From b663c347b6c69b6ae784b9c94514a6c5b9d90a6c Mon Sep 17 00:00:00 2001 From: Sheehan Olver Date: Mon, 9 Mar 2020 11:16:22 +0000 Subject: [PATCH] Lecture 20 --- Manifest.toml | 58 +-- README.md | 1 + notes/Lecture20.pdf | Bin 0 -> 246505 bytes output/Lecture20.pdf | Bin 0 -> 246505 bytes output/Lecture20.tex | 653 ++++++++++++++++++++++++ output/figures/Lecture20_1_1.pdf | Bin 0 -> 7181 bytes output/figures/Lecture20_3_1.pdf | Bin 0 -> 7181 bytes src/Lecture20.jmd | 415 +++++++++++++++- src/Lecture21.jmd | 358 +++++++++++++- src/Lecture22.jmd | 798 ++++++++++++++++++++++++++++++ src/Lecture24.jmd | 54 ++ src/M3M6AppliedComplexAnalysis.jl | 1 + 12 files changed, 2307 insertions(+), 31 deletions(-) create mode 100644 notes/Lecture20.pdf create mode 100644 output/Lecture20.pdf create mode 100644 output/Lecture20.tex create mode 100644 output/figures/Lecture20_1_1.pdf create mode 100644 output/figures/Lecture20_3_1.pdf create mode 100644 src/Lecture22.jmd create mode 100644 src/Lecture24.jmd diff --git a/Manifest.toml b/Manifest.toml index 504f9ca..074d6a1 100644 --- a/Manifest.toml +++ b/Manifest.toml @@ -62,9 +62,9 @@ version = "0.0.4" [[ArrayInterface]] deps = ["LinearAlgebra", "Requires", "SparseArrays"] -git-tree-sha1 = "4f1109cf20b2a31b5196f346bba07b2c61587442" +git-tree-sha1 = "81e5dd1f5374aba2badfe967fc6a132c02ab471a" uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9" -version = "2.4.1" +version = "2.5.0" [[ArrayLayouts]] deps = ["FillArrays", "LinearAlgebra"] @@ -149,9 +149,9 @@ version = "0.2.1" [[ChainRulesCore]] deps = ["MuladdMacro"] -git-tree-sha1 = "a60ff425e8a9834db18e6a60ad728519f234dc89" +git-tree-sha1 = "840cdd255e267a4dfcb14cecd872facb2997a326" uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4" -version = "0.7.0" +version = "0.7.1" [[Codecs]] deps = ["Test"] @@ -225,10 +225,10 @@ uuid = "8f4d0f93-b110-5947-807f-2305c1781a2d" version = "1.4.1" [[Contour]] -deps = ["LinearAlgebra", "StaticArrays", "Test"] -git-tree-sha1 = "b974e164358fea753ef853ce7bad97afec15bb80" +deps = ["StaticArrays"] +git-tree-sha1 = "6d56f927b33d3820561b8f89d7de311718683846" uuid = "d38c429a-6771-53c6-b99e-75d170b6e991" -version = "0.5.1" +version = "0.5.2" [[CoordinateTransformations]] deps = ["Compat", "Rotations", "StaticArrays"] @@ -269,9 +269,9 @@ uuid = "ade2ca70-3891-5945-98fb-dc099432e06a" [[DelayDiffEq]] deps = ["DataStructures", "DiffEqBase", "LinearAlgebra", "Logging", "OrdinaryDiffEq", "Parameters", "Printf", "RecursiveArrayTools", "Reexport", "Roots"] -git-tree-sha1 = "788abe1afd7acdf5015e2a0410a321270030a75b" +git-tree-sha1 = "71ee0eba1f9301462f94f9b2c7c7c321a2d501c8" uuid = "bcd4f6db-9728-5f36-b5f7-82caef46ccdb" -version = "5.21.0" +version = "5.23.0" [[DelimitedFiles]] deps = ["Mmap"] @@ -279,9 +279,9 @@ uuid = "8bb1440f-4735-579b-a4ab-409b98df4dab" [[DiffEqBase]] deps = ["ArrayInterface", "ChainRulesCore", "DataStructures", "Distributed", "DocStringExtensions", "FunctionWrappers", "IterativeSolvers", "IteratorInterfaceExtensions", "LinearAlgebra", "MuladdMacro", "Parameters", "Printf", "RecipesBase", "RecursiveArrayTools", "RecursiveFactorization", "Requires", "Roots", "SparseArrays", "StaticArrays", "Statistics", "SuiteSparse", "TableTraits", "TreeViews", "ZygoteRules"] -git-tree-sha1 = "ac60bdc785530ecc650e9d511e7d928d3805bef6" +git-tree-sha1 = "242055886f3712db6208e940e3ba3b863df2923f" uuid = "2b5f629d-d688-5b77-993f-72d75c75574e" -version = "6.18.1" +version = "6.19.0" [[DiffEqCallbacks]] deps = ["DataStructures", "DiffEqBase", "ForwardDiff", "LinearAlgebra", "NLsolve", "OrdinaryDiffEq", "RecipesBase", "RecursiveArrayTools", "StaticArrays"] @@ -554,9 +554,9 @@ version = "0.6.9" [[ImageMagick]] deps = ["FileIO", "ImageCore", "ImageMagick_jll", "InteractiveUtils", "Libdl", "Pkg", "Random"] -git-tree-sha1 = "6952114380957cbd5858164a909c0d5a1f902031" +git-tree-sha1 = "0563e9b247de1d2950ebaf06971b1e771b0cd8ca" uuid = "6218d12a-5da1-5696-b52f-db25d2ecc6d1" -version = "1.1.2" +version = "1.1.3" [[ImageMagick_jll]] deps = ["JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pkg", "Zlib_jll", "libpng_jll"] @@ -839,9 +839,9 @@ version = "0.11.4" [[OpenBLAS_jll]] deps = ["Libdl", "Pkg"] -git-tree-sha1 = "adc45e596df7007d48bf6829efb1dc64fdec3ddc" +git-tree-sha1 = "858f107d79a016d9511e34186fe2af11566ba762" uuid = "4536629a-c528-5b80-bd46-f80d51c5b363" -version = "0.3.7+6" +version = "0.3.7+7" [[OpenSpecFun_jll]] deps = ["CompilerSupportLibraries_jll", "Libdl", "Pkg"] @@ -869,9 +869,9 @@ version = "0.3.0" [[PaddedViews]] deps = ["OffsetArrays"] -git-tree-sha1 = "ec6add2adab1abc9a2dbf27912da3022b0c55592" +git-tree-sha1 = "6619be66cd81258f6d7a391483bd5f93656d6772" uuid = "5432bcbf-9aad-5242-b902-cca2824c8663" -version = "0.5.1" +version = "0.5.2" [[Parameters]] deps = ["OrderedCollections"] @@ -996,9 +996,9 @@ version = "1.0.0" [[Roots]] deps = ["Printf"] -git-tree-sha1 = "dcc013908465ca1019b34b4bf547b6a187d195f9" +git-tree-sha1 = "869a9990ec6347862d59040d00416ecd0683b965" uuid = "f2b01f46-fcfa-551c-844a-d8ac1e96c665" -version = "0.8.4" +version = "1.0.0" [[Rotations]] deps = ["LinearAlgebra", "StaticArrays", "Statistics"] @@ -1036,9 +1036,9 @@ version = "0.9.1" [[SingularIntegralEquations]] deps = ["ApproxFun", "ApproxFunBase", "ApproxFunFourier", "ApproxFunOrthogonalPolynomials", "ApproxFunSingularities", "BandedMatrices", "DomainSets", "DualNumbers", "HypergeometricFunctions", "LinearAlgebra", "LowRankApprox", "SpecialFunctions", "Test"] -git-tree-sha1 = "b45f871f9d06446fa5d778d07762e6c334be003d" +git-tree-sha1 = "9e16152baff2358e10d58001c9896c0436e48e62" uuid = "e094c991-5a90-5477-8896-c1e4c9552a1a" -version = "0.6.1" +version = "0.6.2" [[Sockets]] uuid = "6462fe0b-24de-5631-8697-dd941f90decc" @@ -1061,9 +1061,9 @@ uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" [[SparseDiffTools]] deps = ["Adapt", "ArrayInterface", "Compat", "DataStructures", "FiniteDiff", "ForwardDiff", "InteractiveUtils", "LightGraphs", "LinearAlgebra", "Requires", "SparseArrays", "VertexSafeGraphs"] -git-tree-sha1 = "f65937edd6d1e9ba48da16315c14c12568867dd6" +git-tree-sha1 = "6429f706e7da24a8c1c5e0416c3f0ce9974ab991" uuid = "47a9eef4-7e08-11e9-0b38-333d64bd3804" -version = "1.3.3" +version = "1.4.0" [[SpecialFunctions]] deps = ["OpenSpecFun_jll"] @@ -1083,9 +1083,9 @@ uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" [[StatsBase]] deps = ["DataAPI", "DataStructures", "LinearAlgebra", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics"] -git-tree-sha1 = "be5c7d45daa449d12868f4466dbf5882242cf2d9" +git-tree-sha1 = "19bfcb46245f69ff4013b3df3b977a289852c3a1" uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" -version = "0.32.1" +version = "0.32.2" [[SteadyStateDiffEq]] deps = ["DiffEqBase", "DiffEqCallbacks", "LinearAlgebra", "NLsolve", "Reexport"] @@ -1123,9 +1123,9 @@ version = "1.0.0" [[Tables]] deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"] -git-tree-sha1 = "a54b8ce702aa863eced47bade03123d4dca0db84" +git-tree-sha1 = "242b7fde70b8bc6a30d6476adf17ca3cf1ced6ee" uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c" -version = "1.0.2" +version = "1.0.3" [[Test]] deps = ["Distributed", "InteractiveUtils", "Logging", "Random"] @@ -1175,9 +1175,9 @@ version = "0.1.1" [[Weave]] deps = ["Base64", "Compat", "Dates", "Highlights", "JSON", "Markdown", "Mustache", "Printf", "REPL", "Requires", "Serialization", "YAML"] -git-tree-sha1 = "a1698569cb25ddefb4bf538f39e77afdc7aa720e" +git-tree-sha1 = "c80b1bfc82d19bc2dd0cd0579d08825d021b5b01" uuid = "44d3d7a6-8a23-5bf8-98c5-b353f8df5ec9" -version = "0.9.1" +version = "0.9.3" [[WebIO]] deps = ["AssetRegistry", "Base64", "Distributed", "FunctionalCollections", "JSON", "Logging", "Observables", "Pkg", "Random", "Requires", "Sockets", "UUIDs", "WebSockets", "Widgets"] diff --git a/README.md b/README.md index ba789c2..f4fa007 100644 --- a/README.md +++ b/README.md @@ -48,3 +48,4 @@ Examples of previous projects: 17. [Logarithmic singular integrals](notes/Lecture17.pdf) 18. [Logarithmic singular integral examples](notes/Lecture18.pdf) 19. [Inverting logarithmic singular integrals](notes/Lecture19.pdf) +20. 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z@kdaB#%GIOV^lY3^xn_CRjSUp2$2#8lIVpKj(Yt=9qKeCny^mNAI#TntxuXq9yE1K zsT*ZSDfjE7fvne^&vsfAXPRl8phNh#%K3ycKKAzmd1)q0TJLJum_K>n3)N^|_d9gX zmy>SRF?|*@HJ?Nt$9|EC{qQC7ya`B#E!To+X`f{\left\langle{#1}\right\rangle} +\def\dx{\D x} +\def\dt{\D t} +\def\dz{\D z} + +\def\norm#1{\left\| #1 \right\|} + +\def\pr(#1){\left({#1}\right)} +\def\br[#1]{\left[{#1}\right]} + +\def\abs#1{\left|{#1}\right|} +\def\fpr(#1){\!\pr({#1})} + +\def\sopmatrix#1{ \begin{pmatrix}#1\end{pmatrix} } + +\def\endash{–} +\def\mdblksquare{\blacksquare} +\def\lgblksquare{\blacksquare} +\def\scre{\E} +\def\mapengine#1,#2.{\mapfunction{#1}\ifx\void#2\else\mapengine #2.\fi } + +\def\map[#1]{\mapengine #1,\void.} + +\def\mapenginesep_#1#2,#3.{\mapfunction{#2}\ifx\void#3\else#1\mapengine #3.\fi } + +\def\mapsep_#1[#2]{\mapenginesep_{#1}#2,\void.} + + +\def\vcbr[#1]{\pr(#1)} + + +\def\bvect[#1,#2]{ +{ +\def\dots{\cdots} +\def\mapfunction##1{\ | \ ##1} + \sopmatrix{ + \,#1\map[#2]\, + } +} +} + + + +\def\vect[#1]{ +{\def\dots{\ldots} + \vcbr[{#1}] +} } + +\def\vectt[#1]{ +{\def\dots{\ldots} + \vect[{#1}]^{\top} +} } + +\def\Vectt[#1]{ +{ +\def\mapfunction##1{##1 \cr} +\def\dots{\vdots} + \begin{pmatrix} + \map[#1] + \end{pmatrix} +} } + +\def\addtab#1={#1\;&=} +\def\ccr{\\\addtab} + +\begin{document} + +\textbf{M3M6: Applied Complex Analysis} + +Dr. Sheehan Olver + +s.olver@imperial.ac.uk + +\section{Lecture 20: Orthogonal polynomials} +We now introduce orthogonal polynomials (OPs). These are \textbf{fundamental} for computational mathematics, with applications in + +\begin{itemize} +\item[1. ] Function approximation + + +\item[2. ] Quadrature (calculating integrals) + + +\item[3. ] Solving differential equations + + +\item[4. ] Spectral analysis of Schrödinger operators + +\end{itemize} +We will investigate the properties of \emph{general} OPs, in this lecture: + +\begin{itemize} +\item[1. ] Definition of orthogonal polynomials + + +\item[2. ] Three-term recurrence relationships + + +\item[3. ] Function approximation with orthogonal polynomials + + +\item[4. ] Construction of orthogonal polynomials via Gram\ensuremath{\endash}Schmidt process + +\end{itemize} +\subsection{Definition of orthogonal polynomials} +Let $p_0(x),p_1(x),p_2(x),\ensuremath{\dots}$ be a sequence of polynomials such that $p_n(x)$ is exactly degree $n$, that is, + +\[ +p_n(x) = k_n x^n + O(x^{n-1}) +\] +where $k_n \neq 0$. + +Let $w(x)$ be a continuous weight function on a (possibly infinite) interval $(a,b)$: that is $w(x) \geq 0$ for all $a < x < b$. This induces an inner product + +\[ +\ip := \int_a^b f(x) g(x) w(x) \dx +\] +We say that $\{p_0, p_1,\ldots\}$ are \emph{orthogonal with respect to the weight $w$} if + +\[ +\ip = 0\qqfor n \neq m. +\] +Because $w$ is continuous, we have + +\[ +\norm{p_n}^2 = \ip > 0 . +\] +Orthogonal polymomials are not unique: we can multiply each $p_n$ by a different nonzero constant $\tilde p_n(x) = c_n p_n(x)$, and $\tilde p_n$ will be orthogonal w.r.t. $w$. However, if we specify $k_n$, this is sufficient to uniquely define them: + +\textbf{Proposition (Uniqueness of OPs I)} Given a non-zero $k_n$, there is a unique polynomial $p_n$ orthogonal w.r.t. $w$ to all lower degree polynomials. + +\textbf{Proof} Suppose $r_n(x) = k_n x^n + O(x^{n-1})$ is another OP w.r.t. $w$. We want to show $p_n - r_n$ is zero. But this is a polynomial of degree $ c_k = \ip = \ip - \ip = 0 - 0 = 0 +\] +which shows all $c_k$ are zero. + +\ensuremath{\blacksquare} + +\textbf{Corollary (Uniqueness of OPs I)} If $q_n$ and $p_n$ are orthogonal w.r.t. $w$ to all lower degree polynomials, then $q_n(x) = C p_n(x)$ for some constant $C$. + +\subsubsection{Monic orthogonal polynomials} +If $k_n = 1$, that is, + +\[ +p_n(x) = x^n + O(x^{n-1}) +\] +then we refer to the orthogonal polymomials as monic. + +Monic OPs are unique as we have specified $k_n$. + +\subsubsection{Orthonormal polynomials} +If $\norm{p_n} = 1$, then we refer to the orthogonal polynomials as orthonormal w.r.t. $w$. We will usually use $q_n$ when they are orthonormal. Note it's not unique: we can multiply by $\pm 1$ without changing the norm. + +\textbf{Remark} The classical OPs are \textbf{not} monic or orthonormal (apart from one case). Many people make the mistake of using orthonormal polynomials for computations. But there is a good reason to use classical OPs: their properties result in rational formulae, whereas orthonormal polynomials require square roots. This makes a performance difference. + +\subsection{Function approximation with orthogonal polynomials} +A basic usage of orthogonal polynomials is for polynomial approximation. Suppose $f(x)$ is a degree $n-1$ polynomial. Since $\{p_0(x),\ldots,p_{n-1}(x)\}$ span all degree $n-1$ polynomials, we know that + +\[ +f(x) = \sum_{k=0}^{n-1} f_k p_k(x) +\] +where + +\[ +f_k = {\ip \over \ip} +\] +Sometimes, we want to incorporate the weight into the approximation. This is typically one of two forms, depending on the application: + +\[ +f(x) = w(x) \sum_{k=0}^\infty f_k p_k(x) +\] +or + +\[ + f(x) = \sqrt{w(x)} \sum_{k=0}^\infty f_k p_k(x) +\] +\subsection{Jacobi operators and three-term recurences for general orthogonal polynomials} +\subsubsection{Three-term recurrence relationships} +A central theme: if you know the Jacobi operator / three-term recurrence, you know the polynomials. This is the \textbf{best} way to evaluate expansions in orthogonal polynomials: even for cases where we have explicit formulae (e.g. Chebyshev polynomials $T_n(x) = \cos n \acos x$), using the recurrence avoids evaluating trigonometric functions. + +Every family of orthogonal polynomials has a three-term recurrence relationship: + +\textbf{Theorem (three-term recurrence)} Suppose $\{p_n(x)\}$ are a family of orthogonal polynomials w.r.t. a weight $w(x)$. Then there exists constants $a_n \neq 0$, $b_n$ and $c_n$ such that + + +\begin{align*} +x p_0(x) = a_0 p_0(x) + b_0 p_1(x) \\ +x p_n(x) = c_n p_{n-1}(x) + a_n p_n(x) + b_n p_{n+1}(x) +\end{align*} +\textbf{Proof} The first part follows since $p_0(x)$ and $p_1(x)$ span all degree 1 polynomials. + +The second part follows essentially because multiplication by $x$ is "self-adjoint", that is, + +\[ +\ip = \int_a^b x f(x) g(x) w(x) \dx = \ip +\] +Therefore, if $f_m$ is a degree $m < n-1$ polynomial, we have + +\[ +\ip = \ip = 0. +\] +In particular, if we write + +\[ +x p_n(x) = \sum_{k=0}^{n+1} C_k p_k(x) +\] +then + +\[ +C_k = {\ip< x p_n, p_k> \over \norm{p_k}^2} = 0 +\] +if $k < n-1$. + +\ensuremath{\blacksquare} + +Monic polynomials clearly have $b_n = 1$. Orthonormal polynomials have an even simpler form: + +\textbf{Theorem (orthonormal three-term recurrence)} Suppose $\{q_n(x)\}$ are a family of orthonotms polynomials w.r.t. a weight $w(x)$. Then there exists constants $a_n$ and $b_n$ such that + + +\begin{align*} +x q_0(x) = a_0 q_0(x) + b_0 q_1(x)\\ +x q_n(x) = b_{n-1} q_{n-1}(x) + a_n q_n(x) + b_{n} q_{n+1}(x) +\end{align*} +\textbf{Proof} Follows again by self-adjointness of multiplication by $x$: + +\[ +c_n = \ip = \ip = \ip = b_{n-1} +\] +\ensuremath{\blacksquare} + +\textbf{Corollary (symmetric three-term recurrence implies orthonormal)} Suppose $\{p_n(x)\}$ are a family of orthogonal polynomials w.r.t. a weight $w(x)$ such that + + +\begin{align*} +x p_0(x) = a_0 p_0(x) + b_0 p_1(x)\\ +x p_n(x) = b_{n-1} p_{n-1}(x) + a_n p_n(x) + b_{n} p_{n+1}(x) +\end{align*} +with $b_n \neq 0$. Suppose further that $\norm{p_0} = 1$. Then $p_n$ must be orthonormal. + +\textbf{Proof} This follows from + +\[ +b_n = {\ip \over \norm{p_{n+1}}^2} = {\ip \over \norm{p_{n+1}}^2} = b_n {\norm{p_n}^2 \over \norm{p_{n+1}}^2 } +\] +which implies + +\[ +\norm{p_{n+1}}^2 = \norm{p_n}^2 = \cdots = \norm{p_0}^2 = 1 +\] +\ensuremath{\blacksquare} + +\textbf{Remark} We can scale $w(x)$ by a constant without changing the orthogonality properties, hence we can make $\|p_0\| = 1$ by changing the weight. + +\textbf{Remark} This is beyond the scope of this course, but satisfying a three-term recurrence like this such that coefficients are bounded with $p_0(x) = 1$ is sufficient to show that there exists a $w(x)$ (or more accurately, a Borel measure) such that $p_n(x)$ are orthogonal w.r.t. $w$. The relationship between the coefficients $a_n,b_n$ and the $w(x)$ is an object of study in spectral theory, particularly when the coefficients are periodic, quasi-periodic or random. + +\subsection{Jacobi operators and multiplication by $x$} +We can rewrite the three-term recurrence as + +\[ +x \begin{pmatrix} p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \end{pmatrix} = J\begin{pmatrix} p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \end{pmatrix} +\] +where $J$ is a Jacobi operator, an infinite-dimensional tridiagonal matrix: + +\[ +J = \begin{pmatrix} +a_0 & b_0 \cr +c_1 & a_1 & b_1 \cr +& c_2 & a_2 & b_2 \cr +&& c_3 & a_3 & \ddots \cr +&&&\ddots & \ddots +\end{pmatrix} +\] +When the polynomials are monic, we have $1$ on the superdiagonal. When we have an orthonormal basis, then $J$ is symmetric: + +\[ +J = \begin{pmatrix} +a_0 & b_0 \cr +b_0 & a_1 & b_1 \cr +& b_1 & a_2 & b_2 \cr +&& b_2 & a_3 & \ddots \cr +&&&\ddots & \ddots +\end{pmatrix} +\] +Given a polynomial expansion, $J$ tells us how to multiply by $x$ in coefficient space, that is, if + +\[ +f(x) = \sum_{k=0}^\infty f_k p_k(x) = (p_0(x) , p_1(x) , \ldots ) \begin{pmatrix}f_0\\ f_1\\f_2\\\vdots\end{pmatrix} +\] +then (provided the relevant sums converge absolutely and uniformly) + +\[ +x f(x) = x (p_0(x) , p_1(x) , \ldots ) \begin{pmatrix}f_0\\ f_1\\f_2\\\vdots\end{pmatrix} = + \left(J \begin{pmatrix} p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \end{pmatrix}\right)^\top \begin{pmatrix}f_0\\ f_1\\f_2\\\vdots\end{pmatrix} = (p_0(x) , p_1(x) , \ldots ) X \begin{pmatrix}f_0\\ f_1\\f_2\\\vdots\end{pmatrix} +\] +where $X := J^\top$. + +\subsubsection{Evaluating polynomials} +We can use the three-term recurrence to construct the polynomials. I think it's nicest to express this in terms of linear algebra. Suppose we are given $p_0(x) = k_0$ (where $k_0 = 1$ is pretty much always the case in practice). This can be written in matrix form as + +\[ +(1,0,0,0,0,\ldots) \begin{pmatrix} p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \end{pmatrix} = k_0 +\] +We can combine this with the Jacobi operator to get + +\[ +\underbrace{\begin{pmatrix} +1 \\ +a_0-x & b_0 \\ +c_1 & a_1-x & b_1 \\ +& c_2 & a_2-x & b_2 \cr +&& c_3 & a_3-x & b_3 \cr +&&&\ddots & \ddots & \ddots +\end{pmatrix}}_{L_x} \begin{pmatrix} p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \end{pmatrix} = \begin{pmatrix} k_0\cr 0 \cr 0 \cr \vdots \end{pmatrix} +\] +For $x$ fixed, this is a lower triangular system, that is, the polynomials equal + +\[ +k_0 L_x^{-1} \vc e_0 +\] +This can be solved via forward recurrence: + + +\begin{align*} + p_0(x) &= k_0 \\ + p_1(x) &= {(x-a_0) p_0(x) \over b_0}\\ + p_2(x) &= {(x-a_1) p_0(x) - c_1 p_0(x) \over b_1}\\ + p_3(x) &= {(x-a_2) p_1(x) - c_2 p_1(x) \over b_2}\\ + &\vdots +\end{align*} +We can use this to evaluate functions as well: + +\[ +f(x) = (p_0(x),p_1(x),\ldots) \begin{pmatrix}f_0 \\ f_1\\ \vdots \end{pmatrix} = +k_0 \vc e_0^\top L_x^{-\top} \begin{pmatrix}f_0 \\ f_1\\ \vdots \end{pmatrix} +\] +when $f$ is a degree $n-1$ polynomial, this becomes a problem of inverting an upper triangular matrix, that is, we want to solve the $n \times n$ system + +\[ +\underbrace{\begin{pmatrix} +1 & a_0-x & c_1 \\ +& b_0 & a_1-x & c_2 \\ +& & b_1 & a_2-x & \ddots \\ +& & & b_2 & \ddots & c_{n-2} \\ +&&&&\ddots & a_{n-2}-x \\ +&&&&& b_{n-2} +\end{pmatrix}}_{L_x^\top} \begin{pmatrix} \gamma_0 \\\vdots\\ \gamma_{n-1} \end{pmatrix} +\] +so that $f(x) = \gamma_0$. We we can solve this via back-substitution: + + +\begin{align*} +\gamma_{n-1} &= {f_{n-1} \over b_{n-2}} \\ +\gamma_{n-2} &= {f_{n-2} - (a_{n-2}-x) \gamma_{n-1} \over b_{n-3}} \\ +\gamma_{n-3} &= {f_{n-3} - (a_{n-3}-x) \gamma_{n-2} - c_{n-2} \gamma_{n-1} \over b_{n-4}} \\ +& \vdots \\ +\gamma_1 &= {f_1 - (a_1-x) \gamma_2 - c_2 \gamma_3 \over b_0} \\ +\gamma_0 &= f_0 - (a_0-x) \gamma_1 - c_1 \gamma_2 +\end{align*} +We give examples of these algorithms applied to Chebyshev polynomials in the next lecture. + +\subsection{Gram\ensuremath{\endash}Schmidt algorithm} +In general we don't have nice formulae. But we can always construct them via Gram\ensuremath{\endash}Schmidt: + +\textbf{Proposition (Gram\ensuremath{\endash}Schmidt)} Define + + +\begin{align*} +p_0(x) = 1 \\ +q_0(x) = {1 \over \norm{p_0}}\\ +p_{n+1}(x) = x q_n(x) - \sum_{k=0}^n \ip q_k(x)\\ +q_{n+1}(x) = {p_{n+1}(x) \over \norm{p_n}} +\end{align*} +Then $q_0(x), q_1(x), \ldots$ are orthonormal w.r.t. $w$. + +\textbf{Proof} By linearity we have + +\[ +\ip = \ip} q_k, q_j> = \ip - \ip \ip = 0 +\] +Thus $p_{n+1}$ is orthogonal to all lower degree polynomials. So is $q_{n+1}$, since it is a constant multiple of $p_{n+1}$. + +\ensuremath{\blacksquare} + +Let's make our own family: + + +\begin{lstlisting} +(*@\HLJLk{using}@*) (*@\HLJLn{ApproxFun}@*)(*@\HLJLp{,}@*) (*@\HLJLn{Plots}@*) +(*@\HLJLn{x}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{Fun}@*)(*@\HLJLp{()}@*) +(*@\HLJLn{w}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{exp}@*)(*@\HLJLp{(}@*)(*@\HLJLn{x}@*)(*@\HLJLp{)}@*) +(*@\HLJLn{ip}@*) (*@\HLJLoB{=}@*) (*@\HLJLp{(}@*)(*@\HLJLn{f}@*)(*@\HLJLp{,}@*)(*@\HLJLn{g}@*)(*@\HLJLp{)}@*) (*@\HLJLoB{->}@*) (*@\HLJLnf{sum}@*)(*@\HLJLp{(}@*)(*@\HLJLn{f}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{g}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{w}@*)(*@\HLJLp{)}@*) +(*@\HLJLn{nrm}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{f}@*) (*@\HLJLoB{->}@*) (*@\HLJLnf{sqrt}@*)(*@\HLJLp{(}@*)(*@\HLJLnf{ip}@*)(*@\HLJLp{(}@*)(*@\HLJLn{f}@*)(*@\HLJLp{,}@*)(*@\HLJLn{f}@*)(*@\HLJLp{))}@*) +(*@\HLJLn{n}@*) (*@\HLJLoB{=}@*) (*@\HLJLni{10}@*) +(*@\HLJLn{q}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{Array}@*)(*@\HLJLp{{\{}}@*)(*@\HLJLn{Fun}@*)(*@\HLJLp{{\}}(}@*)(*@\HLJLn{undef}@*)(*@\HLJLp{,}@*)(*@\HLJLn{n}@*)(*@\HLJLp{)}@*) +(*@\HLJLn{p}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{Array}@*)(*@\HLJLp{{\{}}@*)(*@\HLJLn{Fun}@*)(*@\HLJLp{{\}}(}@*)(*@\HLJLn{undef}@*)(*@\HLJLp{,}@*)(*@\HLJLn{n}@*)(*@\HLJLp{)}@*) +(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{Fun}@*)(*@\HLJLp{(}@*)(*@\HLJLni{1}@*)(*@\HLJLp{,}@*) (*@\HLJLoB{-}@*)(*@\HLJLni{1}@*) (*@\HLJLoB{..}@*) (*@\HLJLni{1}@*) (*@\HLJLp{)}@*) +(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*)(*@\HLJLoB{/}@*)(*@\HLJLnf{nrm}@*)(*@\HLJLp{(}@*)(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{])}@*) + +(*@\HLJLk{for}@*) (*@\HLJLn{k}@*)(*@\HLJLoB{=}@*)(*@\HLJLni{1}@*)(*@\HLJLoB{:}@*)(*@\HLJLn{n}@*)(*@\HLJLoB{-}@*)(*@\HLJLni{1}@*) + (*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{x}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLp{]}@*) + (*@\HLJLk{for}@*) (*@\HLJLn{j}@*)(*@\HLJLoB{=}@*)(*@\HLJLni{1}@*)(*@\HLJLoB{:}@*)(*@\HLJLn{k}@*) + (*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{-=}@*) (*@\HLJLnf{ip}@*)(*@\HLJLp{(}@*)(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{],}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{j}@*)(*@\HLJLp{])}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{j}@*)(*@\HLJLp{]}@*) + (*@\HLJLk{end}@*) + (*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*)(*@\HLJLoB{/}@*)(*@\HLJLnf{nrm}@*)(*@\HLJLp{(}@*)(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{])}@*) +(*@\HLJLk{end}@*) + +(*@\HLJLnf{sum}@*)(*@\HLJLp{(}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLni{2}@*)(*@\HLJLp{]}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLni{4}@*)(*@\HLJLp{]}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{w}@*)(*@\HLJLp{)}@*) + +(*@\HLJLn{p}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{plot}@*)(*@\HLJLp{(;}@*) (*@\HLJLn{legend}@*)(*@\HLJLoB{=}@*)(*@\HLJLkc{false}@*)(*@\HLJLp{)}@*) +(*@\HLJLk{for}@*) (*@\HLJLn{k}@*)(*@\HLJLoB{=}@*)(*@\HLJLni{1}@*)(*@\HLJLoB{:}@*)(*@\HLJLni{10}@*) + (*@\HLJLnf{plot!}@*)(*@\HLJLp{(}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLp{])}@*) +(*@\HLJLk{end}@*) +(*@\HLJLn{p}@*) +\end{lstlisting} + +\includegraphics[width=\linewidth]{figures/Lecture20_1_1.pdf} + +The three-term recurrence means we can simplify Gram\ensuremath{\endash}Schmidt, and calculate the recurrence coefficients at the same time: + +\textbf{Proposition (Gram\ensuremath{\endash}Schmidt)} Define + + +\begin{align*} +p_0(x) &= 1 \\ +q_0(x) &= {1 \over \norm{p_0}}\\ +a_n &= \ip \\ +b_{n-1} &= \ip\\ +p_{n+1}(x) &= x q_n(x) - a_n q_n(x) - b_{n-1} q_{n-1}(x)\\ +q_{n+1}(x) &= {p_{n+1}(x) \over \norm{p_n}} +\end{align*} +Then $q_0(x), q_1(x), \ldots$ are orthonormal w.r.t. $w$. + +\textbf{Remark} This can be made a bit more efficient by using $\norm{p_n}$ to calculate $b_n$. + + +\begin{lstlisting} +(*@\HLJLn{x}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{Fun}@*)(*@\HLJLp{()}@*) +(*@\HLJLn{w}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{exp}@*)(*@\HLJLp{(}@*)(*@\HLJLn{x}@*)(*@\HLJLp{)}@*) +(*@\HLJLn{ip}@*) (*@\HLJLoB{=}@*) (*@\HLJLp{(}@*)(*@\HLJLn{f}@*)(*@\HLJLp{,}@*)(*@\HLJLn{g}@*)(*@\HLJLp{)}@*) (*@\HLJLoB{->}@*) (*@\HLJLnf{sum}@*)(*@\HLJLp{(}@*)(*@\HLJLn{f}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{g}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{w}@*)(*@\HLJLp{)}@*) +(*@\HLJLn{nrm}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{f}@*) (*@\HLJLoB{->}@*) (*@\HLJLnf{sqrt}@*)(*@\HLJLp{(}@*)(*@\HLJLnf{ip}@*)(*@\HLJLp{(}@*)(*@\HLJLn{f}@*)(*@\HLJLp{,}@*)(*@\HLJLn{f}@*)(*@\HLJLp{))}@*) +(*@\HLJLn{n}@*) (*@\HLJLoB{=}@*) (*@\HLJLni{10}@*) +(*@\HLJLn{q}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{Array}@*)(*@\HLJLp{{\{}}@*)(*@\HLJLn{Fun}@*)(*@\HLJLp{{\}}(}@*)(*@\HLJLn{undef}@*)(*@\HLJLp{,}@*) (*@\HLJLn{n}@*)(*@\HLJLp{)}@*) +(*@\HLJLn{p}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{Array}@*)(*@\HLJLp{{\{}}@*)(*@\HLJLn{Fun}@*)(*@\HLJLp{{\}}(}@*)(*@\HLJLn{undef}@*)(*@\HLJLp{,}@*) (*@\HLJLn{n}@*)(*@\HLJLp{)}@*) +(*@\HLJLn{a}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{zeros}@*)(*@\HLJLp{(}@*)(*@\HLJLn{n}@*)(*@\HLJLp{)}@*) +(*@\HLJLn{b}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{zeros}@*)(*@\HLJLp{(}@*)(*@\HLJLn{n}@*)(*@\HLJLp{)}@*) +(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{Fun}@*)(*@\HLJLp{(}@*)(*@\HLJLni{1}@*)(*@\HLJLp{,}@*) (*@\HLJLoB{-}@*)(*@\HLJLni{1}@*) (*@\HLJLoB{..}@*) (*@\HLJLni{1}@*) (*@\HLJLp{)}@*) +(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*)(*@\HLJLoB{/}@*)(*@\HLJLnf{nrm}@*)(*@\HLJLp{(}@*)(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{])}@*) + +(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{2}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{x}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) +(*@\HLJLn{a}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{ip}@*)(*@\HLJLp{(}@*)(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{2}@*)(*@\HLJLp{],}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{])}@*) +(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{2}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{-=}@*) (*@\HLJLn{a}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) +(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLni{2}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{2}@*)(*@\HLJLp{]}@*)(*@\HLJLoB{/}@*)(*@\HLJLnf{nrm}@*)(*@\HLJLp{(}@*)(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLni{2}@*)(*@\HLJLp{])}@*) + +(*@\HLJLk{for}@*) (*@\HLJLn{k}@*)(*@\HLJLoB{=}@*)(*@\HLJLni{2}@*)(*@\HLJLoB{:}@*)(*@\HLJLn{n}@*)(*@\HLJLoB{-}@*)(*@\HLJLni{1}@*) + (*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{x}@*)(*@\HLJLoB{*}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLp{]}@*) + (*@\HLJLn{b}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{-}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*)(*@\HLJLnf{ip}@*)(*@\HLJLp{(}@*)(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{],}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{-}@*)(*@\HLJLni{1}@*)(*@\HLJLp{])}@*) + (*@\HLJLn{a}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{ip}@*)(*@\HLJLp{(}@*)(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{],}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLp{])}@*) + (*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{-}@*) (*@\HLJLn{a}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLp{]}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{-}@*) (*@\HLJLn{b}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{-}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{-}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) + (*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*) (*@\HLJLoB{=}@*) (*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{]}@*)(*@\HLJLoB{/}@*)(*@\HLJLnf{nrm}@*)(*@\HLJLp{(}@*)(*@\HLJLn{p}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLoB{+}@*)(*@\HLJLni{1}@*)(*@\HLJLp{])}@*) +(*@\HLJLk{end}@*) + +(*@\HLJLnf{ip}@*)(*@\HLJLp{(}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLni{5}@*)(*@\HLJLp{],}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLni{2}@*)(*@\HLJLp{])}@*) (*@\HLJLcs{{\#}}@*) (*@\HLJLcs{shows}@*) (*@\HLJLcs{orthogonality}@*) (*@\HLJLcs{(to}@*) (*@\HLJLcs{numerical}@*) (*@\HLJLcs{accuracy)}@*) +\end{lstlisting} + +\begin{lstlisting} +5.828670879282072e-16 +\end{lstlisting} + + +Here we see a plot of the first 10 polynomials: + + +\begin{lstlisting} +(*@\HLJLn{p}@*) (*@\HLJLoB{=}@*) (*@\HLJLnf{plot}@*)(*@\HLJLp{(;}@*) (*@\HLJLn{legend}@*)(*@\HLJLoB{=}@*)(*@\HLJLkc{false}@*)(*@\HLJLp{)}@*) +(*@\HLJLk{for}@*) (*@\HLJLn{k}@*)(*@\HLJLoB{=}@*)(*@\HLJLni{1}@*)(*@\HLJLoB{:}@*)(*@\HLJLni{10}@*) + (*@\HLJLnf{plot!}@*)(*@\HLJLp{(}@*)(*@\HLJLn{q}@*)(*@\HLJLp{[}@*)(*@\HLJLn{k}@*)(*@\HLJLp{])}@*) +(*@\HLJLk{end}@*) +(*@\HLJLn{p}@*) +\end{lstlisting} + +\includegraphics[width=\linewidth]{figures/Lecture20_3_1.pdf} + + +\end{document} diff --git a/output/figures/Lecture20_1_1.pdf b/output/figures/Lecture20_1_1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c74e4aacf76d8e5ba1d318d8561354c87bda06ea GIT binary patch literal 7181 zcmc&(c{r5s*O$navSrIM*(uBn2HAJAmo=fmU>Gxuv1Ou#P?$|C>pJz&Mzwhsl_s{pau5;bbIrrzR_kG=eoFi#tXe;ZQgfssvSs!C)%N$^gIykMj!g zMEz2Ls6YVA5HCDB6or>%)z)T3VZA`h@V`9RMFgTCiUvpm5`*&r3?O|_!4PEtaFSKg zI)s2hV*y@?#{aJYklzjQmw{a<5F;)GOMpQCatB)bPc+TTE)>S9XpZ&@hPbGJUR?iD zRR2rS0)_P<_(EWCRz+hphJeB=>YXw$Gc!en8=^dMUMNs4m;jV3fE7ynylEep@^3L{ z{=dTjQ^uo!h=1pN5*3UK!FvKpYidFijd8#Z?ME4+3Wf&(+$b7_6HHDKz%u~P$mA4U zg@!TEgFqFa>S|CWbvQ)P6ZsbhYJs>vKo|`G1I>YZHb?@H5*D-vvQE|>ekf1SGsuDx zOOzKHsgDbXxIlq0R8$cVB{gMNu!L9w5PdKN0Y(Qxf1feP?`HeEcFJl9#BTK 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z-|rFz=>rUTz|jieTAwy%kcS}<5P3K-+JFJ$6o|SK42bL$0Tti?Q3GBt17x5n3hm=d zfT+UMSQYhxJ;50Y?H_apGV*Y!DyyOyFjzu+>S28_C~zL~SFquDls79BF#mgC15Rxq z2o)726^J+Fj|_MYzzrw_`$qT^Rw7L}O5R+T`vO`Wy<(P4OfSM}UA6H(;P := \int_a^b f(x) g(x) w(x) \dx +$$ +We say that $\{p_0, p_1,\ldots\}$ are _orthogonal with respect to the weight $w$_ if +$$ +\ip = 0\qqfor n \neq m. +$$ +Because $w$ is continuous, we have +$$ +\norm{p_n}^2 = \ip > 0 . +$$ + +Orthogonal polymomials are not unique: we can multiply each $p_n$ by a different nonzero constant $\tilde p_n(x) = c_n p_n(x)$, and +$\tilde p_n$ will be orthogonal w.r.t. $w$. However, if we specify $k_n$, this is sufficient to uniquely define them: + +**Proposition (Uniqueness of OPs I)** Given a non-zero $k_n$, there is a unique polynomial $p_n$ orthogonal w.r.t. $w$ +to all lower degree polynomials. + +**Proof** Suppose $r_n(x) = k_n x^n + O(x^{n-1})$ is another OP w.r.t. $w$. We want to show $p_n - r_n$ is zero. +But this is a polynomial of degree $ c_k = \ip = \ip - \ip = 0 - 0 = 0 +$$ +which shows all $c_k$ are zero. + +■ + +**Corollary (Uniqueness of OPs I)** If $q_n$ and $p_n$ are orthogonal w.r.t. $w$ to all lower degree polynomials, +then $q_n(x) = C p_n(x)$ for some constant $C$. + +### Monic orthogonal polynomials + +If $k_n = 1$, that is, +$$ +p_n(x) = x^n + O(x^{n-1}) +$$ +then we refer to the orthogonal polymomials as monic. + +Monic OPs are unique as we have specified $k_n$. + + +### Orthonormal polynomials + +If $\norm{p_n} = 1$, then we refer to the orthogonal polynomials as orthonormal w.r.t. $w$. +We will usually use $q_n$ when they are orthonormal. Note it's not unique: we can multiply by $\pm 1$ without changing the norm. + + +**Remark** The classical OPs are __not__ monic or orthonormal (apart from one case). Many people make the mistake of using +orthonormal polynomials for computations. But there is a good reason to use classical OPs: their properties result in rational formulae, +whereas orthonormal polynomials require square roots. This makes a performance difference. + +## Function approximation with orthogonal polynomials + +A basic usage of orthogonal polynomials is for polynomial approximation. +Suppose $f(x)$ is a degree $n-1$ polynomial. Since $\{p_0(x),\ldots,p_{n-1}(x)\}$ span all degree $n-1$ polynomials, we know that +$$ +f(x) = \sum_{k=0}^{n-1} f_k p_k(x) +$$ +where +$$ +f_k = {\ip \over \ip} +$$ + + +Sometimes, we want to incorporate the weight into the approximation. This is typically one of two forms, depending on the application: +$$ +f(x) = w(x) \sum_{k=0}^\infty f_k p_k(x) +$$ +or +$$ + f(x) = \sqrt{w(x)} \sum_{k=0}^\infty f_k p_k(x) +$$ + + + + +## Jacobi operators and three-term recurences for general orthogonal polynomials +### Three-term recurrence relationships + + +A central theme: if you know the Jacobi operator / three-term recurrence, you know the polynomials. +This is the __best__ way to evaluate expansions in orthogonal polynomials: even for cases where we have explicit +formulae (e.g. Chebyshev polynomials $T_n(x) = \cos n \acos x$), +using the recurrence avoids evaluating trigonometric functions. + +Every family of orthogonal polynomials has a three-term recurrence relationship: + +**Theorem (three-term recurrence)** Suppose $\{p_n(x)\}$ are a family of orthogonal polynomials w.r.t. a weight $w(x)$. +Then there exists constants $a_n \neq 0$, $b_n$ and $c_n$ such that +$$ +\begin{align*} +x p_0(x) = a_0 p_0(x) + b_0 p_1(x) \\ +x p_n(x) = c_n p_{n-1}(x) + a_n p_n(x) + b_n p_{n+1}(x) +\end{align*} +$$ + +**Proof** +The first part follows since $p_0(x)$ and $p_1(x)$ span all degree 1 polynomials. + +The second part follows essentially because multiplication by $x$ is "self-adjoint", that is, +$$ +\ip = \int_a^b x f(x) g(x) w(x) \dx = \ip +$$ +Therefore, if $f_m$ is a degree $m < n-1$ polynomial, we have +$$ +\ip = \ip = 0. +$$ +In particular, if we write +$$ +x p_n(x) = \sum_{k=0}^{n+1} C_k p_k(x) +$$ +then +$$ +C_k = {\ip< x p_n, p_k> \over \norm{p_k}^2} = 0 +$$ +if $k < n-1$. + +■ + + +Monic polynomials clearly have $b_n = 1$. Orthonormal polynomials have an even simpler form: + +**Theorem (orthonormal three-term recurrence)** Suppose $\{q_n(x)\}$ are a family of orthonotms polynomials w.r.t. a weight $w(x)$. +Then there exists constants $a_n$ and $b_n$ such that +$$ +\begin{align*} +x q_0(x) = a_0 q_0(x) + b_0 q_1(x)\\ +x q_n(x) = b_{n-1} q_{n-1}(x) + a_n q_n(x) + b_{n} q_{n+1}(x) +\end{align*} +$$ + +**Proof** +Follows again by self-adjointness of multiplication by $x$: +$$ +c_n = \ip = \ip = \ip = b_{n-1} +$$ +■ + + +**Corollary (symmetric three-term recurrence implies orthonormal)** Suppose $\{p_n(x)\}$ are a family of orthogonal polynomials +w.r.t. a weight $w(x)$ such that +$$ +\begin{align*} +x p_0(x) = a_0 p_0(x) + b_0 p_1(x)\\ +x p_n(x) = b_{n-1} p_{n-1}(x) + a_n p_n(x) + b_{n} p_{n+1}(x) +\end{align*} +$$ +with $b_n \neq 0$. Suppose further that $\norm{p_0} = 1$. Then $p_n$ must be orthonormal. + +**Proof** This follows from +$$ +b_n = {\ip \over \norm{p_{n+1}}^2} = {\ip \over \norm{p_{n+1}}^2} = b_n {\norm{p_n}^2 \over \norm{p_{n+1}}^2 } +$$ +which implies +$$ +\norm{p_{n+1}}^2 = \norm{p_n}^2 = \cdots = \norm{p_0}^2 = 1 +$$ +■ + +**Remark** We can scale $w(x)$ by a constant without changing the orthogonality properties, hence we can make $\|p_0\| = 1$ by changing the weight. + +**Remark** This is beyond the scope of this course, but satisfying a three-term recurrence like this such that coefficients +are bounded with $p_0(x) = 1$ is sufficient to show that there exists a $w(x)$ (or more accurately, a Borel measure) +such that $p_n(x)$ are orthogonal w.r.t. $w$. The relationship between the coefficients $a_n,b_n$ and the $w(x)$ is +an object of study in spectral theory, particularly when the coefficients are periodic, quasi-periodic or random. + +## Jacobi operators and multiplication by $x$ + +We can rewrite the three-term recurrence as +$$ +x \begin{pmatrix} p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \end{pmatrix} = J\begin{pmatrix} p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \end{pmatrix} +$$ +where $J$ is a Jacobi operator, an infinite-dimensional tridiagonal matrix: +$$ +J = \begin{pmatrix} +a_0 & b_0 \cr +c_1 & a_1 & b_1 \cr +& c_2 & a_2 & b_2 \cr +&& c_3 & a_3 & \ddots \cr +&&&\ddots & \ddots +\end{pmatrix} +$$ + +When the polynomials are monic, we have $1$ on the superdiagonal. When we have an orthonormal basis, then $J$ is symmetric: +$$ +J = \begin{pmatrix} +a_0 & b_0 \cr +b_0 & a_1 & b_1 \cr +& b_1 & a_2 & b_2 \cr +&& b_2 & a_3 & \ddots \cr +&&&\ddots & \ddots +\end{pmatrix} +$$ + + +Given a polynomial expansion, $J$ tells us how to multiply by $x$ in coefficient space, that is, if +$$ +f(x) = \sum_{k=0}^\infty f_k p_k(x) = (p_0(x) , p_1(x) , \ldots ) \begin{pmatrix}f_0\\ f_1\\f_2\\\vdots\end{pmatrix} +$$ +then (provided the relevant sums converge absolutely and uniformly) +$$ +x f(x) = x (p_0(x) , p_1(x) , \ldots ) \begin{pmatrix}f_0\\ f_1\\f_2\\\vdots\end{pmatrix} = + \left(J \begin{pmatrix} p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \end{pmatrix}\right)^\top \begin{pmatrix}f_0\\ f_1\\f_2\\\vdots\end{pmatrix} = (p_0(x) , p_1(x) , \ldots ) X \begin{pmatrix}f_0\\ f_1\\f_2\\\vdots\end{pmatrix} +$$ +where $X := J^\top$. + + + +### Evaluating polynomials + + +We can use the three-term recurrence to construct the polynomials. +I think it's nicest to express this in terms of linear algebra. +Suppose we are given $p_0(x) = k_0$ (where $k_0 = 1$ is pretty much always the case in practice). This can be written in matrix form as +$$ +(1,0,0,0,0,\ldots) \begin{pmatrix} p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \end{pmatrix} = k_0 +$$ +We can combine this with the Jacobi operator to get +$$ +\underbrace{\begin{pmatrix} +1 \\ +a_0-x & b_0 \\ +c_1 & a_1-x & b_1 \\ +& c_2 & a_2-x & b_2 \cr +&& c_3 & a_3-x & b_3 \cr +&&&\ddots & \ddots & \ddots +\end{pmatrix}}_{L_x} \begin{pmatrix} p_0(x) \cr p_1(x) \cr p_2(x) \cr \vdots \end{pmatrix} = \begin{pmatrix} k_0\cr 0 \cr 0 \cr \vdots \end{pmatrix} +$$ +For $x$ fixed, this is a lower triangular system, that is, the polynomials equal +$$ +k_0 L_x^{-1} \vc e_0 +$$ +This can be solved via forward recurrence: +$$ +\begin{align*} + p_0(x) &= k_0 \\ + p_1(x) &= {(x-a_0) p_0(x) \over b_0}\\ + p_2(x) &= {(x-a_1) p_0(x) - c_1 p_0(x) \over b_1}\\ + p_3(x) &= {(x-a_2) p_1(x) - c_2 p_1(x) \over b_2}\\ + &\vdots +\end{align*} +$$ + +We can use this to evaluate functions as well: +$$ +f(x) = (p_0(x),p_1(x),\ldots) \begin{pmatrix}f_0 \\ f_1\\ \vdots \end{pmatrix} = +k_0 \vc e_0^\top L_x^{-\top} \begin{pmatrix}f_0 \\ f_1\\ \vdots \end{pmatrix} +$$ +when $f$ is a degree $n-1$ polynomial, this becomes a problem of inverting an upper triangular matrix, +that is, we want to solve the $n \times n$ system +$$ +\underbrace{\begin{pmatrix} +1 & a_0-x & c_1 \\ +& b_0 & a_1-x & c_2 \\ +& & b_1 & a_2-x & \ddots \\ +& & & b_2 & \ddots & c_{n-2} \\ +&&&&\ddots & a_{n-2}-x \\ +&&&&& b_{n-2} +\end{pmatrix}}_{L_x^\top} \begin{pmatrix} \gamma_0 \\\vdots\\ \gamma_{n-1} \end{pmatrix} +$$ +so that $f(x) = \gamma_0$. We we can solve this via back-substitution: +$$ +\begin{align*} +\gamma_{n-1} &= {f_{n-1} \over b_{n-2}} \\ +\gamma_{n-2} &= {f_{n-2} - (a_{n-2}-x) \gamma_{n-1} \over b_{n-3}} \\ +\gamma_{n-3} &= {f_{n-3} - (a_{n-3}-x) \gamma_{n-2} - c_{n-2} \gamma_{n-1} \over b_{n-4}} \\ +& \vdots \\ +\gamma_1 &= {f_1 - (a_1-x) \gamma_2 - c_2 \gamma_3 \over b_0} \\ +\gamma_0 &= f_0 - (a_0-x) \gamma_1 - c_1 \gamma_2 +\end{align*} +$$ +We give examples of these algorithms applied to Chebyshev polynomials in the next lecture. + + +## Gram–Schmidt algorithm + +In general we don't have nice formulae. But we can always construct them via Gram–Schmidt: + +**Proposition (Gram–Schmidt)** Define +$$ +\begin{align*} +p_0(x) = 1 \\ +q_0(x) = {1 \over \norm{p_0}}\\ +p_{n+1}(x) = x q_n(x) - \sum_{k=0}^n \ip q_k(x)\\ +q_{n+1}(x) = {p_{n+1}(x) \over \norm{p_n}} +\end{align*} +$$ +Then $q_0(x), q_1(x), \ldots$ are orthonormal w.r.t. $w$. + +**Proof** By linearity we have +$$ +\ip = \ip} q_k, q_j> = \ip - \ip \ip = 0 +$$ +Thus $p_{n+1}$ is orthogonal to all lower degree polynomials. So is $q_{n+1}$, since it is a constant multiple of $p_{n+1}$. + +■ + +Let's make our own family: +```julia +using ApproxFun, Plots +x = Fun() +w = exp(x) +ip = (f,g) -> sum(f*g*w) +nrm = f -> sqrt(ip(f,f)) +n = 10 +q = Array{Fun}(undef,n) +p = Array{Fun}(undef,n) +p[1] = Fun(1, -1 .. 1 ) +q[1] = p[1]/nrm(p[1]) + +for k=1:n-1 + p[k+1] = x*q[k] + for j=1:k + p[k+1] -= ip(p[k+1],q[j])*q[j] + end + q[k+1] = p[k+1]/nrm(p[k+1]) +end + +sum(q[2]*q[4]*w) + +p = plot(; legend=false) +for k=1:10 + plot!(q[k]) +end +p +``` + + + +The three-term recurrence means we can simplify Gram--Schmidt, and calculate the recurrence coefficients at the same time: + + +**Proposition (Gram–Schmidt)** Define +$$ +\begin{align*} +p_0(x) &= 1 \\ +q_0(x) &= {1 \over \norm{p_0}}\\ +a_n &= \ip \\ +b_{n-1} &= \ip\\ +p_{n+1}(x) &= x q_n(x) - a_n q_n(x) - b_{n-1} q_{n-1}(x)\\ +q_{n+1}(x) &= {p_{n+1}(x) \over \norm{p_n}} +\end{align*} +$$ +Then $q_0(x), q_1(x), \ldots$ are orthonormal w.r.t. $w$. + +**Remark** This can be made a bit more efficient by using $\norm{p_n}$ to calculate $b_n$. +```julia +x = Fun() +w = exp(x) +ip = (f,g) -> sum(f*g*w) +nrm = f -> sqrt(ip(f,f)) +n = 10 +q = Array{Fun}(undef, n) +p = Array{Fun}(undef, n) +a = zeros(n) +b = zeros(n) +p[1] = Fun(1, -1 .. 1 ) +q[1] = p[1]/nrm(p[1]) + +p[2] = x*q[1] +a[1] = ip(p[2],q[1]) +p[2] -= a[1]*q[1] +q[2] = p[2]/nrm(p[2]) + +for k=2:n-1 + p[k+1] = x*q[k] + b[k-1] =ip(p[k+1],q[k-1]) + a[k] = ip(p[k+1],q[k]) + p[k+1] = p[k+1] - a[k]q[k] - b[k-1]q[k-1] + q[k+1] = p[k+1]/nrm(p[k+1]) +end + +ip(q[5],q[2]) # shows orthogonality (to numerical accuracy) +``` +Here we see a plot of the first 10 polynomials: +```julia +p = plot(; legend=false) +for k=1:10 + plot!(q[k]) +end +p +``` \ No newline at end of file diff --git a/src/Lecture21.jmd b/src/Lecture21.jmd index 2804539..0f15e3e 100644 --- a/src/Lecture21.jmd +++ b/src/Lecture21.jmd @@ -6,4 +6,360 @@ s.olver@imperial.ac.uk -# Lecture 21: Classical orthogonal polynomials \ No newline at end of file +# Lecture 21: Classical orthogonal polynomials + + +We will also investigate the properties of _classical_ OPs: + +1. Explicit formulae +3. Rodriguez formulae +2. Ordinary differential equations +2. Derivatives + +A good reference is [Digital Library of Mathematical Functions, Chapter 18](http://dlmf.nist.gov/18). + +2. Definition of classical orthogonal polynomials + - Hermite, Laguerre, and Jacobi polynomials + - Legendre, Chebyshev, and ultraspherical polynomials + - Explicit construction for Chebyshev polynomials + + +## Definition of classical orthogonal polynomials + +Classical orthogonal polynomials are orthogonal with respect to the following three weights: + + +| Name | Interval $(a,b)$ |Weight function $w(x)$ | Standard polynomial | highest order coefficient $k_n$ | +|:-------------|:------------- |:----------------------|:-----|:-----| +| Hermite |$(-\infty,\infty)$ | $\E^{-x^2}$ | $H_n(x)$ | $2^n$ | +| Laguerre | $(0,\infty)$ | $x^\alpha \E^{-x}$ | $L_n^{(\alpha)}(x)$ | See [Table 18.3.1](http://dlmf.nist.gov/18.3) | +| Jacobi | $(-1,1)$ | $(1-x)^{\alpha} (1+x)^\beta$ | $P_n^{(\alpha,\beta)}(x)$ | See [Table 18.3.1](http://dlmf.nist.gov/18.3) | + + +Note out of convention the parameters for Jacobi polynomials are in the "wrong" order. + +We can actually construct these polynomials in Julia: first consider Hermite: + +H₀ = Fun(Hermite(), [1]) +H₁ = Fun(Hermite(), [0,1]) +H₂ = Fun(Hermite(), [0,0,1]) +H₃ = Fun(Hermite(), [0,0,0,1]) +H₄ = Fun(Hermite(), [0,0,0,0,1]) +H₅ = Fun(Hermite(), [0,0,0,0,0,1]) + +xx = -4:0.01:4 +plot(xx, H₀.(xx); label="H_0", ylims=(-400,400)) +plot!(xx, H₁.(xx); label="H_1", ylims=(-400,400)) +plot!(xx, H₂.(xx); label="H_2", ylims=(-400,400)) +plot!(xx, H₃.(xx); label="H_3", ylims=(-400,400)) +plot!(xx, H₄.(xx); label="H_4", ylims=(-400,400)) +plot!(xx, H₅.(xx); label="H_5") + + +We verify their orthogonality: + +w = Fun(GaussWeight(), [1.0]) + +@show sum(H₂*H₅*w) # means integrate +@show sum(H₅*H₅*w); + +Now Jacobi: + +α,β = 0.1,0.2 +P₀ = Fun(Jacobi(β,α), [1]) +P₁ = Fun(Jacobi(β,α), [0,1]) +P₂ = Fun(Jacobi(β,α), [0,0,1]) +P₃ = Fun(Jacobi(β,α), [0,0,0,1]) +P₄ = Fun(Jacobi(β,α), [0,0,0,0,1]) +P₅ = Fun(Jacobi(β,α), [0,0,0,0,0,1]) + +xx = -1:0.01:1 +plot( xx, P₀.(xx); label="P_0^($α,$β)", ylims=(-2,2)) +plot!(xx, P₁.(xx); label="P_1^($α,$β)") +plot!(xx, P₂.(xx); label="P_2^($α,$β)") +plot!(xx, P₃.(xx); label="P_3^($α,$β)") +plot!(xx, P₄.(xx); label="P_4^($α,$β)") +plot!(xx, P₅.(xx); label="P_5^($α,$β)") + +w = Fun(JacobiWeight(β,α), [1.0]) +@show sum(P₂*P₅*w) # means integrate +@show sum(P₅*P₅*w); + +## Legendre, Chebyshev, and ultraspherical polynomials + +There are special families of Jacobi weights with their own name. + +| Name | Jacobi parameters |Weight function $w(x)$ | Standard polynomial | highest order coefficient $k_n$ | +|:-------------|:------------- |:----------------------|:-----|:------| +| Jacobi | $\alpha,\beta$ | $(1-x)^{\alpha} (1+x)^\beta$ | $P_n^{(\alpha,\beta)}(x)$ | See [Table 18.3.1](http://dlmf.nist.gov/18.3) | +| Legendre | $0,0$ | $1$ | $P_n(x)$ | $2^n(1/2)_n/n!$ | +| Chebyshev (first kind) | $-{1 \over 2},-{1 \over 2}$ | $1 \over \sqrt{1-x^2}$ | $T_n(x)$ | $1 (n=0), 2^{n-1} (n \neq 0)$ | +| Chebyshev (second kind) | ${1 \over 2},{1 \over 2}$ | $\sqrt{1-x^2}$ | $U_n(x)$ | $2^n$ +| Ultraspherical | $\lambda-{1 \over 2},\lambda-{1 \over 2}$ | $(1-x^2)^{\lambda - 1/2}, \lambda \neq 0$ | $C_n^{(\lambda)}(x)$ | $2^n(\lambda)_n/n!$ | + +Note that other than Legendre, these polynomials have a different normalization than $P_n^{(\alpha,\beta)}$: + + +T₂ = Fun(Chebyshev(), [0.0,0,1]) +P₂ = Fun(Jacobi(-1/2,-1/2), [0.0,0,1]) +plot(T₂; label="T_2", title="T_2 is C*P_2 for some C") +plot!(P₂; label="P_2") + +But because they are orthogonal w.r.t. the same weight, they must be a constant multiple of each-other. + +### Explicit construction of Chebyshev polynomials (first kind and second kind) + +Chebyshev polynomials are pretty much the only OPs with _simple_ closed form expressions. + +**Proposition (Chebyshev first kind formula)** +$$T_n(x) = \cos n \acos x$$ +or in other words, +$$ +T_n(\cos \theta) = \cos n \theta +$$ + +**Proof** We first show that they are orthogonal w.r.t. $1/\sqrt{1-x^2}$. Too easy: do $x = \cos \theta$, $\dx = -\sin \theta$ to get (for $n \neq m$) +$$ + \int_{-1}^1 {\cos n \acos x \cos m \acos x \dx \over \sqrt{1-x^2}} = -\int_\pi^0 \cos n \theta \cos m \theta \D \theta = \int_0^\pi {\E^{\I (-n-m)\theta} + \E^{\I (n-m)\theta} + \E^{\I (m-n)\theta} + \E^{\I (n+m)\theta} \over 4} \D \theta =0 +$$ + +We then need to show it has the right highest order term $k_n$. Note that $k_0 = k_1 = 1$. Using $z = \E^{\I \theta}$ we see that $\cos n \theta$ has a simple recurrence for $n=2,3,\ldots$: +$$ +\cos n \theta = {z^n + z^{-n} \over 2} = 2 {z + z^{-1} \over 2} {z^{n-1} + z^{1-n} \over 2}- {z^{n-2} + z^{2-n} \over 2} = 2 \cos \theta \cos (n-1)\theta - \cos(n-2)\theta +$$ +thus +$$ +\cos n \acos x = 2 x \cos(n-1) \acos x - \cos(n-2) \acos x +$$ +It follows that +$$ +k_n = 2 x k_{n-1} = 2^{n-1} k_1 = 2^{n-1} +$$ +By uniqueness we have $T_n(x) \cos n \acos x$. + +⬛️ + + + +**Proposition (Chebyshev second kind formula)** +$$U_n(x) = {\sin (n+1) \acos x \over \sin \acos x}$$ +or in other words, +$$ +U_n(\cos \theta) = {\sin (n+1) \theta \over \sin \theta} +$$ + + + +*Example* For the case of Chebyshev polynomials, we have +$$ +J = \begin{pmatrix} +0 & 1 \cr +\half & 0 & \half \cr +& \half & 0 & \half \cr +&& \half & 0 & \ddots \cr +&&&\ddots & \ddots +\end{pmatrix} +$$ +Therefore, the Chebyshev coefficients of $x f(x)$ are given by +$$ +J^\top \vc f = \begin{pmatrix} +0 & \half \cr +1 & 0 & \half \cr +& \half & 0 & \half \cr +&& \half & 0 & \ddots \cr +&&&\ddots & \ddots +\end{pmatrix} \begin{pmatrix} f_0\\ f_1\\f_2\\f_3\\\vdots\end{pmatrix} +$$ +In the case where $f$ is a degree $n-1$ polynomial, we can represent $J^\top$ as an $n+1 \times n$ matrix (this makes sense as $x f(x)$ is one more degree than $f$): + +f = Fun(exp, Chebyshev()) +n = ncoefficients(f) # number of coefficients +@show n +J = zeros(n,n+1) +J[1,2] = 1 +for k=2:n + J[k,k-1] = J[k,k+1] = 1/2 +end +J' + +cfs = J'*f.coefficients # coefficients of x*f +xf = Fun(Chebyshev(), cfs) + +xf(0.1) - 0.1*f(0.1) + + +**Example** We can construct $T_0(x),\ldots,T_{n-1}(x)$ via +\begin{align*} + p_0(x) &= 1\\ + p_1(x) &= x T_0(x) \\ + T_2(x) &= 2x T_0(x) - T_0(x) \\ + T_3(x) &= 2x T_1(x) - T_1(x) + &\vdots +\end{align*} +Believe it or not, this is much faster than using $\cos k \acos x$: + + + +function recurrence_Chebyshev(n,x) + T = zeros(n) + T[1] = 1.0 + T[2] = x*T[1] + for k = 2:n-1 + T[k+1] = 2x*T[k] - T[k-1] + end + T +end + +trig_Chebyshev(n,x) = [cos(k*acos(x)) for k=0:n-1] + +n = 10 +recurrence_Chebyshev(n, 0.1) - trig_Chebyshev(n,0.1) |>norm + +n = 10000 +@time recurrence_Chebyshev(n, 0.1) +@time trig_Chebyshev(n,0.1); + + + +*Example* For Chebyshev, we want to solve the system +$$ +\underbrace{\begin{pmatrix} +1 & -x & \half \\ +& 1 & -x & \half \\ +& & \half & -x & \ddots \\ +& & & \half & \ddots & \half \\ +&&&&\ddots & -x \\ +&&&&& \half +\end{pmatrix}}_{L_x^\top} \begin{pmatrix} \gamma_0 \\\vdots\\ \gamma_{n-1} \end{pmatrix} +$$ +via + +\begin{align*} +\gamma_{n-1} &= 2f_{n-1} \\ +\gamma_{n-2} &= 2f_{n-2} + 2x \gamma_{n-1} \\ +\gamma_{n-3} &= 2 f_{n-3} + 2x \gamma_{n-2} - \gamma_{n-1} \\ +& \vdots \\ +\gamma_1 &= f_1 + x \gamma_2 - \half \gamma_3 \\ +\gamma_0 &= f_0 + x \gamma_1 - \half \gamma_2 +\end{align*} + +then $f(x) = \gamma_0$. + +function clenshaw_Chebyshev(f,x) + n = length(f) + γ = zeros(n) + γ[n] = 2f[n] + γ[n-1] = 2f[n-1] +2x*f[n] + for k = n-2:-1:1 + γ[k] = 2f[k] + 2x*γ[k+1] - γ[k+2] + end + γ[2] = f[2] + x*γ[3] - γ[4]/2 + γ[1] = f[1] + x*γ[2] - γ[3]/2 + γ[1] +end + +f = Fun(exp, Chebyshev()) +clenshaw_Chebyshev(f.coefficients, 0.1) - exp(0.1) + +With some high performance computing tweeks, this can be made more accurate: this is the algorithm used for evaluating functions in ApproxFun: + +f(0.1) - exp(0.1) + + + +## Approximation with Chebyshev polynomials + + + + +*Example* Last lecture, we used the formula, derived via trigonometric manipulations, +$$ +T_1(x) = x T_0(x) \\ +T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x) +$$ +Rearranging, this becomes +$$ + x T_0(x) = T_1(x) \\ +x T_n(x) = {T_{n-1}(x) \over 2} + {T_{n+1}(x) \over 2} +$$ +This tells us that we have the three-term recurrence with $a_n = 0$, $b_0 = 1$, $c_n = b_n = {1 \over 2}$ for $n > 0$. + +T = (n,x) -> cos(n*acos(x)) +x = 0.5 +n = 10 +@show x*T(0,x) - (T(1,x)) +@show x*T(n,x) - (T(n-1,x) + T(n+1,x))/2; + + +Here, we demonstrate this with Chebyshev polynomials: +```julia +f = Fun(x -> 1 + x + x^2 + x^3, Chebyshev()) +f₀, f₁, f₂, f₃ = f.coefficients +``` +```julia +x = 0.1 +@show f₀*1 + f₁*x + f₂*cos(2acos(x)) + f₃*cos(3acos(x)) +@show 1 + x + x^2 + x^3; +``` + + +plot(Fun(exp)) +plot!(Fun(t-> exp(cos(t)), -pi .. pi)) + +x = Fun() +ip = (f,g) -> sum(f*g/sqrt(1-x^2)) + +T₂ = cos.(2 .* acos.(x)) +ip(T₂,f)/ip(T₂,T₂) # gives back f₂ + +Of course, if $p_k$ are othernormal than we don't need the denominator. + +This can be extended to function approximation Provided the sum converges absolutely and uniformly in $x$, we can write +$$ +f(x) = \sum_{k=0}^\infty f_k p_k(x). +$$ +In practice, we can approximate smooth functions by a finite truncation: +$$ +f(x) \approx f_n(x) = \sum_{k=0}^{n-1} f_k p_k(x) +$$ + +Here we see that $\E^x$ can be approximated by a Chebyshev approximation using 14 coefficients and is accurate to 16 digits: + +f = Fun(x -> exp(x), Chebyshev()) +@show f.coefficients +@show ncoefficients(f) + +@show f(0.1) # equivalent to f.coefficients'*[cos(k*acos(x)) for k=0:ncoefficients(f)-1] +@show exp(0.1); + +The accuracy of this approximation is typically dictated by the smoothness of $f$: the more times we can differentiate, the faster it converges. For analytic functions, it's dictated by the domain of analyticity, just like Laurent/Fourier series. In the case above, $\E^x$ is entire hence we get faster than exponential convergence. + + +Chebyshev expansions work even when Taylor series do not. For example, the following function has poles at $\pm {\I \over 5}$, which means the radius of convergence for the Taylor series is $ |x| < {1 \over 5}$, but Chebyshev polynomials continue to work on $[-1,1]$: + +f = Fun( x -> 1/(25x^2 + 1), Chebyshev()) +@show ncoefficients(f) +plot(f) + +This can be explained for Chebyshev expansion by noting that it is cosine expansion / Fourier expansion of an even function: +$$ +f(x) = \sum_{k=0}^\infty f_k T_k(x) \Leftrightarrow f(\cos \theta) = \sum_{k=0}^\infty f_k \cos k \theta +$$ +From Lecture 4 we saw that Fourier coefficients decay exponentially (thence the approximation converges exponentially fast) uf $f\left({z + z^{-1} \over 2}\right)$ is analytic in an ellipse. In the case of $f(x) = {1 \over 25 x^2 + 1}$, we find that +$$ +f(z) = {4 z^2 \over 25 + 54 z^2 + 25 z^4} +$$ +which has poles at +$ +\pm 0.8198040\I,\pm1.2198\I: +$ + +f = x -> 1/(25x^2 + 1) +portrait(-3..3, -3..3, z -> f((z+1/z)/2)) + +Hence we predict a rate of decay of about $1.2198^{-k}$: + +f = Fun( x -> 1/(25x^2 + 1), Chebyshev()) +plot(abs.(f.coefficients) .+ 1E-40; yscale=:log10, label="Chebyshev coefficients") +plot!( 1.2198.^(-(0:ncoefficients(f))); label="R^(-k)") + diff --git a/src/Lecture22.jmd b/src/Lecture22.jmd new file mode 100644 index 0000000..d0b9a29 --- /dev/null +++ b/src/Lecture22.jmd @@ -0,0 +1,798 @@ +__M3M6: Applied Complex Analysis__ + +Dr. Sheehan Olver + +s.olver@imperial.ac.uk + + + +# Lecture 22: Orthogonal polynomials and differential equations + +# M3M6: Methods of Mathematical Physics + +$$ +\def\dashint{{\int\!\!\!\!\!\!-\,}} +\def\infdashint{\dashint_{\!\!\!-\infty}^{\,\infty}} +\def\D{\,{\rm d}} +\def\E{{\rm e}} +\def\dx{\D x} +\def\dt{\D t} +\def\dz{\D z} +\def\C{{\mathbb C}} +\def\R{{\mathbb R}} +\def\CC{{\cal C}} +\def\HH{{\cal H}} +\def\I{{\rm i}} +\def\qqqquad{\qquad\qquad} +\def\qqfor{\qquad\hbox{for}\qquad} +\def\qqwhere{\qquad\hbox{where}\qquad} +\def\Res_#1{\underset{#1}{\rm Res}}\, +\def\sech{{\rm sech}\,} +\def\acos{\,{\rm acos}\,} +\def\vc#1{{\mathbf #1}} +\def\ip<#1,#2>{\left\langle#1,#2\right\rangle} +\def\norm#1{\left\|#1\right\|} +\def\half{{1 \over 2}} +$$ + +Dr. Sheehan Olver +
+s.olver@imperial.ac.uk + +
+Website: https://github.com/dlfivefifty/M3M6LectureNotes + + +# Lecture 16: Solving differential equations with orthogonal polynomials + + +This lecture we do the following: + +1. Recurrence relationships for Chebyshev and ultrashperical polynomials + - Conversion + - Three-term recurrence and Jacobi operators +2. Application: solving differential equations + - First order constant coefficients differential equations + - Second order constant coefficient differential equations with boundary conditions + - Non-constant coefficients + + + +That is, we introduce recurrences related to ultraspherical polynomials. This allows us to represent general linear differential equations as almost-banded systems. + +## Recurrence relationships for Chebyshev and ultraspherical polynomials + + +We have discussed general properties, but now we want to discuss some classical orthogonal polynomials, beginning with Chebyshev (first kind) $T_n(x)$, which is orthogonal w.r.t. +$$1\over \sqrt{1-x^2}$$ +and ultraspherical $C_n^{(\lambda)}(x)$, which is orthogonal w.r.t. +$$(1-x^2)^{\lambda - \half}$$ +for $\lambda > 0$. Note that Chebyshev (second kind) satisfies $U_n(x) = C_n^{(1)}(x)$. + +For Chebyshev, recall we have the normalization constant (here we use a superscript $T_n(x) = k_n^{\rm T} x^n + O(x^{n-1})$) +$$ +k_0^{\rm T} = 1, k_n^{\rm T} = 2^{n-1} +$$ +For Ultraspherical $C_n^{(\lambda)}$, this is +$$ +k_n^{(\lambda)} = {2^n (\lambda)_n \over n!} = {2^n \lambda (\lambda+1) (\lambda+2) \cdots (\lambda+n-1) \over n!} +$$ +where $(\lambda)_n$ is the Pochhammer symbol. Note for $U_n(x) = C_n^{(1)}(x)$ this simplifies to $k_n^{\rm U} = k_n^{(1)} = 2^n$. + +We have already found the recurrence for Chebyshev: +$$ +x T_n(x) = {T_{n-1}(x) \over 2} + {T_{n+1}(x) \over 2} +$$ +We will show that we can use this to find the recurrence for _all_ ultraspherical polynomials. But first we need some special recurrences. + +**Remark** Jacobi, Laguerre, and Hermite all have similar relationships, which will be discussed further in the problem sheet. + +### Derivatives + +It turns out that the derivative of $T_n(x)$ is precisely a multiple of $U_{n-1}(x)$, and similarly the derivative of $C_n^{(\lambda)}$ is a multiple of $C_{n-1}^{(\lambda+1)}$. + +**Proposition (Chebyshev derivative)** $$T_n'(x) = n U_{n-1}(x)$$ + +**Proof** +We first show that $T_n'(x)$ is othogonal w.r.t. $\sqrt{1-x^2}$ to all polynomials of degree $m < n-1$, denoted $f_m$, using integration by parts: +$$ +\ip_{\rm U} = \int_{-1}^1 T_n'(x) f_m(x) \sqrt{1-x^2} \dx = -\int_{-1}^1 T_n(x) (f_m'(x)(1-x^2) + xf_m) {1 \over \sqrt{1-x^2}} \dx = - \ip_{\rm T} = 0 +$$ +since $f_m'(1-x^2) + f_m $ is degree $m-1 +2 = m+1 < n$. + +The constant works out since +$$ +T_n'(x) = {\D \over \dx} (2^{n-1} x^n) + O(x^{n-2}) = n 2^{n-1} x^{n-1} + O(x^{n-2}) +$$ +⬛️ + +The exact same proof shows the following: + +**Proposition (Ultraspherical derivative)** +$${\D \over \dx} C_n^{(\lambda)}(x) = 2 \lambda C_{n-1}^{(\lambda+1)}(x)$$ + +Like the three-term recurrence and Jacobi operators, it is useful to express this in matrix form. That is, for the derivatives of $T_n(x)$ we get +$$ +{\D \over \dx} \begin{pmatrix} T_0(x) \\ T_1(x) \\ T_2(x) \\ \vdots \end{pmatrix}= \begin{pmatrix} +0 \cr +1 \cr +& 2 \cr +&& 3 \cr +&&&\ddots +\end{pmatrix} \begin{pmatrix} U_0(x) \\ U_1(x) \\ U_2(x) \\ \vdots \end{pmatrix} +$$ +which let's us know that, for +$$ +f(x) = (T_0(x),T_1(x),\ldots) \begin{pmatrix} f_0\\f_1\\\vdots \end{pmatrix} +$$ +we have a derivative operator in coefficient space as +$$ +f'(x) = (U_0(x),U_1(x),\ldots)\begin{pmatrix} +0 & 1 \cr +&& 2 \cr +&&& 3 \cr +&&&&\ddots +\end{pmatrix} \begin{pmatrix} f_0\\f_1\\\vdots \end{pmatrix} +$$ + +_Demonstration_ Here we see that applying a matrix to a vector of coefficients successfully calculates the derivative: + +f = Fun(x -> cos(x^2), Chebyshev()) # f is expanded in Chebyshev coefficients +n = ncoefficients(f) # This is the number of coefficients +D = zeros(n-1,n) +for k=1:n-1 + D[k,k+1] = k +end +D + +Here `D*f.coefficients` gives the vector of coefficients corresponding to the derivative, but now the coefficients are in the $U_n(x)$ basis, that is, `Ultraspherical(1)`: + +fp = Fun(Ultraspherical(1), D*f.coefficients) + +fp(0.1) + +Indeed, it matches the "true" derivative: + +f'(0.1) + +-2*0.1*sin(0.1^2) + +Note that in ApproxFun.jl we can construct these operators rather nicely: + +D = Derivative() +(D*f)(0.1) + +Here we see that we can write produce the ∞-dimensional version as follows: + +D : Chebyshev() → Ultraspherical(1) + +### Conversion + + + +We can convert between any two polynomial bases using a lower triangular operator, because their span's are equivalent. In the case of Chebyshev and ultraspherical polynomials, they have the added property that they are banded. + +**Proposition (Chebyshev T-to-U conversion)** +\begin{align*} + T_0(x) &= U_0(x) \\ + T_1(x) &= {U_1(x) \over 2} \\ + T_n(x) &= {U_n(x) \over 2} - {U_{n-2}(x) \over 2} +\end{align*} + +**Proof** + +Before we do the proof, note that the fact that there are limited non-zero entries follows immediately: if $m < n-2$ we have +$$ +\ip_{\rm U} = \ip_{\rm T} = 0 +$$ + +To actually determine the entries, we use the trigonometric formulae. Recall for $x = (z + z^{-1})/2$, $z = \E^{\I \theta}$, we have +\begin{align*} +T_n(x) &= \cos n \theta = {z^{-n} + z^n \over 2}\\ +U_n(x) &= {\sin (n+1) \theta \over \sin \theta} = {z^{n+1} - z^{-n-1} \over z - z^{-1}} = z^{-n} + z^{2-n} + \cdots + \cdots + z^{n-2} + z^n +\end{align*} +The result follows immediately. + +⬛️ + +**Corollary (Ultrapherical λ-to-(λ+1) conversion)** +$$ +C_n^{(\lambda)}(x) = {\lambda \over n+ \lambda} (C_n^{(\lambda+1)}(x) - C_{n-2}^{(\lambda+1)}(x)) +$$ + +**Proof** This follows from differentiating the previous result. For example: +\begin{align*} + {\D\over \dx} T_0(x) &= {\D\over \dx} U_0(x) \\ + {\D\over \dx} T_1(x) &= {\D\over \dx} {U_1(x) \over 2} \\ +{\D\over \dx} T_n(x) &= {\D\over \dx} \left({U_n(x) \over 2} - {U_{n-2} \over 2} \right) +\end{align*} +becomes +\begin{align*} + 0 &= 0\\ + U_0(x) &= C_1^{(2)}(x) \\ + n U_{n-1}(x) &= C_{n-1}^{(2)}(x) - C_{n-3}^{(2)}(x) +\end{align*} + +Differentiating this repeatedly completes the proof. + +⬛️ + + +Note we can write this in matrix form, for example, we have +$$ +\underbrace{\begin{pmatrix}1 \cr + 0 & \half\cr + -\half & 0 & \half \cr + &\ddots &\ddots & \ddots\end{pmatrix} }_{S_0^\top} \begin{pmatrix} + U_0(x) \\ U_1(x) \\ U_2(x) \\ \vdots \end{pmatrix} = \begin{pmatrix} T_0(x) \\ T_1(x) \\ T_2(x) \\ \vdots \end{pmatrix} +$$ + +therefore, +$$ +f(x) = (T_0(x),T_1(x),\ldots) \begin{pmatrix} f_0\\f_1\\\vdots \end{pmatrix} = (U_0(x),U_1(x),\ldots) S_0 \begin{pmatrix} f_0\\f_1\\\vdots \end{pmatrix} +$$ + +Again, we can construct this nicely in ApproxFun: + +S₀ = I : Chebyshev() → Ultraspherical(1) + +f = Fun(exp, Chebyshev()) +g = S₀*f + +g(0.1) - exp(0.1) + +### Ultraspherical Three-term recurrence + +**Theorem (three-term recurrence for Chebyshev U)** +\begin{align*} +x U_0(x) &= {U_1(x) \over 2} \\ +x U_n(x) &= {U_{n-1}(x) \over 2} + {U_{n+1}(x) \over 2} +\end{align*} + +**Proof** +Differentiating +\begin{align*} + x T_0(x) &= T_1(x) \\ +x T_n(x) &= {T_{n-1}(x) \over 2} + {T_{n+1}(x) \over 2} +\end{align*} +we get +\begin{align*} + T_0(x) &= U_0(x) \\ + T_n(x) + n x U_{n-1}(x) &= {(n-1) U_{n-2}(x) \over 2} + {(n+1) U_n(x) \over 2} +\end{align*} +substituting in the conversion $T_n(x) = (U_n(x) - U_{n-2}(x))/2$ we get +\begin{align*} + T_0(x) &= U_0(x) \\ + n x U_{n-1}(x) &= {(n-1) U_{n-2}(x) \over 2} + {(n+1) U_n(x) \over 2} - (U_n(x) - U_{n-2}(x))/2 = {n U_{n-2}(x) \over 2} + {n U_n(x) \over 2} +\end{align*} + +⬛️ + +Differentiating this theorem again and applying the conversion we get the following + +**Corollary (three-term recurrence for ultrashperical)** +\begin{align*} +x C_0^{(\lambda)}(x) &= {1 \over 2\lambda } C_1^{(\lambda)}(x) \\ + x C_n^{(\lambda)}(x) &= {n+2\lambda-1 \over 2(n+\lambda)} C_{n-1}^{(\lambda)}(x) + {n+1 \over 2(n+\lambda)} C_{n+1}^{(\lambda)}(x) +\end{align*} + + +Here's an example of the Jacobi operator (which is the transpose of the multiplciation by $x$ operator): + +Multiplication(Fun(), Ultraspherical(2))' + +## Application: solving differential equations + +The preceding results allowed us to represent + +1. Differentiation +2. Conversion +3. Multiplication + +as banded operators. We will see that we can combine these, along with + +4\. Evaluation + +to solve ordinary differential equations. + +### First order, constant coefficient differential equations + +Consider the simplest ODE: +\begin{align*} +u(0) &= 0 \\ +u'(x) - u(x) &= 0 +\end{align*} +and suppose represent $u(x)$ in its Chebyshev expansion, with to be determined coefficents. In other words, we want to calculate coefficients $u_k$ such that +$$ +u(x) = \sum_{k=0}^\infty u_k T_k(x) = (T_0(x), T_1(x), \ldots) \begin{pmatrix} u_0 \\ u_1 \\ \vdots \end{pmatrix} +$$ +In this case we know that $u(x) = \E^x$, but we would still need other means to calculate $u_k$ (They are definitely not as simple as Taylor series coefficients). + +We can express the constraints as acting on the coefficients. For example, we have +$$ +u(0) = (T_0(0), T_1(0), \ldots) \begin{pmatrix} u_0\\u_1\\\vdots \end{pmatrix} = (1,0,-1,0,1,\ldots) \begin{pmatrix} u_0\\u_1\\\vdots \end{pmatrix} +$$ +We also have +$$u'(x) = (U_0(x),U_1(x),\ldots) \begin{pmatrix} +0 & 1 \cr +&& 2 \cr +&&& 3 \cr +&&&&\ddots +\end{pmatrix}\begin{pmatrix} u_0\\u_1\\\vdots \end{pmatrix} +$$ +To represent $u'(x) - u(x)$, we need to make sure the bases are compatible. In other words, we want to express $u(x)$ in it's $U_k(x)$ basis using the conversion operator $S_0$: +$$u(x) = (U_0(x),U_1(x),\ldots) \begin{pmatrix} + 1 &0 & -\half \cr +& \half & 0 & -\half \cr +&&\ddots & \ddots & \ddots +\end{pmatrix}\begin{pmatrix} u_0\\u_1\\\vdots \end{pmatrix} +$$ + +Which gives us, +$$ +u'(x) - u(x) = (U_0(x),U_1(x),\ldots) \begin{pmatrix} + -1 &1 & \half \cr +& -\half & 2 & \half \cr +&& -\half & 3 & \half \cr +&&&\ddots & \ddots & \ddots +\end{pmatrix} \begin{pmatrix} u_0\\u_1\\\vdots \end{pmatrix} +$$ + + +Combing the differential part and the evaluation part, we arrive at an (infinite) system of equations for the coefficients $u_0,u_1,\dots$: +$$ +\begin{pmatrix} + 1 & 0 & -1 & 0 & 1 & \cdots \\ + -1 &1 & \half \cr +& -\half & 2 & \half \cr +&& -\half & 3 & \half \cr +&&&\ddots & \ddots & \ddots +\end{pmatrix} \begin{pmatrix} u_0\\u_1\\\vdots \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ \vdots \end{pmatrix} +$$ + +How to solve this system is outside the scope of this course (though a simple approach is to truncate the infinite system to finite systems). We can however do this in ApproxFun: + +B = Evaluation(0.0) : Chebyshev() +D = Derivative() : Chebyshev() → Ultraspherical(1) +S₀ = I : Chebyshev() → Ultraspherical(1) +L = [B; + D - S₀] + +We can solve this system as follows: + +u = L \ [1; 0] +plot(u) + +It matches the "true" result: + +u(0.1) - exp(0.1) + +Note we can incorporate right-hand sides as well, for example, to solve $u'(x) - u(x) = f(x)$, by expanding $f$ in its Chebyshev U series. + +### Second-order constanst coefficient equations + +This approach extends to second-order constant-coefficient equations by using ultraspherical polynomials. Consider +\begin{align*} +u(-1) &= 1\\ +u(1) &= 0\\ +u''(x) + u'(x) + u(x) &= 0 +\end{align*} +Evaluation works as in the first-order case. To handle second-derivatives, we need $C^{(2)}$ polynomials: + +D₀ = Derivative() : Chebyshev() → Ultraspherical(1) +D₁ = Derivative() : Ultraspherical(1) → Ultraspherical(2) +D₁*D₀ # 2 zeros not printed in (1,1) and (1,2) entry + +For the identity operator, we use two conversion operators: + +S₀ = I : Chebyshev() → Ultraspherical(1) +S₁ = I : Ultraspherical(1) → Ultraspherical(2) +S₁*S₀ + +And for the first derivative, we use a derivative and then a conversion: + +S₁*D₀ # or could have been D₁*S₀ + +Putting everything together we get: + +B₋₁ = Evaluation(-1) : Chebyshev() +B₁ = Evaluation(1) : Chebyshev() +# u(-1) +# u(1) +# u'' + u' +u + +L = [B₋₁; + B₁; + D₁*D₀ + S₁*D₀ + S₁*S₀] + +u = L \ [1.0,0.0,0.0] +plot(u) + +### Variable coefficients + +Consider the Airy ODE +\begin{align*} +u(-1) &= 1\\ +u(1) &= 0\\ +u''(x) - xu(x) &= 0 +\end{align*} + +to handle, this, we need only use the Jacobi operator to represent multiplication by $x$: + +x = Fun() +Jᵗ = Multiplication(x) : Chebyshev() → Chebyshev() # transpose of the Jacobi operator + +We set op ther system as follows: + +L = [B₋₁; # u(-1) + B₁ ; # u(1) + D₁*D₀ - S₁*S₀*Jᵗ] # u'' - x*u + +u = L \ [1.0;0.0;0.0] +plot(u; legend=false) + +If we introduce a small parameter, that is, solve +\begin{align*} +u(-1) &= 1\\ +u(1) &= 0\\ +\epsilon u''(x) - xu(x) &= 0 +\end{align*} +we can see pretty hard to compute solutions: + +ε = 1E-6 +L = [B₋₁; + B₁ ; + ε*D₁*D₀ - S₁*S₀*Jᵗ] + +u = L \ [1.0;0.0;0.0] +plot(u; legend=false) + +Because of the banded structure, this can be solved fast: + +ε = 1E-10 +L = [B₋₁; + B₁ ; + ε*D₁*D₀ - S₁*S₀*Jᵗ] + + +@time u = L \ [1.0;0.0;0.0] +@show ncoefficients(u); + +To handle other variable coefficients, first consider a polynomial $p(x)$. If Multiplication by $x$ is represented by multiplying the coefficients by $J^\top$, then multiplication by $p$ is represented by $p(J^\top)$: + +M = -I + Jᵗ + (Jᵗ)^2 # represents -1+x+x^2 + +ε = 1E-6 +L = [B₋₁; + B₁ ; + ε*D₁*D₀ - S₁*S₀*M] + +@time u = L \ [1.0;0.0;0.0] + +@show ε*u''(0.1) - (-1+0.1+0.1^2)*u(0.1) +plot(u) + +For other smooth functions, we first approximate in a polynomial basis, and without loss of generality we use Chebyshev T basis. For example, consider +\begin{align*} +u(-1) &= 1\\ +u(1) &= 0\\ +\epsilon u''(x) - \E^x u(x) &= 0 +\end{align*} +where +$$ +\E^x \approx p(x) = \sum_{k=0}^{m-1} p_k T_k(x) +$$ +Evaluating at a point $x$, recall Clenshaw's algorithm: +\begin{align*} +\gamma_{n-1} &= 2p_{n-1} \\ +\gamma_{n-2} &= 2p_{n-2} + 2x \gamma_{n-1} \\ +\gamma_{n-3} &= 2 p_{n-3} + 2x \gamma_{n-2} - \gamma_{n-1} \\ +& \vdots \\ +\gamma_1 &= p_1 + x \gamma_2 - \half \gamma_3 \\ +p(x) = \gamma_0 &= p_0 + x \gamma_1 - \half \gamma_2 +\end{align*} +If multiplication by $x$ becomes $J^\top$, then multiplication by $p(x)$ becomes $p(J^\top)$, and hence we calculate:\ +\begin{align*} +\Gamma_{n-1} &= 2p_{n-1}I \\ +\Gamma_{n-2} &= 2p_{n-2}I + 2J^\top \Gamma_{n-1} \\ +\Gamma_{n-3} &= 2 p_{n-3}I + 2J^\top \Gamma_{n-2} - \Gamma_{n-1} \\ +& \vdots \\ +\Gamma_1 &= p_1I + J^\top \Gamma_2 - \half \Gamma_3 \\ +p(J^\top) = \Gamma_0 &= p_0 + x \Gamma_1 - \half \Gamma_2 +\end{align*} + +Here is an example: + +p = Fun(exp, Chebyshev()) # polynomial approximation to exp(x) +M = Multiplication(p) : Chebyshev() # constructed using Clenshaw: + +ApproxFun.bandwidths(M) # still banded + +ε = 1E-6 +L = [B₋₁; + B₁ ; + ε*D₁*D₀ + S₁*S₀*M] + +@time u = L \ [1.0;0.0;0.0] + +@show ε*u''(0.1) + exp(0.1)*u(0.1) +plot(u) + +# M3M6: Methods of Mathematical Physics + +$$ +\def\dashint{{\int\!\!\!\!\!\!-\,}} +\def\infdashint{\dashint_{\!\!\!-\infty}^{\,\infty}} +\def\D{\,{\rm d}} +\def\E{{\rm e}} +\def\dx{\D x} +\def\dt{\D t} +\def\dz{\D z} +\def\C{{\mathbb C}} +\def\R{{\mathbb R}} +\def\CC{{\cal C}} +\def\HH{{\cal H}} +\def\I{{\rm i}} +\def\qqqquad{\qquad\qquad} +\def\qqand{\qquad\hbox{for}\qquad} +\def\qqfor{\qquad\hbox{for}\qquad} +\def\qqwhere{\qquad\hbox{where}\qquad} +\def\Res_#1{\underset{#1}{\rm Res}}\, +\def\sech{{\rm sech}\,} +\def\acos{\,{\rm acos}\,} +\def\vc#1{{\mathbf #1}} +\def\ip<#1,#2>{\left\langle#1,#2\right\rangle} +\def\norm#1{\left\|#1\right\|} +\def\half{{1 \over 2}} +$$ + +Dr. Sheehan Olver +
+s.olver@imperial.ac.uk + +
+Website: https://github.com/dlfivefifty/M3M6LectureNotes + + +# Lecture 17: Differential equations satisfied by orthogonal polynomials + + +This lecture we do the following: + +1. Differential equations for orthogonal polynomials + - Sturm–Liouville equations + - Weighted differentiation for ultraspherical polynomials + - Differential equation for ultraspherical polynomials +2. Application: Eigenstates of Schrödinger operators with quadratic potentials +2. Rodriguez formulae + + + +The three classical weights are (Hermite) $w(x) = \E^{-x^2}$, (Laguerre) $w_\alpha(x) = x^\alpha \E^{-x}$ and (Jacobi) $w_{\alpha,\beta}(x) = (1-x)^\alpha (1+x)^\beta$. Note all weights form a simple hierarchy: when differentiated, they give a linear polynomial times the previous weight in the hierarchy. For Hermite, +$$ +{\D \over \dx} w(x) = -2x w(x) +$$ +for Laguerre, +$$ +{\D \over \dx} w^{(\alpha)}(x) = (\alpha - x) w^{(\alpha-1)}(x) +$$ +and for Jacobi +$$ +{\D \over \dx} w^{(\alpha,\beta)}(x) = (\beta(1-x) - \alpha(1+x)) w^{(\alpha-1,\beta-1)}(x) +$$ +These relationships lead to simple differential equations that have the classical orthogonal polynomials as eigenfunctions. + +### Sturm–Liouville operator +We first consider a simple class of operators that are self-adjoint: + +**Proposition (Sturm–Liouville self-adjointness)** Consider the weighted inner product +$$ +\ip_w = \int_a^b f(x) g(x) w(x) \dx +$$ +then for any continuously differentiable function $q(x)$ satisfying $q(a) = q(b) = 0$, the operator +$$ +Lu = {1 \over w(x)} {\D \over \dx}\left[ q(x) {\D u\over \dx} \right] +$$ +is self-adjoint in the sense +$$ +\ip_w = \ip_w +$$ + +**Proof** Simple integration by parts argument: +$$ +\ip_w = \int_a^b {\D \over \dx}\left[ q(x) {\D u\over \dx}\right] g(x)\dx = -\int_a^b q(x) {\D u\over \dx} {\D g\over \dx} \dx = \int_a^b u(x) {\D \over \dx} q(x) {\D g\over \dx} \dx = \int_a^b u(x) {1 \over w(x)} {\D \over \dx}\left[q(x) {\D g\over \dx}\right] w(x) \dx = \ip_w +$$ + +⬛️ + +We claim that the classical orthogonal polynomials are eigenfunctions of a Sturm–Liouville problem, that is, in each case there exists a $q(x)$ so that +$$ +L p_n(x) = \lambda_n p_n(x) +$$ +where $\lambda_n$ is the (real) eigenvalue. We will derive this for the ultraspherical polynomials. + + +### Weighted differentiation for ultraspherical polynomials + +We have already seen that Chebyshev and ultraspherical polynomials have simple expressions for derivatives where we decrement the degree and increment the parameter: +\begin{align*} +{\D \over \dx } T_n(x) = n U_{n-1}(x) = n C_{n-1}^{(1)}(x) \\ +{\D \over \dx } C_n^{(\lambda)}(x) = 2 \lambda C_{n-1}^{(\lambda+1)}(x) +\end{align*} +In this section, we see that differentiating the weighted polynomials actually decrements the parameter and increments the degree: + +**Proposition (weighted differentiation)** +\begin{align*} +{\D \over \dx }[\sqrt{1-x^2} U_n(x)] = - {n+1 \over \sqrt{1-x^2}} T_{n+1}(x) \\ +{\D \over \dx }[(1-x^2)^{\lambda-\half} C_n^{(\lambda)}(x)] = -{(n+1) (n+2 \lambda-1) \over 2 (\lambda-1) } (1-x^2)^{\lambda - {3 \over 2}} C_{n+1}^{(\lambda-1)}(x) +\end{align*} + +**Proof** We show the first result by showing that the left-hand side is orthogonal to all polynomials of degree less than $n+1$ by integration by parts: +$$ +\ip< \sqrt{1-x^2} {\D \over \dx }[\sqrt{1-x^2} U_n(x)], p_m(x)>_{\rm T} = -\int_{-1}^1 \sqrt{1-x^2} U_n(x) p_m' \dx =0 +$$ +Note that +$$ +\sqrt{1-x^2} {\D \over \dx } \sqrt{1-x^2} f(x) = (1-x^2) f'(x) - x f(x) +$$ +Thus we just have to verify the constant in front: +$$ +\sqrt{1-x^2} {\D \over \dx }[\sqrt{1-x^2} U_n(x) = (-n -1) 2^n x^{n+1} +$$ + +The other ultraspherical polynomial follow similarly. +⬛️ + + +### Eigenvalue equation for Ultraspherical polynomials + +Note that differentiating increments the parameter and decrements the degree while weight differentiation decements the parameter and increments the degree. Therefore combining them brings us back to where we started. + +In the case of Chebyshev polynomials, this gives us a Sturm–Liouville equation: +$$ +\sqrt{1-x^2} {\D \over \dx} \sqrt{1-x^2} {\D T_n \over \dx} = +n \sqrt{1-x^2} {\D \over \dx} \sqrt{1-x^2} U_{n-1}(x) = -n^2 T_n(x) +$$ + +Note that the chain rule gives us a simple expression as +$$ +(1-x^2) {\D^2 T_n \over \dx^2} -x {\D T_n \over \dx} = -n^2 T_n(x) +$$ + +Similarly, +$$ +(1-x^2)^{\half - \lambda} {\D \over \dx}(1-x^2)^{\lambda + \half} {\D C_n^{(\lambda)} \over \dx} = -n (n+2 \lambda) C_n^{(\lambda)}(x) +$$ +or in other words, +$$ +(1-x^2) {\D^2 C_n^{(\lambda)} \over \dx^2} - (2\lambda+1) x {\D C_n^{(\lambda)} \over \dx} = -{n (n+2 \lambda) \over 2\lambda}C_n^{(\lambda)}(x) +$$ + +## Rodriguez formula + +Because of the special structure of our weights, we have special Rodriguez formulae of the form +$$ + p_n(x) = {1 \over \kappa_n w(x)} {\D^n \over \dx^n} w(x) F(x)^n +$$ +where $w(x)$ is the weight and $F(x) = (1-x^2)$ (Jacobi), $x$ (Laguerre) or $1$ (Hermite) and $\kappa_n$ is a normalization constant. + +**Proposition (Hermite Rodriguez)** +$$ +H_n(x)= (-1)^n \E^{x^2} {\D^n \over \dx^n} \E^{-x^2} +$$ + +**Proof** +We first show that it's a degree $n$ polynomial. This proceeds by induction: +$$ + H_0(x) = \E^{x^2} {\D^0 \over \dx^0}\E^{-x^2} = 1 +$$ +$$ + H_{n+1}(x) = -\E^{x^2}{\D \over \dx}\left[\E^{-x^2} H_{n}(x)\right] = 2x H_{n}(x) + H_n'(x) +$$ +and then we have +$$ + {\D^n \over \dx^n}[p_m(x) \E^{-x^2}]= {\D^{n-1} \over \dx^{n-1}} (p_m'(x)-2x p_m(x)) \E^{-x^2} +$$ +Orthogonality follows from integration by parts: +$$ +\ip_{\rm H} = (-1)^n \int {\D^n \E^{-x^2} \over \dx^n} p_m \dx = \int \E^{-x^2} {\D^n p_m \over \dx^n} \dx = 0 +$$ +if $m < n$. + +Now we just need to show we have the right constant. But we have +$$ + {\D^n \over \dx^n}[\E^{-x^2}] = {\D^{n-1} \over \dx^{n-1}}[-2x \E^{-x^2}] = {\D^{n-2} \over \dx^{n-2}}[(4x^2 + O(x)) \E^{-x^2}] = \cdots = (-1)^n 2^n x^n + $$ + +⬛️ + +Note this tells us the Hermite recurrence: Here we have the simple expressions +$$ +H_n'(x) = 2n H_{n-1}(x) \qqand {\D \over \dx}[\E^{-x^2} H_n(x)] = -\E^{-x^2} H_{n+1}(x) +$$ +These follow from the same arguments as before since $w'(x) = -2x w(x)$. But using the Rodriguez formula, we get + +$$ +2n H_{n-1}(x) = H_{n}'(x) = (-1)^{n} 2 x \E^{x^2} {\D^{n} \over \dx^{n}} \E^{-x^2} + (-1)^n \E^{x^2} {\D^{n+1} \over \dx^{n+1}} \E^{-x^2} = 2x H_n(x) - H_{n+1}(x) +$$ +which means +$$ +x H_n(x) = nH_{n-1}(x) +{H_{n+1}(x) \over 2} +$$ + + +## Application: Eigenstates of Schrödinger operators with quadratic potentials + +Using the derivative formulae tells us a Sturm–Liouville operator for Hermite polynomials: +$$ +\E^{x^2} {\D \over \dx} \E^{-x^2} {\D H_n \over \dx} = 2n \E^{x^2} {\D \over \dx} \E^{-x^2} H_{n-1}(x) = -2nH_n(x) +$$ +or rewritten, this gives us +$$ +{\D^2 H_n \over \dx^2} -2x {\D H_n \over \dx} = -2nH_n(x) +$$ + + +W therefore have +$$ +{\D^2 \over \dx^2}[\E^{-{x^2 \over 2}} H_n(x)] = \E^{-{x^2 \over 2}} (H_n''(x) -2x H_n'(x) + (x^2-1) H_n(x)) = \E^{-{x^2 \over 2}} (x^2-1-2n) H_n(x) +$$ +In other words, for the Hermite function $\psi_n(x)$ we have +$$ +{\D^2 \psi_n \over \dx^2} -x^2 \psi_n = -(2n+1) \psi_n +$$ +and therefore $\psi_n$ are the eigenfunctions. + +Wait, we want to normalize 😩. In Schrödinger equations the square of the wave $\psi(x)^2$ represents a probability distribution, which should integrate to 1. Here's a trick: we know that +$$ +x \begin{pmatrix} H_0(x) \\ H_1(x) \\ H_2(x) \\ \vdots \end{pmatrix} = \underbrace{\begin{pmatrix} 0 & {1 \over 2} \\ +1 & 0 & \half \\ +& 2 & 0 & \half \\ +&& 3 & 0 & \ddots \\ +&&& \ddots & \ddots +\end{pmatrix}}_J\begin{pmatrix} H_0(x) \\ H_1(x) \\ H_2(x) \\ \vdots \end{pmatrix} +$$ +We want to conjugate by a diagonal matrix so that +$$ +\begin{pmatrix}1 \\ & d_1 \\ &&d_2 \\&&&\ddots \end{pmatrix} J \begin{pmatrix}1 \\ & d_1^{-1} \\ &&d_2^{-1} \\&&&\ddots \end{pmatrix} = \begin{pmatrix} 0 & {1 \over 2d_1} \\ +d_1 & 0 & {d_1 \over 2 d_2} \\ +& {2d_2 \over d_1} & 0 & {d_2 \over 2 d_3} \\ +&& {3d_3 \over d_2} & 0 & \ddots \\ +&&& \ddots & \ddots +\end{pmatrix} +$$ +becomes symmetric. This becomes a sequence of equations: +\begin{align*} +d_1 &= {1 \over 2 d_1} \Rightarrow d_1^2 = {1 \over 2} \\ +2d_2d_1^{-1} &= {d_1 \over 2 d_2} \Rightarrow d_2^2 = {d_1^2 \over 4} = {1 \over 8} = {1 \over 2^2 2!} \\ +3d_3d_2^{-1} &= {d_2 \over 2 d_3} \Rightarrow d_3^2 = {d_2^2 \over 3\times 2} = {1 \over 2^3 3!} \\ +&\vdots \\ +d_n^2 = {1 \over 2^n n!} +\end{align*} + +Thus by Lecture 16 the norm of $d_n H_n(x)$ is constant. If we also normalize using +$$ + \int_{-\infty}^\infty \E^{-x^2} \dx = \sqrt{\pi} +$$ +we get the normalized eigenfunctions +$$ + \psi_n(x) = {H_n(x)\E^{-x^2/2} \over \sqrt{\sqrt{\pi} 2^n n!} } +$$ + +p = plot() +for n = 0:5 + H = Fun(Hermite(), [zeros(n);1]) + ψ = Fun(x -> H(x)exp(-x^2/2), -10.0 .. 10.0)/sqrt(sqrt(π)*2^n*factorial(1.0n)) + plot!(ψ; label="n = $n") +end +p + +It's convention to shift them by the eigenvalue: + +p = plot(pad(Fun(x -> x^2, -10 .. 10), 100); ylims=(0,25)) +for n = 0:10 + H = Fun(Hermite(), [zeros(n);1]) + ψ = Fun(x -> H(x)exp(-x^2/2), -10.0 .. 10.0)/sqrt(sqrt(π)*2^n*factorial(1.0n)) + plot!(ψ + 2n+1; label="n = $n") +end +p \ No newline at end of file diff --git a/src/Lecture24.jmd b/src/Lecture24.jmd new file mode 100644 index 0000000..9a73e23 --- /dev/null +++ b/src/Lecture24.jmd @@ -0,0 +1,54 @@ + +## Approximation with Hermite polynomials + + +This is often the case with Hermite polynomials: on the real line polynomial approximation is unnatural unless + the function approximated is a polynomial as otherwise the behaviour at ∞ is inconsistent, so what we really want is weighted approximation. + Thus we can either use +$$ +f(x) = \E^{-x^2}\sum_{k=0}^\infty f_k H_k(x) +$$ +or +$$ +f(x) = \E^{-x^2/2}\sum_{k=0}^\infty f_k H_k(x) +$$ +Depending on your problem, getting this wrong can be disasterous: + +f = Fun(x -> 1+x +x^2, Hermite()) +f(0.10) + +# nonsense trying to approximating sech(x) by a degree 50 polynomial: +f = Fun(x -> sech(x), Hermite(), 51) +xx = -8:0.01:8 +plot(xx, sech.(xx); ylims=(-10,10), label="sech x") +plot!(xx, f.(xx); label="f") + +# weighted by works sqrt(w(x)) = exp(-x^2/2) +f = Fun(x -> sech(x), GaussWeight(Hermite(),1/2),101) + +plot(xx, sech.(xx); ylims=(-10,10), label="sech x") +plot!(xx, f.(xx); label="f") + +# weighted by w(x) = exp(-x^2) breaks again +f = Fun(x -> sech(x), GaussWeight(Hermite()),101) + +plot(xx, sech.(xx); ylims=(-10,10), label="sech x") +plot(xx, f.(xx); label="f") + +Note that correctly weighted Hermite, that is, with $\sqrt{w(x)} = \E^{-x^2/2}$ look "nice": + +p = plot() +for k=0:6 + H_k = Fun(GaussWeight(Hermite(),1/2),[zeros(k);1]) + plot!(xx, H_k.(xx); label="H_$k") +end +p + +Compare this to weighting by $w(x) = \E^{-x^2}$: + +p = plot() +for k=0:6 + H_k = Fun(GaussWeight(Hermite()),[zeros(k);1]) + plot!(xx, H_k.(xx); label="H_$k") +end +p diff --git a/src/M3M6AppliedComplexAnalysis.jl b/src/M3M6AppliedComplexAnalysis.jl index 77f832b..f86a455 100644 --- a/src/M3M6AppliedComplexAnalysis.jl +++ b/src/M3M6AppliedComplexAnalysis.jl @@ -19,6 +19,7 @@ weave("src/Lecture16.jmd", doctype="md2tex", informat="markdown", out_path=pwd() weave("src/Lecture17.jmd", doctype="md2tex", informat="markdown", out_path=pwd()*"/output/", template="src/template.tpl") weave("src/Lecture18.jmd", doctype="md2tex", informat="markdown", out_path=pwd()*"/output/", template="src/template.tpl") weave("src/Lecture19.jmd", doctype="md2tex", informat="markdown", out_path=pwd()*"/output/", template="src/template.tpl") +weave("src/Lecture20.jmd", doctype="md2tex", informat="markdown", out_path=pwd()*"/output/", template="src/template.tpl") weave("src/Solutions1.jmd", doctype="md2tex", informat="markdown", out_path=pwd()*"/output/", template="src/template.tpl") weave("src/Solutions2.jmd", doctype="md2tex", informat="markdown", out_path=pwd()*"/output/", template="src/template.tpl")