{ "cells": [ { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [], "source": [ "using Plots, ComplexPhasePortrait, ApproxFun, SingularIntegralEquations, DifferentialEquations\n", "gr();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# M3M6: Methods of Mathematical Physics\n", "\n", "$$\n", "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n", "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n", "\\def\\D{\\,{\\rm d}}\n", "\\def\\E{{\\rm e}}\n", "\\def\\dx{\\D x}\n", "\\def\\dt{\\D t}\n", "\\def\\dz{\\D z}\n", "\\def\\C{{\\mathbb C}}\n", "\\def\\R{{\\mathbb R}}\n", "\\def\\CC{{\\cal C}}\n", "\\def\\HH{{\\cal H}}\n", "\\def\\I{{\\rm i}}\n", "\\def\\qqqquad{\\qquad\\qquad}\n", "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n", "\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n", "\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n", "\\def\\sech{{\\rm sech}\\,}\n", "\\def\\acos{\\,{\\rm acos}\\,}\n", "\\def\\vc#1{{\\mathbf #1}}\n", "\\def\\ip<#1,#2>{\\left\\langle#1,#2\\right\\rangle}\n", "\\def\\norm#1{\\left\\|#1\\right\\|}\n", "$$\n", "\n", "Dr. Sheehan Olver\n", "
\n", "s.olver@imperial.ac.uk\n", "\n", "
\n", "Website: https://github.com/dlfivefifty/M3M6LectureNotes\n", "\n", "# Chapter 3: Orthogonal Polynomials\n", "\n", "We now introduce orthogonal polynomials (OPs). These are ___fundamental___ for computational mathematics, with applications in\n", "1. Function approximation\n", "2. Quadrature (calculating integrals)\n", "2. Solving differential equations\n", "3. Spectral analysis of Schrödinger operators\n", "\n", "We will investigate the properties of _general_ OPs:\n", "\n", "1. Construction via Gram–Schmidt \n", "2. Three-term recurrence relationships\n", "3. Gaussian quadrature\n", "3. Relationship with Cauchy and logarithmic transforms\n", "\n", "We will also investigate the properties of _classical_ OPs:\n", "\n", "1. Explicit formulae\n", "3. Rodriguez formulae\n", "2. Ordinary differential equations\n", "2. Derivatives\n", "\n", "A good reference is [Digital Library of Mathematical Functions, Chapter 18](http://dlmf.nist.gov/18).\n", "\n", "\n", "\n", "# Lecture 14: Constructing orthogonal polynomials\n", "\n", "\n", "This lecture we do the following:\n", "1. Definition of orthogonal polynomials\n", "2. Definition of classical orthogonal polynomials\n", " - Hermite, Laguerre, and Jacobi polynomials\n", " - Legendre, Chebyshev, and ultraspherical polynomials\n", " - Explicit construction for Chebyshev polynomials\n", "2. Construction of orthogonal polynomials via Gram–Schmidt process \n", "3. Function approximation with orthogonal polynomials \n", "\n", "\n", "## Definition of orthogonal polynomials\n", "\n", "Let $p_0(x),p_1(x),p_2(x),…$ be a sequence of polynomials such that $p_n(x)$ is exactly degree $n$, that is,\n", "$$\n", "p_n(x) = k_n x^n + O(x^{n-1})\n", "$$\n", "where $k_n \\neq 0$.\n", "\n", "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\n", "$$\n", "\\ip := \\int_a^b f(x) g(x) w(x) \\dx\n", "$$\n", "We say that $\\{p_0, p_1,\\ldots\\}$ are _orthogonal with respect to the weight $w$_ if \n", "$$\n", "\\ip = 0\\qqfor n \\neq m.\n", "$$\n", "Because $w$ is continuous, we have\n", "$$\n", "\\norm{p_n}^2 = \\ip > 0 .\n", "$$\n", "\n", "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:\n", "\n", "**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.\n", "\n", "**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\n", "$$\n", "which shows all $c_k$ are zero.\n", "\n", "**Corollary (Uniqueness of OPs I)** If $q_n$ and $p_n$ are orthogonal w.r.t. $w$ to all lower degree polynomials, then $q_n(x) = C p_n(x)$ for some constant $C$. \n", "\n", "### Monic orthogonal polynomials\n", "\n", "If $k_n = 1$, that is, \n", "$$\n", "p_n(x) = x^n + O(x^{n-1})\n", "$$\n", "then we refer to the orthogonal polymomials as monic.\n", "\n", "Monic OPs are unique as we have specified $k_n$.\n", "\n", "\n", "### Orthonormal polynomials\n", "\n", "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.\n", "\n", "\n", "**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 big performance difference.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Definition of classical orthogonal polynomials\n", "\n", "Classical orthogonal polynomials are orthogonal with respect to the following three weights:\n", "\n", "\n", "| Name | Interval $(a,b)$ |Weight function $w(x)$ | Standard polynomial | highest order coefficient $k_n$ |\n", "|:-------------|:------------- |:----------------------|:-----|:-----|\n", "| Hermite |$(-\\infty,\\infty)$ | $\\E^{-x^2}$ | $H_n(x)$ | $2^n$ |\n", "| Laguerre | $(0,\\infty)$ | $x^\\alpha \\E^{-x}$ | $L_n^{(\\alpha)}(x)$ | See [Table 18.3.1](http://dlmf.nist.gov/18.3) |\n", "| 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) |\n", "\n", "\n", "Note out of convention the parameters for Jacobi polynomials are in the \"wrong\" order.\n", "\n", "We can actually construct these polynomials in Julia: first consider Hermite:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-4\n", "\n", "\n", "-2\n", "\n", "\n", "0\n", "\n", "\n", "2\n", "\n", "\n", "4\n", "\n", "\n", "-400\n", "\n", "\n", "-200\n", "\n", "\n", "0\n", "\n", "\n", "200\n", "\n", "\n", "400\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "H_0\n", "\n", "\n", "\n", "H_1\n", "\n", "\n", "\n", "H_2\n", "\n", "\n", "\n", "H_3\n", "\n", "\n", "\n", "H_4\n", "\n", "\n", "\n", "H_5\n", "\n", "\n" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H₀ = Fun(Hermite(), [1])\n", "H₁ = Fun(Hermite(), [0,1])\n", "H₂ = Fun(Hermite(), [0,0,1])\n", "H₃ = Fun(Hermite(), [0,0,0,1])\n", "H₄ = Fun(Hermite(), [0,0,0,0,1])\n", "H₅ = Fun(Hermite(), [0,0,0,0,0,1])\n", "\n", "xx = -4:0.01:4\n", "plot(xx, H₀.(xx); label=\"H_0\", ylims=(-400,400))\n", "plot!(xx, H₁.(xx); label=\"H_1\", ylims=(-400,400))\n", "plot!(xx, H₂.(xx); label=\"H_2\", ylims=(-400,400))\n", "plot!(xx, H₃.(xx); label=\"H_3\", ylims=(-400,400))\n", "plot!(xx, H₄.(xx); label=\"H_4\", ylims=(-400,400))\n", "plot!(xx, H₅.(xx); label=\"H_5\")\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We verify their orthogonality:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "sum(H₂ * H₅ * w) = 0.0\n", "sum(H₅ * H₅ * w) = 6806.222787477181\n" ] } ], "source": [ "w = Fun(GaussWeight(), [1.0])\n", "\n", "@show sum(H₂*H₅*w) # means integrate\n", "@show sum(H₅*H₅*w);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now Jacobi:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-1.0\n", "\n", "\n", "-0.5\n", "\n", "\n", "0.0\n", "\n", "\n", "0.5\n", "\n", "\n", "1.0\n", "\n", "\n", "-2\n", "\n", "\n", "-1\n", "\n", "\n", "0\n", "\n", "\n", "1\n", "\n", "\n", "2\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "P_0^(0.1,0.2)\n", "\n", "\n", "\n", "P_1^(0.1,0.2)\n", "\n", "\n", "\n", "P_2^(0.1,0.2)\n", "\n", "\n", "\n", "P_3^(0.1,0.2)\n", "\n", "\n", "\n", "P_4^(0.1,0.2)\n", "\n", "\n", "\n", "P_5^(0.1,0.2)\n", "\n", "\n" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "α,β = 0.1,0.2\n", "P₀ = Fun(Jacobi(β,α), [1])\n", "P₁ = Fun(Jacobi(β,α), [0,1])\n", "P₂ = Fun(Jacobi(β,α), [0,0,1])\n", "P₃ = Fun(Jacobi(β,α), [0,0,0,1])\n", "P₄ = Fun(Jacobi(β,α), [0,0,0,0,1])\n", "P₅ = Fun(Jacobi(β,α), [0,0,0,0,0,1])\n", "\n", "xx = -1:0.01:1\n", "plot( xx, P₀.(xx); label=\"P_0^($α,$β)\", ylims=(-2,2))\n", "plot!(xx, P₁.(xx); label=\"P_1^($α,$β)\")\n", "plot!(xx, P₂.(xx); label=\"P_2^($α,$β)\")\n", "plot!(xx, P₃.(xx); label=\"P_3^($α,$β)\")\n", "plot!(xx, P₄.(xx); label=\"P_4^($α,$β)\")\n", "plot!(xx, P₅.(xx); label=\"P_5^($α,$β)\")" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "sum(P₂ * P₅ * w) = 2.1250362580715887e-17\n", "sum(P₅ * P₅ * w) = 0.2171335824839316\n" ] } ], "source": [ "w = Fun(JacobiWeight(β,α), [1.0])\n", "@show sum(P₂*P₅*w) # means integrate\n", "@show sum(P₅*P₅*w);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Legendre, Chebyshev, and ultraspherical polynomials\n", "\n", "There are special families of Jacobi weights with their own name. \n", "\n", "| Name | Jacobi parameters |Weight function $w(x)$ | Standard polynomial | highest order coefficient $k_n$ |\n", "|:-------------|:------------- |:----------------------|:-----|:------|\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) | \n", "| Legendre | $0,0$ | $1$ | $P_n(x)$ | $2^n(1/2)_n/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)$ |\n", "| Chebyshev (second kind) | ${1 \\over 2},{1 \\over 2}$ | $\\sqrt{1-x^2}$ | $U_n(x)$ | $2^n$\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!$ |\n", "\n", "Note that other than Legendre, these polynomials have a different normalization than $P_n^{(\\alpha,\\beta)}$:\n" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-1.0\n", "\n", "\n", "-0.5\n", "\n", "\n", "0.0\n", "\n", "\n", "0.5\n", "\n", "\n", "1.0\n", "\n", "\n", "-1.0\n", "\n", "\n", "-0.5\n", "\n", "\n", "0.0\n", "\n", "\n", "0.5\n", "\n", "\n", "1.0\n", "\n", "\n", "T_2 is C*P_2 for some C\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "T_2\n", "\n", "\n", "\n", "P_2\n", "\n", "\n" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T₂ = Fun(Chebyshev(), [0.0,0,1])\n", "P₂ = Fun(Jacobi(-1/2,-1/2), [0.0,0,1])\n", "plot(T₂; label=\"T_2\", title=\"T_2 is C*P_2 for some C\")\n", "plot!(P₂; label=\"P_2\") " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "But because they are orthogonal w.r.t. the same weight, they must be a constant multiple of each-other." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Explicit construction of Chebyshev polynomials (first kind and second kind)\n", "\n", "Chebyshev polynomials are pretty much the only OPs with _simple_ closed form expressions.\n", "\n", "**Proposition (Chebyshev first kind formula)** \n", "$$T_n(x) = \\cos n \\acos x$$\n", "or in other words,\n", "$$\n", "T_n(\\cos \\theta) = \\cos n \\theta\n", "$$\n", "\n", "**Proof** We first show that they are orthogonal w.r.t. $1/\\sqrt{1-x^2}$. Too easy: do $x = \\cos \\theta$, $\\dx = -\\sin \\theta$ to get (for $n \\neq m$)\n", "$$\n", " \\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\n", "$$\n", "\n", "We then need to show it has the right highest order term $k_n$. Note that $k_0 = k_1 = 1$. Using $z = \\E^{\\I \\theta}$ we see that $\\cos n \\theta$ has a simple recurrence for $n=2,3,\\ldots$:\n", "$$\n", "\\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 \n", "$$\n", "thus \n", "$$\n", "\\cos n \\acos x = 2 x \\cos(n-1) \\acos x - \\cos(n-2) \\acos x\n", "$$\n", "It follows that \n", "$$\n", "k_n = 2 x k_{n-1} = 2^{n-1} k_1 = 2^{n-1}\n", "$$\n", "By uniqueness we have $T_n(x) \\cos n \\acos x$.\n", "\n", "⬛️\n", "\n", "\n", "\n", "**Proposition (Chebyshev second kind formula)** \n", "$$U_n(x) = {\\sin (n+1) \\acos x \\over \\sin \\acos x}$$\n", "or in other words,\n", "$$\n", "U_n(\\cos \\theta) = {\\sin (n+1) \\theta \\over \\sin \\theta}\n", "$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Gram–Schmidt algorithm\n", "\n", "In general we don't have nice formulae. But we can always construct them via Gram–Schmidt:\n", "\n", "**Proposition (Gram–Schmidt)** Define\n", "\\begin{align*}\n", "p_0(x) = 1 \\\\\n", "q_0(x) = {1 \\over \\norm{p_0}}\\\\\n", "p_{n+1}(x) = x q_n(x) - \\sum_{k=0}^n \\ip q_k(x)\\\\\n", "q_{n+1}(x) = {p_{n+1}(x) \\over \\norm{p_n}}\n", "\\end{align*}\n", "Then $q_0(x), q_1(x), \\ldots$ are orthonormal w.r.t. $w$.\n", "\n", "**Proof** By linearity we have\n", "$$\n", "\\ip = \\ip q_k, q_j> = \\ip - \\ip \\ip = 0\n", "$$\n", "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}$.\n", "\n", "⬛️\n", "\n", "Let's make our own family:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "x = Fun()\n", "w = exp(x)\n", "ip = (f,g) -> sum(f*g*w)\n", "nrm = f -> sqrt(ip(f,f))\n", "n = 10\n", "q = Array{Fun}(undef,n)\n", "p = Array{Fun}(undef,n)\n", "p[1] = Fun(1, -1 .. 1 )\n", "q[1] = p[1]/nrm(p[1])\n", "\n", "for k=1:n-1\n", " p[k+1] = x*q[k] \n", " for j=1:k\n", " p[k+1] -= ip(p[k+1],q[j])*q[j]\n", " end\n", " q[k+1] = p[k+1]/nrm(p[k+1])\n", "end" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.338624086051233e-16" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sum(q[2]*q[4]*w)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-1.0\n", "\n", "\n", "-0.5\n", "\n", "\n", "0.0\n", "\n", "\n", "0.5\n", "\n", "\n", "1.0\n", "\n", "\n", "-4\n", "\n", "\n", "-2\n", "\n", "\n", "0\n", "\n", "\n", "2\n", "\n", "\n", "4\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p = plot(; legend=false)\n", "for k=1:10\n", " plot!(q[k])\n", "end\n", "p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "## Function approximation with orthogonal polynomials\n", "\n", "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\n", "$$\n", "f(x) = \\sum_{k=0}^{n-1} f_k p_k(x)\n", "$$\n", "where\n", "$$\n", "f_k = {\\ip \\over \\ip}\n", "$$\n", "\n", "Here, we demonstrate this with Chebyshev polynomials:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(f₀, f₁, f₂, f₃) = (1.5, 1.7499999999999998, 0.5, 0.25000000000000006)\n", "f₀ * 1 + f₁ * x + f₂ * cos(2 * acos(x)) + f₃ * cos(3 * acos(x)) = 1.111\n", "1 + x + x ^ 2 + x ^ 3 = 1.111\n" ] } ], "source": [ "f = Fun(x -> 1 + x + x^2 + x^3, Chebyshev())\n", "f₀, f₁, f₂, f₃ = f.coefficients\n", "@show f₀, f₁, f₂, f₃\n", "\n", "x = 0.1\n", "@show f₀*1 + f₁*x + f₂*cos(2acos(x)) + f₃*cos(3acos(x))\n", "@show 1 + x + x^2 + x^3;" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-3\n", "\n", "\n", "-2\n", "\n", "\n", "-1\n", "\n", "\n", "0\n", "\n", "\n", "1\n", "\n", "\n", "2\n", "\n", "\n", "3\n", "\n", "\n", "0.5\n", "\n", "\n", "1.0\n", "\n", "\n", "1.5\n", "\n", "\n", "2.0\n", "\n", "\n", "2.5\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "y1\n", "\n", "\n", "\n", "y2\n", "\n", "\n" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(Fun(exp))\n", "plot!(Fun(t-> exp(cos(t)), -pi .. pi))" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.5" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = Fun()\n", "ip = (f,g) -> sum(f*g/sqrt(1-x^2))\n", "\n", "T₂ = cos.(2 .* acos.(x))\n", "ip(T₂,f)/ip(T₂,T₂) # gives back f₂" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Of course, if $p_k$ are othernormal than we don't need the denominator.\n", "\n", "This can be extended to function approximation Provided the sum converges absolutely and uniformly in $x$, we can write\n", "$$\n", "f(x) = \\sum_{k=0}^\\infty f_k p_k(x).\n", "$$\n", "In practice, we can approximate smooth functions by a finite truncation:\n", "$$\n", "f(x) \\approx f_n(x) = \\sum_{k=0}^{n-1} f_k p_k(x)\n", "$$\n", "\n", "Here we see that $\\E^x$ can be approximated by a Chebyshev approximation using 14 coefficients and is accurate to 16 digits:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "f.coefficients = [1.26607, 1.13032, 0.271495, 0.0443368, 0.00547424, 0.000542926, 4.49773e-5, 3.19844e-6, 1.99212e-7, 1.10368e-8, 5.5059e-10, 2.49796e-11, 1.03911e-12, 3.98969e-14]\n", "ncoefficients(f) = 14\n", "f(0.1) = 1.1051709180756477\n", "exp(0.1) = 1.1051709180756477\n" ] } ], "source": [ "f = Fun(x -> exp(x), Chebyshev())\n", "@show f.coefficients\n", "@show ncoefficients(f)\n", "\n", "@show f(0.1) # equivalent to f.coefficients'*[cos(k*acos(x)) for k=0:ncoefficients(f)-1]\n", "@show exp(0.1);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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.\n", "\n", "\n", "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]$:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "ncoefficients(f) = 189\n" ] }, { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-1.0\n", "\n", "\n", "-0.5\n", "\n", "\n", "0.0\n", "\n", "\n", "0.5\n", "\n", "\n", "1.0\n", "\n", "\n", "0.2\n", "\n", "\n", "0.4\n", "\n", "\n", "0.6\n", "\n", "\n", "0.8\n", "\n", "\n", "1.0\n", "\n", "\n", "\n", "\n", "\n", "\n", "y1\n", "\n", "\n" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f = Fun( x -> 1/(25x^2 + 1), Chebyshev())\n", "@show ncoefficients(f)\n", "plot(f)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This can be explained for Chebyshev expansion by noting that it is cosine expansion / Fourier expansion of an even function:\n", "$$\n", "f(x) = \\sum_{k=0}^\\infty f_k T_k(x) \\Leftrightarrow f(\\cos \\theta) = \\sum_{k=0}^\\infty f_k \\cos k \\theta\n", "$$\n", "From Lecture 4 we saw that Fourier coefficients decay exponentially (thence the approximation converges exponentially fast) uf $f\\left({z + z^{-1} \\over 2}\\right)$ is analytic in an ellipse. In the case of $f(x) = {1 \\over 25 x^2 + 1}$, we find that \n", "$$\n", "f(z) = {4 z^2 \\over 25 + 54 z^2 + 25 z^4}\n", "$$\n", "which has poles at \n", "$\n", "\\pm 0.8198040\\I,\\pm1.2198\\I:\n", "$" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "400×400 Array{RGB{Float64},2} with eltype RGB{Float64}:\n", " RGB{Float64}(0.617696,1.0,0.0) … RGB{Float64}(0.617696,0.0,1.0)\n", " RGB{Float64}(0.617696,1.0,0.0) RGB{Float64}(0.617696,0.0,1.0)\n", " RGB{Float64}(0.627713,1.0,0.0) RGB{Float64}(0.627713,0.0,1.0)\n", " RGB{Float64}(0.63773,1.0,0.0) RGB{Float64}(0.63773,0.0,1.0) \n", " RGB{Float64}(0.63773,1.0,0.0) RGB{Float64}(0.63773,0.0,1.0) \n", " RGB{Float64}(0.647746,1.0,0.0) … RGB{Float64}(0.647746,0.0,1.0)\n", " RGB{Float64}(0.647746,1.0,0.0) RGB{Float64}(0.647746,0.0,1.0)\n", " RGB{Float64}(0.657763,1.0,0.0) RGB{Float64}(0.657763,0.0,1.0)\n", " RGB{Float64}(0.657763,1.0,0.0) RGB{Float64}(0.657763,0.0,1.0)\n", " RGB{Float64}(0.66778,1.0,0.0) RGB{Float64}(0.66778,0.0,1.0) \n", " RGB{Float64}(0.66778,1.0,0.0) … RGB{Float64}(0.66778,0.0,1.0) \n", " RGB{Float64}(0.677796,1.0,0.0) RGB{Float64}(0.677796,0.0,1.0)\n", " RGB{Float64}(0.677796,1.0,0.0) RGB{Float64}(0.677796,0.0,1.0)\n", " ⋮ ⋱ \n", " RGB{Float64}(0.677796,0.0,1.0) RGB{Float64}(0.677796,1.0,0.0)\n", " RGB{Float64}(0.66778,0.0,1.0) RGB{Float64}(0.66778,1.0,0.0) \n", " RGB{Float64}(0.66778,0.0,1.0) … RGB{Float64}(0.66778,1.0,0.0) \n", " RGB{Float64}(0.657763,0.0,1.0) RGB{Float64}(0.657763,1.0,0.0)\n", " RGB{Float64}(0.657763,0.0,1.0) RGB{Float64}(0.657763,1.0,0.0)\n", " RGB{Float64}(0.647746,0.0,1.0) RGB{Float64}(0.647746,1.0,0.0)\n", " RGB{Float64}(0.647746,0.0,1.0) RGB{Float64}(0.647746,1.0,0.0)\n", " RGB{Float64}(0.63773,0.0,1.0) … RGB{Float64}(0.63773,1.0,0.0) \n", " RGB{Float64}(0.63773,0.0,1.0) RGB{Float64}(0.63773,1.0,0.0) \n", " RGB{Float64}(0.627713,0.0,1.0) RGB{Float64}(0.627713,1.0,0.0)\n", " RGB{Float64}(0.617696,0.0,1.0) RGB{Float64}(0.617696,1.0,0.0)\n", " RGB{Float64}(0.617696,0.0,1.0) RGB{Float64}(0.617696,1.0,0.0)" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f = x -> 1/(25x^2 + 1)\n", "portrait(-3..3, -3..3, z -> f((z+1/z)/2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hence we predict a rate of decay of about $1.2198^{-k}$:" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "0\n", "\n", "\n", "50\n", "\n", "\n", "100\n", "\n", "\n", "150\n", "\n", "\n", "10\n", "\n", "\n", "-\n", "\n", "\n", "40 \n", "\n", "\n", "10\n", "\n", "\n", "-\n", "\n", "\n", "30 \n", "\n", "\n", "10\n", "\n", "\n", "-\n", "\n", "\n", "20 \n", "\n", "\n", "10\n", "\n", "\n", "-\n", "\n", "\n", "10 \n", "\n", "\n", "10\n", "\n", "\n", "0 \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "Chebyshev coefficients\n", "\n", "\n", "\n", "R^(-k)\n", "\n", "\n" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f = Fun( x -> 1/(25x^2 + 1), Chebyshev())\n", "plot(abs.(f.coefficients) .+ 1E-40; yscale=:log10, label=\"Chebyshev coefficients\")\n", "plot!( 1.2198.^(-(0:ncoefficients(f))); label=\"R^(-k)\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "### Weighted approximation\n", "\n", "Sometimes, we want to incorporate the weight into the approximation. This is typically one of two forms, depending on the application: \n", "$$\n", "f(x) = w(x) \\sum_{k=0}^\\infty f_k p_k(x)\n", "$$\n", "or\n", "$$\n", " f(x) = \\sqrt{w(x)} \\sum_{k=0}^\\infty f_k p_k(x)\n", "$$\n", "\n", "This is often the case with Hermite polynomials: on the real line polynomial approximation is unnatural unless the function approximated is a polynomial as otherwise the behaviour at ∞ is inconsistent, so what we really want is weighted approximation. Thus we can either use\n", "$$\n", "f(x) = \\E^{-x^2}\\sum_{k=0}^\\infty f_k H_k(x)\n", "$$\n", "or\n", "$$\n", "f(x) = \\E^{-x^2/2}\\sum_{k=0}^\\infty f_k H_k(x)\n", "$$\n", "Depending on your problem, getting this wrong can be disasterous:" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1.109999999999997" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f = Fun(x -> 1+x +x^2, Hermite())\n", "f(0.10)" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-6\n", "\n", "\n", "-3\n", "\n", "\n", "0\n", "\n", "\n", "3\n", "\n", "\n", "6\n", "\n", "\n", "-10\n", "\n", "\n", "-5\n", "\n", "\n", "0\n", "\n", "\n", "5\n", "\n", "\n", "10\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "sech x\n", "\n", "\n", "\n", "f\n", "\n", "\n" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# nonsense trying to approximating sech(x) by a degree 50 polynomial:\n", "f = Fun(x -> sech(x), Hermite(), 51)\n", "xx = -8:0.01:8\n", "plot(xx, sech.(xx); ylims=(-10,10), label=\"sech x\")\n", "plot!(xx, f.(xx); label=\"f\")" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-6\n", "\n", "\n", "-3\n", "\n", "\n", "0\n", "\n", "\n", "3\n", "\n", "\n", "6\n", "\n", "\n", "-10\n", "\n", "\n", "-5\n", "\n", "\n", "0\n", "\n", "\n", "5\n", "\n", "\n", "10\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "sech x\n", "\n", "\n", "\n", "f\n", "\n", "\n" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# weighted by works sqrt(w(x)) = exp(-x^2/2)\n", "f = Fun(x -> sech(x), GaussWeight(Hermite(),1/2),101)\n", "\n", "plot(xx, sech.(xx); ylims=(-10,10), label=\"sech x\")\n", "plot!(xx, f.(xx); label=\"f\")" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-6\n", "\n", "\n", "-3\n", "\n", "\n", "0\n", "\n", "\n", "3\n", "\n", "\n", "6\n", "\n", "\n", "-\n", "\n", "\n", "5.0×10\n", "\n", "\n", "30\n", "\n", "\n", "-\n", "\n", "\n", "2.5×10\n", "\n", "\n", "30\n", "\n", "\n", "0\n", "\n", "\n", "2.5×10\n", "\n", "\n", "30\n", "\n", "\n", "5.0×10\n", "\n", "\n", "30\n", "\n", "\n", "\n", "\n", "\n", "\n", "f\n", "\n", "\n" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# weighted by w(x) = exp(-x^2) breaks again\n", "f = Fun(x -> sech(x), GaussWeight(Hermite()),101)\n", "\n", "plot(xx, sech.(xx); ylims=(-10,10), label=\"sech x\")\n", "plot(xx, f.(xx); label=\"f\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that correctly weighted Hermite, that is, with $\\sqrt{w(x)} = \\E^{-x^2/2}$ look \"nice\":" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-6\n", "\n", "\n", "-3\n", "\n", "\n", "0\n", "\n", "\n", "3\n", "\n", "\n", "6\n", "\n", "\n", "-100\n", "\n", "\n", "-50\n", "\n", "\n", "0\n", "\n", "\n", "50\n", "\n", "\n", "100\n", "\n", "\n", "150\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "H_0\n", "\n", "\n", "\n", "H_1\n", "\n", "\n", "\n", "H_2\n", "\n", "\n", "\n", "H_3\n", "\n", "\n", "\n", "H_4\n", "\n", "\n", "\n", "H_5\n", "\n", "\n", "\n", "H_6\n", "\n", "\n" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p = plot()\n", "for k=0:6\n", " H_k = Fun(GaussWeight(Hermite(),1/2),[zeros(k);1])\n", " plot!(xx, H_k.(xx); label=\"H_$k\")\n", "end\n", "p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Compare this to weighting by $w(x) = \\E^{-x^2}$:" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "-6\n", "\n", "\n", "-3\n", "\n", "\n", "0\n", "\n", "\n", "3\n", "\n", "\n", "6\n", "\n", "\n", "-100\n", "\n", "\n", "-50\n", "\n", "\n", "0\n", "\n", "\n", "50\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "H_0\n", "\n", "\n", "\n", "H_1\n", "\n", "\n", "\n", "H_2\n", "\n", "\n", "\n", "H_3\n", "\n", "\n", "\n", "H_4\n", "\n", "\n", "\n", "H_5\n", "\n", "\n", "\n", "H_6\n", "\n", "\n" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p = plot()\n", "for k=0:6\n", " H_k = Fun(GaussWeight(Hermite()),[zeros(k);1])\n", " plot!(xx, H_k.(xx); label=\"H_$k\")\n", "end\n", "p" ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.0.2", "language": "julia", "name": "julia-1.0" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.0.2" } }, "nbformat": 4, "nbformat_minor": 2 }