "signature": "(self, save=None):", "funcdef": "def"}, "pyerrors.obs.Obs.dump": {"fullname": "pyerrors.obs.Obs.dump", "modulename": "pyerrors.obs", "qualname": "Obs.dump", "kind": "function", "doc": "

    Dump the Obs to a file 'name' of chosen format.

    \n\n
    Parameters
    \n\n
      \n
    • filename (str):\nname of the file to be saved.
      • \n
    • datatype (str):\nFormat of the exported file. Supported formats include\n\"json.gz\" and \"pickle\"
      • \n
    • description (str):\nDescription for output file, only relevant for json.gz format.
      • \n
    • path (str):\nspecifies a custom path for the file (default '.')
      • \n
    \n", "signature": "(self, filename, datatype='json.gz', description='', **kwargs):", "funcdef": "def"}, "pyerrors.obs.Obs.export_jackknife": {"fullname": "pyerrors.obs.Obs.export_jackknife", "modulename": "pyerrors.obs", "qualname": "Obs.export_jackknife", "kind": "function", "doc": "

    Export jackknife samples from the Obs

    \n\n
    Returns
    \n\n
      \n
    • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N jackknife samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived jackknife samples\nshould agree with samples from a full jackknife analysis up to O(1/N).
      • \n
    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.export_bootstrap": {"fullname": "pyerrors.obs.Obs.export_bootstrap", "modulename": "pyerrors.obs", "qualname": "Obs.export_bootstrap", "kind": "function", "doc": "

    Export bootstrap samples from the Obs

    \n\n
    Parameters
    \n\n
      \n
    • samples (int):\nNumber of bootstrap samples to generate.
      • \n
    • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples.\nIf not provided the bootstrap samples are generated bashed on the md5 hash of the enesmble name.
      • \n
    • save_rng (str):\nSave the random numbers to a file if a path is specified.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • numpy.ndarray: Returns a numpy array of length N + 1 where N is the number of samples\nfor the given ensemble and replicum. The zeroth entry of the array contains\nthe mean value of the Obs, entries 1 to N contain the N import_bootstrap samples\nderived from the Obs. The current implementation only works for observables\ndefined on exactly one ensemble and replicum. The derived bootstrap samples\nshould agree with samples from a full bootstrap analysis up to O(1/N).
      • \n
    \n", "signature": "(self, samples=500, random_numbers=None, save_rng=None):", "funcdef": "def"}, "pyerrors.obs.Obs.sqrt": {"fullname": "pyerrors.obs.Obs.sqrt", "modulename": "pyerrors.obs", "qualname": "Obs.sqrt", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.log": {"fullname": "pyerrors.obs.Obs.log", "modulename": "pyerrors.obs", "qualname": "Obs.log", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.exp": {"fullname": "pyerrors.obs.Obs.exp", "modulename": "pyerrors.obs", "qualname": "Obs.exp", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sin": {"fullname": "pyerrors.obs.Obs.sin", "modulename": "pyerrors.obs", "qualname": "Obs.sin", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cos": {"fullname": "pyerrors.obs.Obs.cos", "modulename": "pyerrors.obs", "qualname": "Obs.cos", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tan": {"fullname": "pyerrors.obs.Obs.tan", "modulename": "pyerrors.obs", "qualname": "Obs.tan", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsin": {"fullname": "pyerrors.obs.Obs.arcsin", "modulename": "pyerrors.obs", "qualname": "Obs.arcsin", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccos": {"fullname": "pyerrors.obs.Obs.arccos", "modulename": "pyerrors.obs", "qualname": "Obs.arccos", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctan": {"fullname": "pyerrors.obs.Obs.arctan", "modulename": "pyerrors.obs", "qualname": "Obs.arctan", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.sinh": {"fullname": "pyerrors.obs.Obs.sinh", "modulename": "pyerrors.obs", "qualname": "Obs.sinh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.cosh": {"fullname": "pyerrors.obs.Obs.cosh", "modulename": "pyerrors.obs", "qualname": "Obs.cosh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.tanh": {"fullname": "pyerrors.obs.Obs.tanh", "modulename": "pyerrors.obs", "qualname": "Obs.tanh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arcsinh": {"fullname": "pyerrors.obs.Obs.arcsinh", "modulename": "pyerrors.obs", "qualname": "Obs.arcsinh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arccosh": {"fullname": "pyerrors.obs.Obs.arccosh", "modulename": "pyerrors.obs", "qualname": "Obs.arccosh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.arctanh": {"fullname": "pyerrors.obs.Obs.arctanh", "modulename": "pyerrors.obs", "qualname": "Obs.arctanh", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.Obs.N_sigma": {"fullname": "pyerrors.obs.Obs.N_sigma", "modulename": "pyerrors.obs", "qualname": "Obs.N_sigma", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.S": {"fullname": "pyerrors.obs.Obs.S", "modulename": "pyerrors.obs", "qualname": "Obs.S", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_ddvalue": {"fullname": "pyerrors.obs.Obs.e_ddvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_ddvalue", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_drho": {"fullname": "pyerrors.obs.Obs.e_drho", "modulename": "pyerrors.obs", "qualname": "Obs.e_drho", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_dtauint": {"fullname": "pyerrors.obs.Obs.e_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_dtauint", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_dvalue": {"fullname": "pyerrors.obs.Obs.e_dvalue", "modulename": "pyerrors.obs", "qualname": "Obs.e_dvalue", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_n_dtauint": {"fullname": "pyerrors.obs.Obs.e_n_dtauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_dtauint", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_n_tauint": {"fullname": "pyerrors.obs.Obs.e_n_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_n_tauint", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_rho": {"fullname": "pyerrors.obs.Obs.e_rho", "modulename": "pyerrors.obs", "qualname": "Obs.e_rho", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_tauint": {"fullname": "pyerrors.obs.Obs.e_tauint", "modulename": "pyerrors.obs", "qualname": "Obs.e_tauint", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.e_windowsize": {"fullname": "pyerrors.obs.Obs.e_windowsize", "modulename": "pyerrors.obs", "qualname": "Obs.e_windowsize", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.Obs.tau_exp": {"fullname": "pyerrors.obs.Obs.tau_exp", "modulename": "pyerrors.obs", "qualname": "Obs.tau_exp", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.CObs": {"fullname": "pyerrors.obs.CObs", "modulename": "pyerrors.obs", "qualname": "CObs", "kind": "class", "doc": "

    Class for a complex valued observable.

    \n"}, "pyerrors.obs.CObs.__init__": {"fullname": "pyerrors.obs.CObs.__init__", "modulename": "pyerrors.obs", "qualname": "CObs.__init__", "kind": "function", "doc": "

    \n", "signature": "(real, imag=0.0)"}, "pyerrors.obs.CObs.tag": {"fullname": "pyerrors.obs.CObs.tag", "modulename": "pyerrors.obs", "qualname": "CObs.tag", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.CObs.real": {"fullname": "pyerrors.obs.CObs.real", "modulename": "pyerrors.obs", "qualname": "CObs.real", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.CObs.imag": {"fullname": "pyerrors.obs.CObs.imag", "modulename": "pyerrors.obs", "qualname": "CObs.imag", "kind": "variable", "doc": "

    \n"}, "pyerrors.obs.CObs.gamma_method": {"fullname": "pyerrors.obs.CObs.gamma_method", "modulename": "pyerrors.obs", "qualname": "CObs.gamma_method", "kind": "function", "doc": "

    Executes the gamma_method for the real and the imaginary part.

    \n", "signature": "(self, **kwargs):", "funcdef": "def"}, "pyerrors.obs.CObs.is_zero": {"fullname": "pyerrors.obs.CObs.is_zero", "modulename": "pyerrors.obs", "qualname": "CObs.is_zero", "kind": "function", "doc": "

    Checks whether both real and imaginary part are zero within machine precision.

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.CObs.conjugate": {"fullname": "pyerrors.obs.CObs.conjugate", "modulename": "pyerrors.obs", "qualname": "CObs.conjugate", "kind": "function", "doc": "

    \n", "signature": "(self):", "funcdef": "def"}, "pyerrors.obs.gamma_method": {"fullname": "pyerrors.obs.gamma_method", "modulename": "pyerrors.obs", "qualname": "gamma_method", "kind": "function", "doc": "

    Vectorized version of the gamma_method applicable to lists or arrays of Obs.

    \n\n

    See docstring of pe.Obs.gamma_method for details.

    \n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.gm": {"fullname": "pyerrors.obs.gm", "modulename": "pyerrors.obs", "qualname": "gm", "kind": "function", "doc": "

    Vectorized version of the gamma_method applicable to lists or arrays of Obs.

    \n\n

    See docstring of pe.Obs.gamma_method for details.

    \n", "signature": "(x, **kwargs):", "funcdef": "def"}, "pyerrors.obs.derived_observable": {"fullname": "pyerrors.obs.derived_observable", "modulename": "pyerrors.obs", "qualname": "derived_observable", "kind": "function", "doc": "

    Construct a derived Obs according to func(data, **kwargs) using automatic differentiation.

    \n\n
    Parameters
    \n\n
      \n
    • func (object):\narbitrary function of the form func(data, **kwargs). For the\nautomatic differentiation to work, all numpy functions have to have\nthe autograd wrapper (use 'import autograd.numpy as anp').
      • \n
    • data (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
      • \n
    • num_grad (bool):\nif True, numerical derivatives are used instead of autograd\n(default False). To control the numerical differentiation the\nkwargs of numdifftools.step_generators.MaxStepGenerator\ncan be used.
      • \n
    • man_grad (list):\nmanually supply a list or an array which contains the jacobian\nof func. Use cautiously, supplying the wrong derivative will\nnot be intercepted.
      • \n
    \n\n
    Notes
    \n\n

    For simple mathematical operations it can be practical to use anonymous\nfunctions. For the ratio of two observables one can e.g. use

    \n\n

    new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2])

    \n", "signature": "(func, data, array_mode=False, **kwargs):", "funcdef": "def"}, "pyerrors.obs.reweight": {"fullname": "pyerrors.obs.reweight", "modulename": "pyerrors.obs", "qualname": "reweight", "kind": "function", "doc": "

    Reweight a list of observables.

    \n\n
    Parameters
    \n\n
      \n
    • weight (Obs):\nReweighting factor. An Observable that has to be defined on a superset of the\nconfigurations in obs[i].idl for all i.
      • \n
    • obs (list):\nlist of Obs, e.g. [obs1, obs2, obs3].
      • \n
    • all_configs (bool):\nif True, the reweighted observables are normalized by the average of\nthe reweighting factor on all configurations in weight.idl and not\non the configurations in obs[i].idl. Default False.
      • \n
    \n", "signature": "(weight, obs, **kwargs):", "funcdef": "def"}, "pyerrors.obs.correlate": {"fullname": "pyerrors.obs.correlate", "modulename": "pyerrors.obs", "qualname": "correlate", "kind": "function", "doc": "

    Correlate two observables.

    \n\n
    Parameters
    \n\n
      \n
    • obs_a (Obs):\nFirst observable
      • \n
    • obs_b (Obs):\nSecond observable
      • \n
    \n\n
    Notes
    \n\n

    Keep in mind to only correlate primary observables which have not been reweighted\nyet. The reweighting has to be applied after correlating the observables.\nCurrently only works if ensembles are identical (this is not strictly necessary).

    \n", "signature": "(obs_a, obs_b):", "funcdef": "def"}, "pyerrors.obs.covariance": {"fullname": "pyerrors.obs.covariance", "modulename": "pyerrors.obs", "qualname": "covariance", "kind": "function", "doc": "

    Calculates the error covariance matrix of a set of observables.

    \n\n

    WARNING: This function should be used with care, especially for observables with support on multiple\n ensembles with differing autocorrelations. See the notes below for details.

    \n\n

    The gamma method has to be applied first to all observables.

    \n\n
    Parameters
    \n\n
      \n
    • obs (list or numpy.ndarray):\nList or one dimensional array of Obs
      • \n
    • visualize (bool):\nIf True plots the corresponding normalized correlation matrix (default False).
      • \n
    • correlation (bool):\nIf True the correlation matrix instead of the error covariance matrix is returned (default False).
      • \n
    • smooth (None or int):\nIf smooth is an integer 'E' between 2 and the dimension of the matrix minus 1 the eigenvalue\nsmoothing procedure of hep-lat/9412087 is applied to the correlation matrix which leaves the\nlargest E eigenvalues essentially unchanged and smoothes the smaller eigenvalues to avoid extremely\nsmall ones.
      • \n
    \n\n
    Notes
    \n\n

    The error covariance is defined such that it agrees with the squared standard error for two identical observables\n$$\\operatorname{cov}(a,a)=\\sum_{s=1}^N\\delta_a^s\\delta_a^s/N^2=\\Gamma_{aa}(0)/N=\\operatorname{var}(a)/N=\\sigma_a^2$$\nin the absence of autocorrelation.\nThe error covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. The covariance at windowsize 0 is guaranteed to be positive semi-definite\n$$\\sum_{i,j}v_i\\Gamma_{ij}(0)v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i,j}v_i\\delta_i^s\\delta_j^s v_j=\\frac{1}{N}\\sum_{s=1}^N\\sum_{i}|v_i\\delta_i^s|^2\\geq 0\\,,$$ for every $v\\in\\mathbb{R}^M$, while such an identity does not hold for larger windows/lags.\nFor observables defined on a single ensemble our approximation is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.\n$$\\tau_{\\mathrm{int}, ij}=\\sqrt{\\tau_{\\mathrm{int}, i}\\times \\tau_{\\mathrm{int}, j}}$$\nThis construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).

    \n", "signature": "(obs, visualize=False, correlation=False, smooth=None, **kwargs):", "funcdef": "def"}, "pyerrors.obs.import_jackknife": {"fullname": "pyerrors.obs.import_jackknife", "modulename": "pyerrors.obs", "qualname": "import_jackknife", "kind": "function", "doc": "

    Imports jackknife samples and returns an Obs

    \n\n
    Parameters
    \n\n
      \n
    • jacks (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N jackknife samples as first to Nth entry.
      • \n
    • name (str):\nname of the ensemble the samples are defined on.
      • \n
    \n", "signature": "(jacks, name, idl=None):", "funcdef": "def"}, "pyerrors.obs.import_bootstrap": {"fullname": "pyerrors.obs.import_bootstrap", "modulename": "pyerrors.obs", "qualname": "import_bootstrap", "kind": "function", "doc": "

    Imports bootstrap samples and returns an Obs

    \n\n
    Parameters
    \n\n
      \n
    • boots (numpy.ndarray):\nnumpy array containing the mean value as zeroth entry and\nthe N bootstrap samples as first to Nth entry.
      • \n
    • name (str):\nname of the ensemble the samples are defined on.
      • \n
    • random_numbers (np.ndarray):\nArray of shape (samples, length) containing the random numbers to generate the bootstrap samples,\nwhere samples is the number of bootstrap samples and length is the length of the original Monte Carlo\nchain to be reconstructed.
      • \n
    \n", "signature": "(boots, name, random_numbers):", "funcdef": "def"}, "pyerrors.obs.merge_obs": {"fullname": "pyerrors.obs.merge_obs", "modulename": "pyerrors.obs", "qualname": "merge_obs", "kind": "function", "doc": "

    Combine all observables in list_of_obs into one new observable

    \n\n
    Parameters
    \n\n
      \n
    • list_of_obs (list):\nlist of the Obs object to be combined
      • \n
    \n\n
    Notes
    \n\n

    It is not possible to combine obs which are based on the same replicum

    \n", "signature": "(list_of_obs):", "funcdef": "def"}, "pyerrors.obs.cov_Obs": {"fullname": "pyerrors.obs.cov_Obs", "modulename": "pyerrors.obs", "qualname": "cov_Obs", "kind": "function", "doc": "

    Create an Obs based on mean(s) and a covariance matrix

    \n\n
    Parameters
    \n\n
      \n
    • mean (list of floats or float):\nN mean value(s) of the new Obs
      • \n
    • cov (list or array):\n2d (NxN) Covariance matrix, 1d diagonal entries or 0d covariance
      • \n
    • name (str):\nidentifier for the covariance matrix
      • \n
    • grad (list or array):\nGradient of the Covobs wrt. the means belonging to cov.
      • \n
    \n", "signature": "(means, cov, name, grad=None):", "funcdef": "def"}, "pyerrors.roots": {"fullname": "pyerrors.roots", "modulename": "pyerrors.roots", "kind": "module", "doc": "

    \n"}, "pyerrors.roots.find_root": {"fullname": "pyerrors.roots.find_root", "modulename": "pyerrors.roots", "qualname": "find_root", "kind": "function", "doc": "

    Finds the root of the function func(x, d) where d is an Obs.

    \n\n
    Parameters
    \n\n
      \n
    • d (Obs):\nObs passed to the function.
      • \n

    • func (object):\nFunction to be minimized. Any numpy functions have to use the autograd.numpy wrapper.\nExample:

      \n\n
      \n
      import autograd.numpy as anp\ndef root_func(x, d):\n   return anp.exp(-x ** 2) - d\n
      \n
      • \n

    • guess (float):\nInitial guess for the minimization.

      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (Obs):\nObs valued root of the function.
      • \n
    \n", "signature": "(d, func, guess=1.0, **kwargs):", "funcdef": "def"}, "pyerrors.special": {"fullname": "pyerrors.special", "modulename": "pyerrors.special", "kind": "module", "doc": "

    \n"}, "pyerrors.special.beta": {"fullname": "pyerrors.special.beta", "modulename": "pyerrors.special", "qualname": "beta", "kind": "function", "doc": "

    beta(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    beta(a, b, out=None)

    \n\n

    Beta function.

    \n\n

    This function is defined in 1 as

    \n\n

    $$B(a, b) = \\int_0^1 t^{a-1}(1-t)^{b-1}dt\n = \\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)},$$

    \n\n

    where \\( \\Gamma \\) is the gamma function.

    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nReal-valued arguments
      • \n
    • out (ndarray, optional):\nOptional output array for the function result
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the beta function
      • \n
    \n\n
    See Also
    \n\n

    gamma: the gamma function
    \nbetainc: the regularized incomplete beta function
    \nbetaln: the natural logarithm of the absolute\nvalue of the beta function

    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n

    The beta function relates to the gamma function by the\ndefinition given above:

    \n\n
    \n
    >>> sc.beta(2, 3)\n0.08333333333333333\n>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)\n0.08333333333333333\n
    \n
    \n\n

    As this relationship demonstrates, the beta function\nis symmetric:

    \n\n
    \n
    >>> sc.beta(1.7, 2.4)\n0.16567527689031739\n>>> sc.beta(2.4, 1.7)\n0.16567527689031739\n
    \n
    \n\n

    This function satisfies \\( B(1, b) = 1/b \\):

    \n\n
    \n
    >>> sc.beta(1, 4)\n0.25\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      NIST Digital Library of Mathematical Functions,\nEq. 5.12.1. https://dlmf.nist.gov/5.12 ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betainc": {"fullname": "pyerrors.special.betainc", "modulename": "pyerrors.special", "qualname": "betainc", "kind": "function", "doc": "

    betainc(x1, x2, x3, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    betainc(a, b, x, out=None)

    \n\n

    Regularized incomplete beta function.

    \n\n

    Computes the regularized incomplete beta function, defined as 1:

    \n\n

    $$I_x(a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^x\nt^{a-1}(1-t)^{b-1}dt,$$

    \n\n

    for \\( 0 \\leq x \\leq 1 \\).

    \n\n

    This function is the cumulative distribution function for the beta\ndistribution; its range is [0, 1].

    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nPositive, real-valued parameters
      • \n
    • x (array_like):\nReal-valued such that \\( 0 \\leq x \\leq 1 \\),\nthe upper limit of integration
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the regularized incomplete beta function
      • \n
    \n\n
    See Also
    \n\n

    beta: beta function
    \nbetaincinv: inverse of the regularized incomplete beta function
    \nbetaincc: complement of the regularized incomplete beta function
    \nscipy.stats.beta: beta distribution

    \n\n
    Notes
    \n\n

    The term regularized in the name of this function refers to the\nscaling of the function by the gamma function terms shown in the\nformula. When not qualified as regularized, the name incomplete\nbeta function often refers to just the integral expression,\nwithout the gamma terms. One can use the function beta from\nscipy.special to get this \"nonregularized\" incomplete beta\nfunction by multiplying the result of betainc(a, b, x) by\nbeta(a, b).

    \n\n
    References
    \n\n
    Examples
    \n\n

    Let \\( B(a, b) \\) be the beta function.

    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n

    The coefficient in terms of gamma is equal to\n\\( 1/B(a, b) \\). Also, when \\( x=1 \\)\nthe integral is equal to \\( B(a, b) \\).\nTherefore, \\( I_{x=1}(a, b) = 1 \\) for any \\( a, b \\).

    \n\n
    \n
    >>> sc.betainc(0.2, 3.5, 1.0)\n1.0\n
    \n
    \n\n

    It satisfies\n\\( I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b)) \\),\nwhere \\( F \\) is the hypergeometric function hyp2f1:

    \n\n
    \n
    >>> a, b, x = 1.4, 3.1, 0.5\n>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))\n0.8148904036225295\n>>> sc.betainc(a, b, x)\n0.8148904036225296\n
    \n
    \n\n

    This functions satisfies the relationship\n\\( I_x(a, b) = 1 - I_{1-x}(b, a) \\):

    \n\n
    \n
    >>> sc.betainc(2.2, 3.1, 0.4)\n0.49339638807619446\n>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)\n0.49339638807619446\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/8.17 ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.betaln": {"fullname": "pyerrors.special.betaln", "modulename": "pyerrors.special", "qualname": "betaln", "kind": "function", "doc": "

    betaln(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    betaln(a, b, out=None)

    \n\n

    Natural logarithm of absolute value of beta function.

    \n\n

    Computes ln(abs(beta(a, b))).

    \n\n
    Parameters
    \n\n
      \n
    • a, b (array_like):\nPositive, real-valued parameters
      • \n
    • out (ndarray, optional):\nOptional output array for function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Value of the betaln function
      • \n
    \n\n
    See Also
    \n\n

    gamma: the gamma function
    \nbetainc: the regularized incomplete beta function
    \nbeta: the beta function

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import betaln, beta\n
    \n
    \n\n

    Verify that, for moderate values of a and b, betaln(a, b)\nis the same as log(beta(a, b)):

    \n\n
    \n
    >>> betaln(3, 4)\n-4.0943445622221\n
    \n
    \n\n
    \n
    >>> np.log(beta(3, 4))\n-4.0943445622221\n
    \n
    \n\n

    In the following beta(a, b) underflows to 0, so we can't compute\nthe logarithm of the actual value.

    \n\n
    \n
    >>> a = 400\n>>> b = 900\n>>> beta(a, b)\n0.0\n
    \n
    \n\n

    We can compute the logarithm of beta(a, b) by using betaln:

    \n\n
    \n
    >>> betaln(a, b)\n-804.3069951764146\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.polygamma": {"fullname": "pyerrors.special.polygamma", "modulename": "pyerrors.special", "qualname": "polygamma", "kind": "function", "doc": "

    Polygamma functions.

    \n\n

    Defined as \\( \\psi^{(n)}(x) \\) where \\( \\psi \\) is the\ndigamma function. See [dlmf]_ for details.

    \n\n
    Parameters
    \n\n
      \n
    • n (array_like):\nThe order of the derivative of the digamma function; must be\nintegral
      • \n
    • x (array_like):\nReal valued input
      • \n
    \n\n
    Returns
    \n\n
      \n
    • ndarray: Function results
      • \n
    \n\n
    See Also
    \n\n

    digamma

    \n\n
    References
    \n\n

    .. [dlmf] NIST, Digital Library of Mathematical Functions,\n https://dlmf.nist.gov/5.15

    \n\n
    Examples
    \n\n
    \n
    >>> from scipy import special\n>>> x = [2, 3, 25.5]\n>>> special.polygamma(1, x)\narray([ 0.64493407,  0.39493407,  0.03999467])\n>>> special.polygamma(0, x) == special.psi(x)\narray([ True,  True,  True], dtype=bool)\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.psi": {"fullname": "pyerrors.special.psi", "modulename": "pyerrors.special", "qualname": "psi", "kind": "function", "doc": "

    psi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    psi(z, out=None)

    \n\n

    The digamma function.

    \n\n

    The logarithmic derivative of the gamma function evaluated at z.

    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex argument.
      • \n
    • out (ndarray, optional):\nArray for the computed values of psi.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • digamma (scalar or ndarray):\nComputed values of psi.
      • \n
    \n\n
    Notes
    \n\n

    For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.

    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n

    Verify psi(z) = psi(z + 1) - 1/z:

    \n\n
    \n
    >>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      2. \n\n
    2. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      3. \n\n
    3. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      4. \n\n
    4. \n

      Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩

      \n
      5. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.digamma": {"fullname": "pyerrors.special.digamma", "modulename": "pyerrors.special", "qualname": "digamma", "kind": "function", "doc": "

    psi(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    psi(z, out=None)

    \n\n

    The digamma function.

    \n\n

    The logarithmic derivative of the gamma function evaluated at z.

    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex argument.
      • \n
    • out (ndarray, optional):\nArray for the computed values of psi.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • digamma (scalar or ndarray):\nComputed values of psi.
      • \n
    \n\n
    Notes
    \n\n

    For large values not close to the negative real axis, psi is\ncomputed using the asymptotic series (5.11.2) from 1. For small\narguments not close to the negative real axis, the recurrence\nrelation (5.5.2) from 2 is used until the argument is large\nenough to use the asymptotic series. For values close to the\nnegative real axis, the reflection formula (5.5.4) from 3 is\nused first. Note that psi has a family of zeros on the\nnegative real axis which occur between the poles at nonpositive\nintegers. Around the zeros the reflection formula suffers from\ncancellation and the implementation loses precision. The sole\npositive zero and the first negative zero, however, are handled\nseparately by precomputing series expansions using 4, so the\nfunction should maintain full accuracy around the origin.

    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> from scipy.special import psi\n>>> z = 3 + 4j\n>>> psi(z)\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n

    Verify psi(z) = psi(z + 1) - 1/z:

    \n\n
    \n
    >>> psi(z + 1) - 1/z\n(1.55035981733341+1.0105022091860445j)\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      2. \n\n
    2. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      3. \n\n
    3. \n

      NIST Digital Library of Mathematical Functions\nhttps://dlmf.nist.gov/5 ↩

      \n
      4. \n\n
    4. \n

      Fredrik Johansson and others.\n\"mpmath: a Python library for arbitrary-precision floating-point arithmetic\"\n(Version 0.19) http://mpmath.org/ ↩

      \n
      5. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gamma": {"fullname": "pyerrors.special.gamma", "modulename": "pyerrors.special", "qualname": "gamma", "kind": "function", "doc": "

    gamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    gamma(z, out=None)

    \n\n

    gamma function.

    \n\n

    The gamma function is defined as

    \n\n

    $$\\Gamma(z) = \\int_0^\\infty t^{z-1} e^{-t} dt$$

    \n\n

    for \\( \\Re(z) > 0 \\) and is extended to the rest of the complex\nplane by analytic continuation. See [dlmf]_ for more details.

    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex valued argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the gamma function
      • \n
    \n\n
    Notes
    \n\n

    The gamma function is often referred to as the generalized\nfactorial since \\( \\Gamma(n + 1) = n! \\) for natural numbers\n\\( n \\). More generally it satisfies the recurrence relation\n\\( \\Gamma(z + 1) = z \\cdot \\Gamma(z) \\) for complex \\( z \\),\nwhich, combined with the fact that \\( \\Gamma(1) = 1 \\), implies\nthe above identity for \\( z = n \\).

    \n\n
    References
    \n\n

    .. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5.2#E1

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import gamma, factorial\n
    \n
    \n\n
    \n
    >>> gamma([0, 0.5, 1, 5])\narray([         inf,   1.77245385,   1.        ,  24.        ])\n
    \n
    \n\n
    \n
    >>> z = 2.5 + 1j\n>>> gamma(z)\n(0.77476210455108352+0.70763120437959293j)\n>>> gamma(z+1), z*gamma(z)  # Recurrence property\n((1.2292740569981171+2.5438401155000685j),\n (1.2292740569981158+2.5438401155000658j))\n
    \n
    \n\n
    \n
    >>> gamma(0.5)**2  # gamma(0.5) = sqrt(pi)\n3.1415926535897927\n
    \n
    \n\n

    Plot gamma(x) for real x

    \n\n
    \n
    >>> x = np.linspace(-3.5, 5.5, 2251)\n>>> y = gamma(x)\n
    \n
    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')\n>>> k = np.arange(1, 7)\n>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,\n...          label='(x-1)!, x = 1, 2, ...')\n>>> plt.xlim(-3.5, 5.5)\n>>> plt.ylim(-10, 25)\n>>> plt.grid()\n>>> plt.xlabel('x')\n>>> plt.legend(loc='lower right')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaln": {"fullname": "pyerrors.special.gammaln", "modulename": "pyerrors.special", "qualname": "gammaln", "kind": "function", "doc": "

    gammaln(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    gammaln(x, out=None)

    \n\n

    Logarithm of the absolute value of the gamma function.

    \n\n

    Defined as

    \n\n

    $$\\ln(\\lvert\\Gamma(x)\\rvert)$$

    \n\n

    where \\( \\Gamma \\) is the gamma function. For more details on\nthe gamma function, see [dlmf]_.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the log of the absolute value of gamma
      • \n
    \n\n
    See Also
    \n\n

    gammasgn: sign of the gamma function
    \nloggamma: principal branch of the logarithm of the gamma function

    \n\n
    Notes
    \n\n

    It is the same function as the Python standard library function\nmath.lgamma().

    \n\n

    When used in conjunction with gammasgn, this function is useful\nfor working in logspace on the real axis without having to deal\nwith complex numbers via the relation exp(gammaln(x)) =\ngammasgn(x) * gamma(x).

    \n\n

    For complex-valued log-gamma, use loggamma instead of gammaln.

    \n\n
    References
    \n\n

    .. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import scipy.special as sc\n
    \n
    \n\n

    It has two positive zeros.

    \n\n
    \n
    >>> sc.gammaln([1, 2])\narray([0., 0.])\n
    \n
    \n\n

    It has poles at nonpositive integers.

    \n\n
    \n
    >>> sc.gammaln([0, -1, -2, -3, -4])\narray([inf, inf, inf, inf, inf])\n
    \n
    \n\n

    It asymptotically approaches x * log(x) (Stirling's formula).

    \n\n
    \n
    >>> x = np.array([1e10, 1e20, 1e40, 1e80])\n>>> sc.gammaln(x)\narray([2.20258509e+11, 4.50517019e+21, 9.11034037e+41, 1.83206807e+82])\n>>> x * np.log(x)\narray([2.30258509e+11, 4.60517019e+21, 9.21034037e+41, 1.84206807e+82])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammainc": {"fullname": "pyerrors.special.gammainc", "modulename": "pyerrors.special", "qualname": "gammainc", "kind": "function", "doc": "

    gammainc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    gammainc(a, x, out=None)

    \n\n

    Regularized lower incomplete gamma function.

    \n\n

    It is defined as

    \n\n

    $$P(a, x) = \\frac{1}{\\Gamma(a)} \\int_0^x t^{a - 1}e^{-t} dt$$

    \n\n

    for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.

    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nPositive parameter
      • \n
    • x (array_like):\nNonnegative argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the lower incomplete gamma function
      • \n
    \n\n
    See Also
    \n\n

    gammaincc: regularized upper incomplete gamma function
    \ngammaincinv: inverse of the regularized lower incomplete gamma function
    \ngammainccinv: inverse of the regularized upper incomplete gamma function

    \n\n
    Notes
    \n\n

    The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammaincc is the regularized upper\nincomplete gamma function.

    \n\n

    The implementation largely follows that of [boost]_.

    \n\n
    References
    \n\n

    .. [dlmf] NIST Digital Library of Mathematical functions\n https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html

    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n

    It is the CDF of the gamma distribution, so it starts at 0 and\nmonotonically increases to 1.

    \n\n
    \n
    >>> sc.gammainc(0.5, [0, 1, 10, 100])\narray([0.        , 0.84270079, 0.99999226, 1.        ])\n
    \n
    \n\n

    It is equal to one minus the upper incomplete gamma function.

    \n\n
    \n
    >>> a, x = 0.5, 0.4\n>>> sc.gammainc(a, x)\n0.6289066304773024\n>>> 1 - sc.gammaincc(a, x)\n0.6289066304773024\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammaincc": {"fullname": "pyerrors.special.gammaincc", "modulename": "pyerrors.special", "qualname": "gammaincc", "kind": "function", "doc": "

    gammaincc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    gammaincc(a, x, out=None)

    \n\n

    Regularized upper incomplete gamma function.

    \n\n

    It is defined as

    \n\n

    $$Q(a, x) = \\frac{1}{\\Gamma(a)} \\int_x^\\infty t^{a - 1}e^{-t} dt$$

    \n\n

    for \\( a > 0 \\) and \\( x \\geq 0 \\). See [dlmf]_ for details.

    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nPositive parameter
      • \n
    • x (array_like):\nNonnegative argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the upper incomplete gamma function
      • \n
    \n\n
    See Also
    \n\n

    gammainc: regularized lower incomplete gamma function
    \ngammaincinv: inverse of the regularized lower incomplete gamma function
    \ngammainccinv: inverse of the regularized upper incomplete gamma function

    \n\n
    Notes
    \n\n

    The function satisfies the relation gammainc(a, x) +\ngammaincc(a, x) = 1 where gammainc is the regularized lower\nincomplete gamma function.

    \n\n

    The implementation largely follows that of [boost]_.

    \n\n
    References
    \n\n

    .. [dlmf] NIST Digital Library of Mathematical functions\n https://dlmf.nist.gov/8.2#E4\n.. [boost] Maddock et. al., \"Incomplete Gamma Functions\",\n https://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html

    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n

    It is the survival function of the gamma distribution, so it\nstarts at 1 and monotonically decreases to 0.

    \n\n
    \n
    >>> sc.gammaincc(0.5, [0, 1, 10, 100, 1000])\narray([1.00000000e+00, 1.57299207e-01, 7.74421643e-06, 2.08848758e-45,\n       0.00000000e+00])\n
    \n
    \n\n

    It is equal to one minus the lower incomplete gamma function.

    \n\n
    \n
    >>> a, x = 0.5, 0.4\n>>> sc.gammaincc(a, x)\n0.37109336952269756\n>>> 1 - sc.gammainc(a, x)\n0.37109336952269756\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.gammasgn": {"fullname": "pyerrors.special.gammasgn", "modulename": "pyerrors.special", "qualname": "gammasgn", "kind": "function", "doc": "

    gammasgn(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    gammasgn(x, out=None)

    \n\n

    Sign of the gamma function.

    \n\n

    It is defined as

    \n\n

    $$\\text{gammasgn}(x) =\n\\begin{cases}\n +1 & \\Gamma(x) > 0 \\\n -1 & \\Gamma(x) < 0\n\\end{cases}$$

    \n\n

    where \\( \\Gamma \\) is the gamma function; see gamma. This\ndefinition is complete since the gamma function is never zero;\nsee the discussion after [dlmf]_.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Sign of the gamma function
      • \n
    \n\n
    See Also
    \n\n

    gamma: the gamma function
    \ngammaln: log of the absolute value of the gamma function
    \nloggamma: analytic continuation of the log of the gamma function

    \n\n
    Notes
    \n\n

    The gamma function can be computed as gammasgn(x) *\nnp.exp(gammaln(x)).

    \n\n
    References
    \n\n

    .. [dlmf] NIST Digital Library of Mathematical Functions\n https://dlmf.nist.gov/5.2#E1

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import scipy.special as sc\n
    \n
    \n\n

    It is 1 for x > 0.

    \n\n
    \n
    >>> sc.gammasgn([1, 2, 3, 4])\narray([1., 1., 1., 1.])\n
    \n
    \n\n

    It alternates between -1 and 1 for negative integers.

    \n\n
    \n
    >>> sc.gammasgn([-0.5, -1.5, -2.5, -3.5])\narray([-1.,  1., -1.,  1.])\n
    \n
    \n\n

    It can be used to compute the gamma function.

    \n\n
    \n
    >>> x = [1.5, 0.5, -0.5, -1.5]\n>>> sc.gammasgn(x) * np.exp(sc.gammaln(x))\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n>>> sc.gamma(x)\narray([ 0.88622693,  1.77245385, -3.5449077 ,  2.3632718 ])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.rgamma": {"fullname": "pyerrors.special.rgamma", "modulename": "pyerrors.special", "qualname": "rgamma", "kind": "function", "doc": "

    rgamma(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    rgamma(z, out=None)

    \n\n

    Reciprocal of the gamma function.

    \n\n

    Defined as \\( 1 / \\Gamma(z) \\), where \\( \\Gamma \\) is the\ngamma function. For more on the gamma function see gamma.

    \n\n
    Parameters
    \n\n
      \n
    • z (array_like):\nReal or complex valued input
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Function results
      • \n
    \n\n
    See Also
    \n\n

    gamma,, gammaln,, loggamma

    \n\n
    Notes
    \n\n

    The gamma function has no zeros and has simple poles at\nnonpositive integers, so rgamma is an entire function with zeros\nat the nonpositive integers. See the discussion in [dlmf]_ for\nmore details.

    \n\n
    References
    \n\n

    .. [dlmf] Nist, Digital Library of Mathematical functions,\n https://dlmf.nist.gov/5.2#i

    \n\n
    Examples
    \n\n
    \n
    >>> import scipy.special as sc\n
    \n
    \n\n

    It is the reciprocal of the gamma function.

    \n\n
    \n
    >>> sc.rgamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n>>> 1 / sc.gamma([1, 2, 3, 4])\narray([1.        , 1.        , 0.5       , 0.16666667])\n
    \n
    \n\n

    It is zero at nonpositive integers.

    \n\n
    \n
    >>> sc.rgamma([0, -1, -2, -3])\narray([0., 0., 0., 0.])\n
    \n
    \n\n

    It rapidly underflows to zero along the positive real axis.

    \n\n
    \n
    >>> sc.rgamma([10, 100, 179])\narray([2.75573192e-006, 1.07151029e-156, 0.00000000e+000])\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.multigammaln": {"fullname": "pyerrors.special.multigammaln", "modulename": "pyerrors.special", "qualname": "multigammaln", "kind": "function", "doc": "

    Returns the log of multivariate gamma, also sometimes called the\ngeneralized gamma.

    \n\n
    Parameters
    \n\n
      \n
    • a (ndarray):\nThe multivariate gamma is computed for each item of a.
      • \n
    • d (int):\nThe dimension of the space of integration.
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (ndarray):\nThe values of the log multivariate gamma at the given points a.
      • \n
    \n\n
    Notes
    \n\n

    The formal definition of the multivariate gamma of dimension d for a real\na is

    \n\n

    $$\\Gamma_d(a) = \\int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA$$

    \n\n

    with the condition \\( a > (d-1)/2 \\), and \\( A > 0 \\) being the set of\nall the positive definite matrices of dimension d. Note that a is a\nscalar: the integrand only is multivariate, the argument is not (the\nfunction is defined over a subset of the real set).

    \n\n

    This can be proven to be equal to the much friendlier equation

    \n\n

    $$\\Gamma_d(a) = \\pi^{d(d-1)/4} \\prod_{i=1}^{d} \\Gamma(a - (i-1)/2).$$

    \n\n
    References
    \n\n

    R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in\nprobability and mathematical statistics).

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import multigammaln, gammaln\n>>> a = 23.5\n>>> d = 10\n>>> multigammaln(a, d)\n454.1488605074416\n
    \n
    \n\n

    Verify that the result agrees with the logarithm of the equation\nshown above:

    \n\n
    \n
    >>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()\n454.1488605074416\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.kn": {"fullname": "pyerrors.special.kn", "modulename": "pyerrors.special", "qualname": "kn", "kind": "function", "doc": "

    Modified Bessel function of the second kind of integer order n

    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j0": {"fullname": "pyerrors.special.j0", "modulename": "pyerrors.special", "qualname": "j0", "kind": "function", "doc": "

    j0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    j0(x, out=None)

    \n\n

    Bessel function of the first kind of order 0.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 0 at x.
      • \n
    \n\n
    See Also
    \n\n

    jv: Bessel function of real order and complex argument.
    \nspherical_jn: spherical Bessel functions.

    \n\n
    Notes
    \n\n

    The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval the following rational approximation is used:

    \n\n

    $$J_0(x) \\approx (w - r_1^2)(w - r_2^2) \\frac{P_3(w)}{Q_8(w)},$$

    \n\n

    where \\( w = x^2 \\) and \\( r_1 \\), \\( r_2 \\) are the zeros of\n\\( J_0 \\), and \\( P_3 \\) and \\( Q_8 \\) are polynomials of degrees 3\nand 8, respectively.

    \n\n

    In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.

    \n\n

    This function is a wrapper for the Cephes 1 routine j0.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import j0\n>>> j0(1.)\n0.7651976865579665\n
    \n
    \n\n

    Calculate the function at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> j0(np.array([-2., 0., 4.]))\narray([ 0.22389078,  1.        , -0.39714981])\n
    \n
    \n\n

    Plot the function from -20 to 20.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y0": {"fullname": "pyerrors.special.y0", "modulename": "pyerrors.special", "qualname": "y0", "kind": "function", "doc": "

    y0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    y0(x, out=None)

    \n\n

    Bessel function of the second kind of order 0.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 0 at x.
      • \n
    \n\n
    See Also
    \n\n

    j0: Bessel function of the first kind of order 0
    \nyv: Bessel function of the first kind

    \n\n
    Notes
    \n\n

    The domain is divided into the intervals [0, 5] and (5, infinity). In the\nfirst interval a rational approximation \\( R(x) \\) is employed to\ncompute,

    \n\n

    $$Y_0(x) = R(x) + \\frac{2 \\log(x) J_0(x)}{\\pi},$$

    \n\n

    where \\( J_0 \\) is the Bessel function of the first kind of order 0.

    \n\n

    In the second interval, the Hankel asymptotic expansion is employed with\ntwo rational functions of degree 6/6 and 7/7.

    \n\n

    This function is a wrapper for the Cephes 1 routine y0.

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import y0\n>>> y0(1.)\n0.08825696421567697\n
    \n
    \n\n

    Calculate at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> y0(np.array([0.5, 2., 3.]))\narray([-0.44451873,  0.51037567,  0.37685001])\n
    \n
    \n\n

    Plot the function from 0 to 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.j1": {"fullname": "pyerrors.special.j1", "modulename": "pyerrors.special", "qualname": "j1", "kind": "function", "doc": "

    j1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    j1(x, out=None)

    \n\n

    Bessel function of the first kind of order 1.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function of the first kind of order 1 at x.
      • \n
    \n\n
    See Also
    \n\n

    jv: Bessel function of the first kind
    \nspherical_jn: spherical Bessel functions.

    \n\n
    Notes
    \n\n

    The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 24 term Chebyshev expansion is used. In the second, the\nasymptotic trigonometric representation is employed using two rational\nfunctions of degree 5/5.

    \n\n

    This function is a wrapper for the Cephes 1 routine j1.\nIt should not be confused with the spherical Bessel functions (see\nspherical_jn).

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import j1\n>>> j1(1.)\n0.44005058574493355\n
    \n
    \n\n

    Calculate the function at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> j1(np.array([-2., 0., 4.]))\narray([-0.57672481,  0.        , -0.06604333])\n
    \n
    \n\n

    Plot the function from -20 to 20.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-20., 20., 1000)\n>>> y = j1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.y1": {"fullname": "pyerrors.special.y1", "modulename": "pyerrors.special", "qualname": "y1", "kind": "function", "doc": "

    y1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    y1(x, out=None)

    \n\n

    Bessel function of the second kind of order 1.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function of the second kind of order 1 at x.
      • \n
    \n\n
    See Also
    \n\n

    j1: Bessel function of the first kind of order 1
    \nyn: Bessel function of the second kind
    \nyv: Bessel function of the second kind

    \n\n
    Notes
    \n\n

    The domain is divided into the intervals [0, 8] and (8, infinity). In the\nfirst interval a 25 term Chebyshev expansion is used, and computing\n\\( J_1 \\) (the Bessel function of the first kind) is required. In the\nsecond, the asymptotic trigonometric representation is employed using two\nrational functions of degree 5/5.

    \n\n

    This function is a wrapper for the Cephes 1 routine y1.

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import y1\n>>> y1(1.)\n-0.7812128213002888\n
    \n
    \n\n

    Calculate at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> y1(np.array([0.5, 2., 3.]))\narray([-1.47147239, -0.10703243,  0.32467442])\n
    \n
    \n\n

    Plot the function from 0 to 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> y = y1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.jn": {"fullname": "pyerrors.special.jn", "modulename": "pyerrors.special", "qualname": "jn", "kind": "function", "doc": "

    jv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    jv(v, z, out=None)

    \n\n

    Bessel function of the first kind of real order and complex argument.

    \n\n
    Parameters
    \n\n
      \n
    • v (array_like):\nOrder (float).
      • \n
    • z (array_like):\nArgument (float or complex).
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • J (scalar or ndarray):\nValue of the Bessel function, \\( J_v(z) \\).
      • \n
    \n\n
    See Also
    \n\n

    jve: \\( J_v \\) with leading exponential behavior stripped off.
    \nspherical_jn: spherical Bessel functions.
    \nj0: faster version of this function for order 0.
    \nj1: faster version of this function for order 1.

    \n\n
    Notes
    \n\n

    For positive v values, the computation is carried out using the AMOS\n1 zbesj routine, which exploits the connection to the modified\nBessel function \\( I_v \\),

    \n\n

    $$J_v(z) = \\exp(v\\pi\\imath/2) I_v(-\\imath z)\\qquad (\\Im z > 0)

    \n\n

    J_v(z) = \\exp(-v\\pi\\imath/2) I_v(\\imath z)\\qquad (\\Im z < 0)$$

    \n\n

    For negative v values the formula,

    \n\n

    $$J_{-v}(z) = J_v(z) \\cos(\\pi v) - Y_v(z) \\sin(\\pi v)$$

    \n\n

    is used, where \\( Y_v(z) \\) is the Bessel function of the second\nkind, computed using the AMOS routine zbesy. Note that the second\nterm is exactly zero for integer v; to improve accuracy the second\nterm is explicitly omitted for v values such that v = floor(v).

    \n\n

    Not to be confused with the spherical Bessel functions (see spherical_jn).

    \n\n
    References
    \n\n
    Examples
    \n\n

    Evaluate the function of order 0 at one point.

    \n\n
    \n
    >>> from scipy.special import jv\n>>> jv(0, 1.)\n0.7651976865579666\n
    \n
    \n\n

    Evaluate the function at one point for different orders.

    \n\n
    \n
    >>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)\n(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)\n
    \n
    \n\n

    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:

    \n\n
    \n
    >>> jv([0, 1, 1.5], 1.)\narray([0.76519769, 0.44005059, 0.24029784])\n
    \n
    \n\n

    Evaluate the function at several points for order 0 by providing an\narray for z.

    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> jv(0, points)\narray([ 0.22389078,  1.        , -0.26005195])\n
    \n
    \n\n

    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:

    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> jv(orders, points)\narray([[ 0.22389078,  1.        , -0.26005195],\n       [-0.57672481,  0.        ,  0.33905896]])\n
    \n
    \n\n

    Plot the functions of order 0 to 3 from -10 to 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, jv(i, x), label=f'$J_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.yn": {"fullname": "pyerrors.special.yn", "modulename": "pyerrors.special", "qualname": "yn", "kind": "function", "doc": "

    yn(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    yn(n, x, out=None)

    \n\n

    Bessel function of the second kind of integer order and real argument.

    \n\n
    Parameters
    \n\n
      \n
    • n (array_like):\nOrder (integer).
      • \n
    • x (array_like):\nArgument (float).
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • Y (scalar or ndarray):\nValue of the Bessel function, \\( Y_n(x) \\).
      • \n
    \n\n
    See Also
    \n\n

    yv: For real order and real or complex argument.
    \ny0: faster implementation of this function for order 0
    \ny1: faster implementation of this function for order 1

    \n\n
    Notes
    \n\n

    Wrapper for the Cephes 1 routine yn.

    \n\n

    The function is evaluated by forward recurrence on n, starting with\nvalues computed by the Cephes routines y0 and y1. If n = 0 or 1,\nthe routine for y0 or y1 is called directly.

    \n\n
    References
    \n\n
    Examples
    \n\n

    Evaluate the function of order 0 at one point.

    \n\n
    \n
    >>> from scipy.special import yn\n>>> yn(0, 1.)\n0.08825696421567697\n
    \n
    \n\n

    Evaluate the function at one point for different orders.

    \n\n
    \n
    >>> yn(0, 1.), yn(1, 1.), yn(2, 1.)\n(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)\n
    \n
    \n\n

    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:

    \n\n
    \n
    >>> yn([0, 1, 2], 1.)\narray([ 0.08825696, -0.78121282, -1.65068261])\n
    \n
    \n\n

    Evaluate the function at several points for order 0 by providing an\narray for z.

    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([0.5, 3., 8.])\n>>> yn(0, points)\narray([-0.44451873,  0.37685001,  0.22352149])\n
    \n
    \n\n

    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:

    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> yn(orders, points)\narray([[-0.44451873,  0.37685001,  0.22352149],\n       [-1.47147239,  0.32467442, -0.15806046]])\n
    \n
    \n\n

    Plot the functions of order 0 to 3 from 0 to 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(0., 10., 1000)\n>>> for i in range(4):\n...     ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')\n>>> ax.set_ylim(-3, 1)\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i0": {"fullname": "pyerrors.special.i0", "modulename": "pyerrors.special", "qualname": "i0", "kind": "function", "doc": "

    i0(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    i0(x, out=None)

    \n\n

    Modified Bessel function of order 0.

    \n\n

    Defined as,

    \n\n

    $$I_0(x) = \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{(k!)^2} = J_0(\\imath x),$$

    \n\n

    where \\( J_0 \\) is the Bessel function of the first kind of order 0.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float)
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • I (scalar or ndarray):\nValue of the modified Bessel function of order 0 at x.
      • \n
    \n\n
    See Also
    \n\n

    iv: Modified Bessel function of any order
    \ni0e: Exponentially scaled modified Bessel function of order 0

    \n\n
    Notes
    \n\n

    The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.

    \n\n

    This function is a wrapper for the Cephes 1 routine i0.

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import i0\n>>> i0(1.)\n1.2660658777520082\n
    \n
    \n\n

    Calculate at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> i0(np.array([-2., 0., 3.5]))\narray([2.2795853 , 1.        , 7.37820343])\n
    \n
    \n\n

    Plot the function from -10 to 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i0(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.i1": {"fullname": "pyerrors.special.i1", "modulename": "pyerrors.special", "qualname": "i1", "kind": "function", "doc": "

    i1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    i1(x, out=None)

    \n\n

    Modified Bessel function of order 1.

    \n\n

    Defined as,

    \n\n

    $$I_1(x) = \\frac{1}{2}x \\sum_{k=0}^\\infty \\frac{(x^2/4)^k}{k! (k + 1)!}\n = -\\imath J_1(\\imath x),$$

    \n\n

    where \\( J_1 \\) is the Bessel function of the first kind of order 1.

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nArgument (float)
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • I (scalar or ndarray):\nValue of the modified Bessel function of order 1 at x.
      • \n
    \n\n
    See Also
    \n\n

    iv: Modified Bessel function of the first kind
    \ni1e: Exponentially scaled modified Bessel function of order 1

    \n\n
    Notes
    \n\n

    The range is partitioned into the two intervals [0, 8] and (8, infinity).\nChebyshev polynomial expansions are employed in each interval.

    \n\n

    This function is a wrapper for the Cephes 1 routine i1.

    \n\n
    References
    \n\n
    Examples
    \n\n

    Calculate the function at one point:

    \n\n
    \n
    >>> from scipy.special import i1\n>>> i1(1.)\n0.5651591039924851\n
    \n
    \n\n

    Calculate the function at several points:

    \n\n
    \n
    >>> import numpy as np\n>>> i1(np.array([-2., 0., 6.]))\narray([-1.59063685,  0.        , 61.34193678])\n
    \n
    \n\n

    Plot the function between -10 and 10.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-10., 10., 1000)\n>>> y = i1(x)\n>>> ax.plot(x, y)\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Cephes Mathematical Functions Library,\nhttp://www.netlib.org/cephes/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.iv": {"fullname": "pyerrors.special.iv", "modulename": "pyerrors.special", "qualname": "iv", "kind": "function", "doc": "

    iv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    iv(v, z, out=None)

    \n\n

    Modified Bessel function of the first kind of real order.

    \n\n
    Parameters
    \n\n
      \n
    • v (array_like):\nOrder. If z is of real type and negative, v must be integer\nvalued.
      • \n
    • z (array_like of float or complex):\nArgument.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the modified Bessel function.
      • \n
    \n\n
    See Also
    \n\n

    ive: This function with leading exponential behavior stripped off.
    \ni0: Faster version of this function for order 0.
    \ni1: Faster version of this function for order 1.

    \n\n
    Notes
    \n\n

    For real z and \\( v \\in [-50, 50] \\), the evaluation is carried out\nusing Temme's method 1. For larger orders, uniform asymptotic\nexpansions are applied.

    \n\n

    For complex z and positive v, the AMOS 2 zbesi routine is\ncalled. It uses a power series for small z, the asymptotic expansion\nfor large abs(z), the Miller algorithm normalized by the Wronskian\nand a Neumann series for intermediate magnitudes, and the uniform\nasymptotic expansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large\norders. Backward recurrence is used to generate sequences or reduce\norders when necessary.

    \n\n

    The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,

    \n\n

    $$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$

    \n\n

    (valid when the real part of z is positive). For negative v, the\nformula

    \n\n

    $$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$

    \n\n

    is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.

    \n\n
    References
    \n\n
    Examples
    \n\n

    Evaluate the function of order 0 at one point.

    \n\n
    \n
    >>> from scipy.special import iv\n>>> iv(0, 1.)\n1.2660658777520084\n
    \n
    \n\n

    Evaluate the function at one point for different orders.

    \n\n
    \n
    >>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)\n(1.2660658777520084, 0.565159103992485, 0.2935253263474798)\n
    \n
    \n\n

    The evaluation for different orders can be carried out in one call by\nproviding a list or NumPy array as argument for the v parameter:

    \n\n
    \n
    >>> iv([0, 1, 1.5], 1.)\narray([1.26606588, 0.5651591 , 0.29352533])\n
    \n
    \n\n

    Evaluate the function at several points for order 0 by providing an\narray for z.

    \n\n
    \n
    >>> import numpy as np\n>>> points = np.array([-2., 0., 3.])\n>>> iv(0, points)\narray([2.2795853 , 1.        , 4.88079259])\n
    \n
    \n\n

    If z is an array, the order parameter v must be broadcastable to\nthe correct shape if different orders shall be computed in one call.\nTo calculate the orders 0 and 1 for an 1D array:

    \n\n
    \n
    >>> orders = np.array([[0], [1]])\n>>> orders.shape\n(2, 1)\n
    \n
    \n\n
    \n
    >>> iv(orders, points)\narray([[ 2.2795853 ,  1.        ,  4.88079259],\n       [-1.59063685,  0.        ,  3.95337022]])\n
    \n
    \n\n

    Plot the functions of order 0 to 3 from -5 to 5.

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, iv(i, x), label=f'$I_{i!r}$')\n>>> ax.legend()\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Temme, Journal of Computational Physics, vol 21, 343 (1976) ↩

      \n
      2. \n\n
    2. \n

      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩

      \n
      3. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.ive": {"fullname": "pyerrors.special.ive", "modulename": "pyerrors.special", "qualname": "ive", "kind": "function", "doc": "

    ive(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    ive(v, z, out=None)

    \n\n

    Exponentially scaled modified Bessel function of the first kind.

    \n\n

    Defined as::

    \n\n
    ive(v, z) = iv(v, z) * exp(-abs(z.real))\n
    \n\n

    For imaginary numbers without a real part, returns the unscaled\nBessel function of the first kind iv.

    \n\n
    Parameters
    \n\n
      \n
    • v (array_like of float):\nOrder.
      • \n
    • z (array_like of float or complex):\nArgument.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the exponentially scaled modified Bessel function.
      • \n
    \n\n
    See Also
    \n\n

    iv: Modified Bessel function of the first kind
    \ni0e: Faster implementation of this function for order 0
    \ni1e: Faster implementation of this function for order 1

    \n\n
    Notes
    \n\n

    For positive v, the AMOS 1 zbesi routine is called. It uses a\npower series for small z, the asymptotic expansion for large\nabs(z), the Miller algorithm normalized by the Wronskian and a\nNeumann series for intermediate magnitudes, and the uniform asymptotic\nexpansions for \\( I_v(z) \\) and \\( J_v(z) \\) for large orders.\nBackward recurrence is used to generate sequences or reduce orders when\nnecessary.

    \n\n

    The calculations above are done in the right half plane and continued\ninto the left half plane by the formula,

    \n\n

    $$I_v(z \\exp(\\pm\\imath\\pi)) = \\exp(\\pm\\pi v) I_v(z)$$

    \n\n

    (valid when the real part of z is positive). For negative v, the\nformula

    \n\n

    $$I_{-v}(z) = I_v(z) + \\frac{2}{\\pi} \\sin(\\pi v) K_v(z)$$

    \n\n

    is used, where \\( K_v(z) \\) is the modified Bessel function of the\nsecond kind, evaluated using the AMOS routine zbesk.

    \n\n

    ive is useful for large arguments z: for these, iv easily overflows,\nwhile ive does not due to the exponential scaling.

    \n\n
    References
    \n\n
    Examples
    \n\n

    In the following example iv returns infinity whereas ive still returns\na finite number.

    \n\n
    \n
    >>> from scipy.special import iv, ive\n>>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> iv(3, 1000.), ive(3, 1000.)\n(inf, 0.01256056218254712)\n
    \n
    \n\n

    Evaluate the function at one point for different orders by\nproviding a list or NumPy array as argument for the v parameter:

    \n\n
    \n
    >>> ive([0, 1, 1.5], 1.)\narray([0.46575961, 0.20791042, 0.10798193])\n
    \n
    \n\n

    Evaluate the function at several points for order 0 by providing an\narray for z.

    \n\n
    \n
    >>> points = np.array([-2., 0., 3.])\n>>> ive(0, points)\narray([0.30850832, 1.        , 0.24300035])\n
    \n
    \n\n

    Evaluate the function at several points for different orders by\nproviding arrays for both v for z. Both arrays have to be\nbroadcastable to the correct shape. To calculate the orders 0, 1\nand 2 for a 1D array of points:

    \n\n
    \n
    >>> ive([[0], [1], [2]], points)\narray([[ 0.30850832,  1.        ,  0.24300035],\n       [-0.21526929,  0.        ,  0.19682671],\n       [ 0.09323903,  0.        ,  0.11178255]])\n
    \n
    \n\n

    Plot the functions of order 0 to 3 from -5 to 5.

    \n\n
    \n
    >>> fig, ax = plt.subplots()\n>>> x = np.linspace(-5., 5., 1000)\n>>> for i in range(4):\n...     ax.plot(x, ive(i, x), label=f'$I_{i!r}(z)\\cdot e^{{-|z|}}$')\n>>> ax.legend()\n>>> ax.set_xlabel(r"$z$")\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    1. \n

      Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\nof a Complex Argument and Nonnegative Order\",\nhttp://netlib.org/amos/ ↩

      \n
      2. \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erf": {"fullname": "pyerrors.special.erf", "modulename": "pyerrors.special", "qualname": "erf", "kind": "function", "doc": "

    erf(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    erf(z, out=None)

    \n\n

    Returns the error function of complex argument.

    \n\n

    It is defined as 2/sqrt(pi)*integral(exp(-t**2), t=0..z).

    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nInput array.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (scalar or ndarray):\nThe values of the error function at the given points x.
      • \n
    \n\n
    See Also
    \n\n

    erfc,, erfinv,, erfcinv,, wofz,, erfcx,, erfi

    \n\n
    Notes
    \n\n

    The cumulative of the unit normal distribution is given by\nPhi(z) = 1/2[1 + erf(z/sqrt(2))].

    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erf(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erf(x)$')\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfc": {"fullname": "pyerrors.special.erfc", "modulename": "pyerrors.special", "qualname": "erfc", "kind": "function", "doc": "

    erfc(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    erfc(x, out=None)

    \n\n

    Complementary error function, 1 - erf(x).

    \n\n
    Parameters
    \n\n
      \n
    • x (array_like):\nReal or complex valued argument
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: Values of the complementary error function
      • \n
    \n\n
    See Also
    \n\n

    erf,, erfi,, erfcx,, dawsn,, wofz

    \n\n
    References
    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy import special\n>>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-3, 3)\n>>> plt.plot(x, special.erfc(x))\n>>> plt.xlabel('$x$')\n>>> plt.ylabel('$erfc(x)$')\n>>> plt.show()\n
    \n
    \n\n
    \n
    \n
      \n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfinv": {"fullname": "pyerrors.special.erfinv", "modulename": "pyerrors.special", "qualname": "erfinv", "kind": "function", "doc": "

    erfinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    erfinv(y, out=None)

    \n\n

    Inverse of the error function.

    \n\n

    Computes the inverse of the error function.

    \n\n

    In the complex domain, there is no unique complex number w satisfying\nerf(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, -1 < x < 1, there is a unique real\nnumber satisfying erf(erfinv(x)) = x.

    \n\n
    Parameters
    \n\n
      \n
    • y (ndarray):\nArgument at which to evaluate. Domain: [-1, 1]
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • erfinv (scalar or ndarray):\nThe inverse of erf of y, element-wise
      • \n
    \n\n
    See Also
    \n\n

    erf: Error function of a complex argument
    \nerfc: Complementary error function, 1 - erf(x)
    \nerfcinv: Inverse of the complementary error function

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfinv, erf\n
    \n
    \n\n
    \n
    >>> erfinv(0.5)\n0.4769362762044699\n
    \n
    \n\n
    \n
    >>> y = np.linspace(-1.0, 1.0, num=9)\n>>> x = erfinv(y)\n>>> x\narray([       -inf, -0.81341985, -0.47693628, -0.22531206,  0.        ,\n        0.22531206,  0.47693628,  0.81341985,         inf])\n
    \n
    \n\n

    Verify that erf(erfinv(y)) is y.

    \n\n
    \n
    >>> erf(x)\narray([-1.  , -0.75, -0.5 , -0.25,  0.  ,  0.25,  0.5 ,  0.75,  1.  ])\n
    \n
    \n\n

    Plot the function:

    \n\n
    \n
    >>> y = np.linspace(-1, 1, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfinv(y)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.erfcinv": {"fullname": "pyerrors.special.erfcinv", "modulename": "pyerrors.special", "qualname": "erfcinv", "kind": "function", "doc": "

    erfcinv(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    erfcinv(y, out=None)

    \n\n

    Inverse of the complementary error function.

    \n\n

    Computes the inverse of the complementary error function.

    \n\n

    In the complex domain, there is no unique complex number w satisfying\nerfc(w)=z. This indicates a true inverse function would be multivalued.\nWhen the domain restricts to the real, 0 < x < 2, there is a unique real\nnumber satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).

    \n\n

    It is related to inverse of the error function by erfcinv(1-x) = erfinv(x)

    \n\n
    Parameters
    \n\n
      \n
    • y (ndarray):\nArgument at which to evaluate. Domain: [0, 2]
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • erfcinv (scalar or ndarray):\nThe inverse of erfc of y, element-wise
      • \n
    \n\n
    See Also
    \n\n

    erf: Error function of a complex argument
    \nerfc: Complementary error function, 1 - erf(x)
    \nerfinv: Inverse of the error function

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> import matplotlib.pyplot as plt\n>>> from scipy.special import erfcinv\n
    \n
    \n\n
    \n
    >>> erfcinv(0.5)\n0.4769362762044699\n
    \n
    \n\n
    \n
    >>> y = np.linspace(0.0, 2.0, num=11)\n>>> erfcinv(y)\narray([        inf,  0.9061938 ,  0.59511608,  0.37080716,  0.17914345,\n       -0.        , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,\n              -inf])\n
    \n
    \n\n

    Plot the function:

    \n\n
    \n
    >>> y = np.linspace(0, 2, 200)\n>>> fig, ax = plt.subplots()\n>>> ax.plot(y, erfcinv(y))\n>>> ax.grid(True)\n>>> ax.set_xlabel('y')\n>>> ax.set_title('erfcinv(y)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logit": {"fullname": "pyerrors.special.logit", "modulename": "pyerrors.special", "qualname": "logit", "kind": "function", "doc": "

    logit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    logit(x, out=None)

    \n\n

    Logit ufunc for ndarrays.

    \n\n

    The logit function is defined as logit(p) = log(p/(1-p)).\nNote that logit(0) = -inf, logit(1) = inf, and logit(p)\nfor p<0 or p>1 yields nan.

    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nThe ndarray to apply logit to element-wise.
      • \n
    • out (ndarray, optional):\nOptional output array for the function results
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare logit of the corresponding entry of x.
      • \n
    \n\n
    See Also
    \n\n

    expit

    \n\n
    Notes
    \n\n

    As a ufunc logit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs

    \n\n

    New in version 0.10.0.

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import logit, expit\n
    \n
    \n\n
    \n
    >>> logit([0, 0.25, 0.5, 0.75, 1])\narray([       -inf, -1.09861229,  0.        ,  1.09861229,         inf])\n
    \n
    \n\n

    expit is the inverse of logit:

    \n\n
    \n
    >>> expit(logit([0.1, 0.75, 0.999]))\narray([ 0.1  ,  0.75 ,  0.999])\n
    \n
    \n\n

    Plot logit(x) for x in [0, 1]:

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> x = np.linspace(0, 1, 501)\n>>> y = logit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.ylim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('logit(x)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.expit": {"fullname": "pyerrors.special.expit", "modulename": "pyerrors.special", "qualname": "expit", "kind": "function", "doc": "

    expit(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

    \n\n

    expit(x, out=None)

    \n\n

    Expit (a.k.a. logistic sigmoid) ufunc for ndarrays.

    \n\n

    The expit function, also known as the logistic sigmoid function, is\ndefined as expit(x) = 1/(1+exp(-x)). It is the inverse of the\nlogit function.

    \n\n
    Parameters
    \n\n
      \n
    • x (ndarray):\nThe ndarray to apply expit to element-wise.
      • \n
    • out (ndarray, optional):\nOptional output array for the function values
      • \n
    \n\n
    Returns
    \n\n
      \n
    • scalar or ndarray: An ndarray of the same shape as x. Its entries\nare expit of the corresponding entry of x.
      • \n
    \n\n
    See Also
    \n\n

    logit

    \n\n
    Notes
    \n\n

    As a ufunc expit takes a number of optional\nkeyword arguments. For more information\nsee ufuncs

    \n\n

    New in version 0.10.0.

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import expit, logit\n
    \n
    \n\n
    \n
    >>> expit([-np.inf, -1.5, 0, 1.5, np.inf])\narray([ 0.        ,  0.18242552,  0.5       ,  0.81757448,  1.        ])\n
    \n
    \n\n

    logit is the inverse of expit:

    \n\n
    \n
    >>> logit(expit([-2.5, 0, 3.1, 5.0]))\narray([-2.5,  0. ,  3.1,  5. ])\n
    \n
    \n\n

    Plot expit(x) for x in [-6, 6]:

    \n\n
    \n
    >>> import matplotlib.pyplot as plt\n>>> x = np.linspace(-6, 6, 121)\n>>> y = expit(x)\n>>> plt.plot(x, y)\n>>> plt.grid()\n>>> plt.xlim(-6, 6)\n>>> plt.xlabel('x')\n>>> plt.title('expit(x)')\n>>> plt.show()\n
    \n
    \n", "signature": "(*args, **kwargs):", "funcdef": "def"}, "pyerrors.special.logsumexp": {"fullname": "pyerrors.special.logsumexp", "modulename": "pyerrors.special", "qualname": "logsumexp", "kind": "function", "doc": "

    Compute the log of the sum of exponentials of input elements.

    \n\n
    Parameters
    \n\n
      \n
    • a (array_like):\nInput array.
      • \n

    • axis (None or int or tuple of ints, optional):\nAxis or axes over which the sum is taken. By default axis is None,\nand all elements are summed.

      \n\n

      New in version 0.11.0.

      • \n

    • b (array-like, optional):\nScaling factor for exp(a) must be of the same shape as a or\nbroadcastable to a. These values may be negative in order to\nimplement subtraction.

      \n\n

      New in version 0.12.0.

      • \n

    • keepdims (bool, optional):\nIf this is set to True, the axes which are reduced are left in the\nresult as dimensions with size one. With this option, the result\nwill broadcast correctly against the original array.

      \n\n

      New in version 0.15.0.

      • \n

    • return_sign (bool, optional):\nIf this is set to True, the result will be a pair containing sign\ninformation; if False, results that are negative will be returned\nas NaN. Default is False (no sign information).

      \n\n

      New in version 0.16.0.

      • \n
    \n\n
    Returns
    \n\n
      \n
    • res (ndarray):\nThe result, np.log(np.sum(np.exp(a))) calculated in a numerically\nmore stable way. If b is given then np.log(np.sum(b*np.exp(a)))\nis returned.
      • \n
    • sgn (ndarray):\nIf return_sign is True, this will be an array of floating-point\nnumbers matching res and +1, 0, or -1 depending on the sign\nof the result. If False, only one result is returned.
      • \n
    \n\n
    See Also
    \n\n

    numpy.logaddexp,, numpy.logaddexp2

    \n\n
    Notes
    \n\n

    NumPy has a logaddexp function which is very similar to logsumexp, but\nonly handles two arguments. logaddexp.reduce is similar to this\nfunction, but may be less stable.

    \n\n
    Examples
    \n\n
    \n
    >>> import numpy as np\n>>> from scipy.special import logsumexp\n>>> a = np.arange(10)\n>>> logsumexp(a)\n9.4586297444267107\n>>> np.log(np.sum(np.exp(a)))\n9.4586297444267107\n
    \n
    \n\n

    With weights

    \n\n
    \n
    >>> a = np.arange(10)\n>>> b = np.arange(10, 0, -1)\n>>> logsumexp(a, b=b)\n9.9170178533034665\n>>> np.log(np.sum(b*np.exp(a)))\n9.9170178533034647\n
    \n
    \n\n

    Returning a sign flag

    \n\n
    \n
    >>> logsumexp([1,2],b=[1,-1],return_sign=True)\n(1.5413248546129181, -1.0)\n
    \n
    \n\n

    Notice that logsumexp does not directly support masked arrays. To use it\non a masked array, convert the mask into zero weights:

    \n\n
    \n
    >>> a = np.ma.array([np.log(2), 2, np.log(3)],\n...                  mask=[False, True, False])\n>>> b = (~a.mask).astype(int)\n>>> logsumexp(a.data, b=b), np.log(5)\n1.6094379124341005, 1.6094379124341005\n
    \n
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