Multi-dimensional arrays with broadcasting and lazy computing.

Introduction

xtensor is a C++ library meant for numerical analysis with multi-dimensional array expressions.

xtensor provides

• an extensible expression system enabling lazy broadcasting.
• an API following the idioms of the C++ standard library.
• tools to manipulate array expressions and build upon xtensor.

Containers of xtensor are inspired by NumPy, the Python array programming library. Adaptors for existing data structures to be plugged into our expression system can easily be written.

In fact, xtensor can be used to process numpy data structures inplace using Python's buffer protocol. Similarly, we can operate on Julia and R arrays. For more details on the numpy, Julia and R bindings, check out the xtensor-python, xtensor-julia and xtensor-r projects respectively.

xtensor requires a modern C++ compiler supporting C++14. The following C++ compilers are supported:

• On Windows platforms, Visual C++ 2015 Update 2, or more recent
• On Unix platforms, gcc 4.9 or a recent version of Clang

Installation

xtensor is a header-only library. We provide a package for the conda package manager.

conda install -c conda-forge xtensor

Or you can directly install it from the sources:

cmake -D CMAKE_INSTALL_PREFIX=your_install_prefix
make install

Usage

Basic Usage

Initialize a 2-D array and compute the sum of one of its rows and a 1-D array.

#include <iostream>
#include "xtensor/xarray.hpp"
#include "xtensor/xio.hpp"

xt::xarray<double> arr1
  {{1.0, 2.0, 3.0},
   {2.0, 5.0, 7.0},
   {2.0, 5.0, 7.0}};

xt::xarray<double> arr2
  {5.0, 6.0, 7.0};

xt::xarray<double> res = xt::view(arr1, 1) + arr2;

std::cout << res;

Outputs:

{7, 11, 14}

Initialize a 1-D array and reshape it inplace.

#include <iostream>
#include "xtensor/xarray.hpp"
#include "xtensor/xio.hpp"

xt::xarray<int> arr
  {1, 2, 3, 4, 5, 6, 7, 8, 9};

arr.reshape({3, 3});

std::cout << arr;

Outputs:

{{1, 2, 3},
 {4, 5, 6},
 {7, 8, 9}}

The Numpy to xtensor cheat sheet

If you are familiar with numpy APIs, and you are interested in xtensor, you can check out the numpy to xtensor cheat sheet provided in the documentation.

Lazy Broadcasting with xtensor

We can operate on arrays of different shapes of dimensions in an elementwise fashion. Broadcasting rules of xtensor are similar to those of numpy and libdynd.

Broadcasting rules

In an operation involving two arrays of different dimensions, the array with the lesser dimensions is broadcast across the leading dimensions of the other.

For example, if A has shape (2, 3), and B has shape (4, 2, 3), the result of a broadcasted operation with A and B has shape (4, 2, 3).

   (2, 3) # A
(4, 2, 3) # B
---------
(4, 2, 3) # Result

The same rule holds for scalars, which are handled as 0-D expressions. If A is a scalar, the equation becomes:

       () # A
(4, 2, 3) # B
---------
(4, 2, 3) # Result

If matched up dimensions of two input arrays are different, and one of them has size 1, it is broadcast to match the size of the other. Let's say B has the shape (4, 2, 1) in the previous example, so the broadcasting happens as follows:

   (2, 3) # A
(4, 2, 1) # B
---------
(4, 2, 3) # Result

Universal functions, Laziness and Vectorization

With xtensor, if x, y and z are arrays of broadcastable shapes, the return type of an expression such as x + y * sin(z) is not an array. It is an xexpression object offering the same interface as an N-dimensional array, which does not hold the result. Values are only computed upon access or when the expression is assigned to an xarray object. This allows to operate symbolically on very large arrays and only compute the result for the indices of interest.

We provide utilities to vectorize any scalar function (taking multiple scalar arguments) into a function that will perform on xexpressions, applying the lazy broadcasting rules which we just described. These functions are called xfunctions. They are xtensor's counterpart to numpy's universal functions.

In xtensor, arithmetic operations (+, -, *, /) and all special functions are xfunctions.

Iterating over xexpressions and Broadcasting Iterators

All xexpressions offer two sets of functions to retrieve iterator pairs (and their const counterpart).

• begin() and end() provide instances of xiterators which can be used to iterate over all the elements of the expression. The order in which elements are listed is row-major in that the index of last dimension is incremented first.
• xbegin(shape) and xend(shape) are similar but take a broadcasting shape as an argument. Elements are iterated upon in a row-major way, but certain dimensions are repeated to match the provided shape as per the rules described above. For an expression e, e.xbegin(e.shape()) and e.begin() are equivalent.

Runtime vs Compile-time dimensionality

Two container classes implementing multi-dimensional arrays are provided: xarray and xtensor.

• xarray can be reshaped dynamically to any number of dimensions. It is the container that is the most similar to numpy arrays.
• xtensor has a dimension set at compilation time, which enables many optimizations. For example, shapes and strides of xtensor instances are allocated on the stack instead of the heap.

xarray and xtensor container are both xexpressions and can be involved and mixed in universal functions, assigned to each other etc...

Besides, two access operators are provided:

• The variadic template operator() which can take multiple integral arguments or none.
• And the operator[] which takes a single multi-index argument, which can be of size determined at runtime. operator[] also supports access with braced initializers.

Performance

The dynamic nature of xarray over xtensor has a cost. Since the dimension is unknown at build time, the sequences holding shape and strides of xarray instances are heap-allocated, which makes it significantly more expansive than xtensor. Shape and strides of xtensor are stack allocated which makes them more efficient.

More generally, the library implement a promote_shape mechanism at build time to determine the optimal sequence type to hold the shape of an expression. The shape type of a broadcasting expression whose members have a dimensionality determined at compile time will have a stack allocated shape. If a single member of a broadcasting expression has a dynamic dimension (for example an xarray), it bubbles up to entire broadcasting expression which will have a heap allocated shape. The same hold for views, broadcast expressions, etc...

Therefore, when building an application with xtensor, we recommend using statically dimensioned containers whenever possible to improve the overall performance of the application.

Python bindings

The xtensor-python project provides the implementation of two xtensor containers, pyarray and pytensor which effectively wrap numpy arrays, allowing inplace modification, including reshapes.

Building and Running the Tests

Building the tests requires the GTest testing framework and cmake.

gtest and cmake are available as a packages for most linux distributions. Besides, they can also be installed with the conda package manager (even on windows):

conda install -c conda-forge gtest cmake

Once gtest and cmake are installed, you can build and run the tests:

mkdir build
cd build
cmake -DBUILD_TESTS=ON ../
make xtest

You can also use CMake to download the source of gtest, build it, and use the generated libraries:

mkdir build
cd build
cmake -DBUILD_TESTS=ON -DDOWNLOAD_GTEST=ON ../
make xtest

In the context of continuous integration with Travis CI, tests are run in a conda environment, which can be activated with

cd test
conda env create -f ./test-environment.yml
source activate test-xtensor
cd ..
cmake -DBUILD_TESTS=ON .
make xtest

Building the HTML Documentation

xtensor's documentation is built with three tools

• doxygen
• sphinx
• breathe

While doxygen must be installed separately, you can install breathe by typing

pip install breathe

Breathe can also be installed with conda

conda install -c conda-forge breathe

Finally, build the documentation with

make html

from the docs subdirectory.

License

We use a shared copyright model that enables all contributors to maintain the copyright on their contributions.

This software is licensed under the BSD-3-Clause license. See the LICENSE file for details.