{ "cells": [ { "cell_type": "markdown", "id": "500b0418", "metadata": {}, "source": [ "---\n", "format:\n", " html:\n", " other-links:\n", " - text: This notebook\n", " href: L9 Polynomial Interpolation II.ipynb\n", "---" ] }, { "cell_type": "markdown", "id": "aad6bc09-0de2-4531-9043-0bacc8e41409", "metadata": {}, "source": [ "# Polynomial Interpolation II" ] }, { "cell_type": "markdown", "id": "c79162dc", "metadata": {}, "source": [ "
Note. These notes are mainly a record of what we discussed and are not a substitute for attending the lectures and reading books! If anything is unclear/wrong, let me know and I will update the notes.\n", "
" ] }, { "cell_type": "code", "execution_count": 8, "id": "76076758", "metadata": { "jupyter": { "source_hidden": true } }, "outputs": [ { "data": { "image/png": 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label=L\"y = p(x)\", lw = 3 )\n", " scatter!(plt1, [x], [y], primary = false, markersize = 5)" ] }, { "cell_type": "markdown", "id": "cd95d7b1", "metadata": {}, "source": [ "## Previously.....\n", "\n", "### Lagrange Interpolation Problem:\n", "\n", "Suppose that $X = \\{ x_0 < \\cdots < x_n \\}$ is a set of distinct *interpolation nodes*. \n", "\n", "***Aim***: find $p$ such that $p(x_j) = f_j$ for all $j = 0, \\dots, n$. \n", "\n", "* $f_j$ - data points, \n", "* $f_j = f(x_j)$" ] }, { "cell_type": "markdown", "id": "17fed646", "metadata": {}, "source": [ "
Theorem (Lagrange Interpolation) \n", "\n", "For $x_0 < \\cdots < x_n$ (*nodes*) and a function $f$ defined on $X = \\{x_0,\\dots,x_n\\}$, there exists a unique polynomial *interpolant* $I_Xf$ of degree at most $n$ such that $I_Xf(x_j) = f(x_j)$ for all $j=0,\\dots,n$. \n", "\n", "
\n", "\n", "In the following, we will write $\\mathcal P_n := \\{p(x) = \\sum_{j=0}^n a_j x^j \\colon a_0, \\dots,a_n \\in \\mathbb R\\}$ for the set of polynomials of degree at most $n$." ] }, { "cell_type": "code", "execution_count": 2, "id": "a1ac1132", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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Theorem. \n", "\n", "Suppose $f:[a,b]\\to\\mathbb R$ is $n+1$ times continuously differentiable and $X = \\{x_0 < \\dots < x_n\\}$. Then, for all $x \\in [a,b]$ there exists $\\xi_x \\in [\\min \\{x,x_0\\} , \\max \\{x,x_n\\}]$ such that \n", "\n", "\\begin{align}\n", " f(x) - I_Xf(x) = \\frac{f^{(n+1)}(\\xi_x)}{(n+1)!} (x - x_0)(x-x_1) \\cdots (x-x_n)\n", "\\end{align}\n", "\n", "In particular, we have \n", "\n", "\\begin{align}\n", " \\left\\| f - I_Xf \\right\\|_{L^\\infty([a,b])} \\leq \\|f^{(n+1)}\\|_{L^\\infty([a,b])} \\frac{\\|\\ell_X(x)\\|_{L^\\infty([a,b])}}{(n+1)!} \n", "\\end{align}\n", "\n", "where $\\ell_X(x) := (x-x_0)(x-x_1)\\cdots (x-x_n)$ is the *node polynomial* and $\\| f \\|_{L^\\infty([a,b])} := \\max_{x\\in[a,b]} |f(x)|$.\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "6f516779", "metadata": {}, "source": [ "
Remark. \n", "\n", "Compare with Taylor remainder\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "0db9afcf", "metadata": {}, "source": [ "
Example. \n", "\n", "Find the polynomial interpolation of $e^{-x}$ on $\\{-1, 0, 1\\}$.\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "668b9335", "metadata": {}, "source": [ "For $x \\in [-1,1]$, we have \n", "\\begin{align}\n", " |\\ell_X(x)| &\\leq \\frac{e}{2\\sqrt{n}} \\left( \\frac2e \\right)^n\n", " &\\text{for Equispaced nodes } X = \\Big\\{ \\frac{2j-n}{n} \\Big\\}_{j=0}^n \\\\\n", " |\\ell_X(x)| &\\leq 2^{1-n}\n", " &\\text{for Chebyshev nodes } X = \\Big\\{ \\cos \\frac{j\\pi}{n} \\Big\\}_{j=0}^n\n", "\\end{align}" ] }, { "cell_type": "markdown", "id": "cb125e3f", "metadata": {}, "source": [ "## Equispaced Nodes" ] }, { "cell_type": "code", "execution_count": 3, "id": "6093e418", "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "\u001b[36m\u001b[1m[ \u001b[22m\u001b[39m\u001b[36m\u001b[1mInfo: \u001b[22m\u001b[39mSaved animation to c:\\Users\\math5\\Math 5485\\Pictures\\Runge.gif\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Pictures\\\\Runge.gif\")" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 20;\n", "\n", "anim = @animate for n ∈ 2:N\n", " x = collect( @. ( 2*(0:1:n) - n )/n )\n", " y = @. f₃( x )\n", "\n", " p = fit(ChebyshevT, x, y)\n", "\n", " plt1 = plot(f₃, -1, 1, label=L\"y = f(x)\", lw = 3, linestyle = :dash, title = \"Runge phenomenon\")\n", " plot!(plt1, p, -1, 1, label=L\"y = p(x)\", lw = 3 )\n", " scatter!(plt1, [x], [y], primary = false, ylims=(-1,2), markersize = 5)\n", "\n", " ℓ = fromroots( x )\n", " plt2 = plot(ℓ, -1, 1, title=L\"\\ell_X(x)\", legend = false, lw = 3)\n", " scatter!(plt2, [x], [zeros(n)], primary = false, markersize = 5)\n", "\n", " plot( plt1, plt2, size=(1000, 500))\n", "\n", "end\n", "\n", "gif(anim, \"Pictures/Runge.gif\", fps = 1)" ] }, { "cell_type": "markdown", "id": "db847af5", "metadata": {}, "source": [ "## Chebyshev Nodes" ] }, { "cell_type": "code", "execution_count": 4, "id": "65f11d39", "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "\u001b[36m\u001b[1m[ \u001b[22m\u001b[39m\u001b[36m\u001b[1mInfo: \u001b[22m\u001b[39mSaved animation to c:\\Users\\math5\\Math 5485\\Pictures\\Chebyshev.gif\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Pictures\\\\Chebyshev.gif\")" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 25\n", "\n", "err = zeros(N)\n", "egrid = -1:.01:1\n", "\n", "anim = @animate for n ∈ 2:N\n", " x = @. cos( π*(0:n)/n )\n", " y = @. f₃( x )\n", "\n", " p = fit(ChebyshevT, x, y)\n", "\n", " plt1 = plot(f₃, -1, 1, label=L\"y = f(x)\", lw = 3, linestyle = :dash, title = \"Chebyshev Interpolation\")\n", " plot!(plt1, p, -1, 1, label=L\"y = p(x)\", lw = 3 )\n", " scatter!(plt1, [x], [y], primary = false, ylims=(-.1,1.1), markersize = 5)\n", "\n", " ℓ = fromroots( x )\n", " plt2 = plot(ℓ, -1, 1, title=L\"\\ell_X(x)\", legend = false, lw = 3)\n", " scatter!(plt2, [x], [zeros(n)], primary = false, markersize = 5)\n", "\n", " hline!(plt2, [2.0^(1-n), -2.0^(1-n)], linestyle=:dash )\n", "\n", " plot( plt1, plt2, size=(1000, 500))\n", "\n", " err[n] = maximum( @. abs( p(egrid) - f₃(egrid) ) )\n", "\n", "end\n", "\n", "gif(anim, \"Pictures/Chebyshev.gif\", fps = 1)" ] }, { "cell_type": "markdown", "id": "630dd6ea", "metadata": {}, "source": [ "## How to choose $X$ to minimise $\\| \\ell_X \\|_{L^\\infty([-1,1])}$?\n", "\n", "In this section, we will write $\\| f \\|_{L^\\infty} := \\max_{x \\in [-1,1]} |f(x)|$. \n", "\n", "Recall that \n", "\n", "\\begin{align}\n", " \\left\\| f - I_Xf \\right\\|_{L^\\infty} \\leq \\frac{\\|f^{(n+1)}\\|_{L^\\infty}}{(n+1)!} \\|\\ell_X\\|_{L^\\infty} \n", "\\end{align}\n", "\n", "so it is natural to want to minimise \n", "\n", "\\begin{align}\n", " \\|\\ell_X\\|_{L^\\infty} = \\max_{x \\in [-1,1]} |(x-x_0)(x-x_1)\\dots(x-x_n)|\n", "\\end{align}\n", "\n", "over all choices of $X = \\{x_0,\\dots,x_n\\}$. Equivalently, we are minimising $\\| p \\|_{L^\\infty}$ over all *monic* polynomials $p$ of degree $n+1$: that is, over all $p$ such that $p(x) = x^{n+1} + q(x)$ with $q\\in \\mathcal P_n$. We will show that scaled Chebyshev polynomials (of the first kind) solve this (so-called *Chebyshev*) problem.\n", "\n", "### A Very Brief Introduction to Chebyshev Polynomials\n", "\n", "For $x \\in [-1,1]$ there exists $\\theta \\in [0,\\pi]$ such that $x = \\cos\\theta$. We can therefore define $T_n$ on $[-1,1]$ by the condition that\n", "\n", "\\begin{align}\n", "T_n( \\cos \\theta ) = \\cos n\\theta.\n", "\\end{align}\n", "\n", "for $n = 0,1,2,...$.\n", "\n", "We will use just some of the many interesting properties of $T_n$:\n", "\n", "
Theorem (Chebyshev Polynomials) \n", "\n", "We have the following properties:\n", "\n", "* $T_0(x) = 1$, and $T_1(x) = x$,\n", "* $T_{n+1}(x) = 2x T_{n}(x) - T_{n-1}(x)$ for $n =1,2,\\dots$,\n", "* $T_n$ is thus a polynomial of degree $n$,\n", "* The coefficent of $x^n$ in $T_n$ is $2^{n-1}$ for $n \\geq 1$,\n", "* $\\|T_n\\|_{L^\\infty([-1,1])} = 1$ and $T_{n}\\big( \\cos\\tfrac{j\\pi}{n} \\big) = (-1)^j$ for $j = 0,\\dots,n$,\n", "* $T_{n+1} = 0$ on $\\big\\{ \\cos \\frac{(2j+1)\\pi}{2(n+1)} \\big\\}_{j=0}^{n}$.\n", "\n", "
\n", "\n", "We define $X_{\\mathrm{I}}$ to be the set of $n+1$ zeros of $T_{n+1}$ and $X_{\\mathrm{II}}$ to be the set of $n+1$ extreme points of $T_n$.\n", "\n", "\\begin{align}\n", " X_{\\mathrm I} &= \\Big\\{ \\cos \\frac{(2j+1)\\pi}{2(n+1)} \\Big\\}_{j=0}^{n} \\nonumber\\\\\n", " X_{\\mathrm{II}} &= \\big\\{ \\cos\\tfrac{j\\pi}{n} \\big\\}_{j=0}^n\n", "\\end{align}\n", "\n", "$X_{\\mathrm{I}}$ and $X_{\\mathrm{II}}$ are the *Chebyshev nodes of the first and second kind*, respectively.\n", "\n", "
Proof. \n", "\n", "Firstly, $T_0(\\cos\\theta) =\\cos 0 = 1$ and so $T_0 = 1$. Moreover, $T_{1}(\\cos \\theta) = \\cos\\theta$ and so $T_1 = x$. \n", "\n", "Using the fact that $\\cos(x+y) = \\cos x \\cos y - \\sin x \\sin y$ (which follows from e.g.~$e^{i(x+y)} = e^{ix} e^{iy}$), we find that \n", "\n", "\\begin{align}\n", " T_{n\\pm1}(\\cos\\theta) &= \\cos( n\\theta \\pm \\theta) \\nonumber\\\\\n", " %\n", " &= \\cos \\theta \\cos n\\theta \\mp \\sin\\theta \\sin n\\theta.\n", "\\end{align}\n", "\n", "The sum of these equations is therefore $T_{n+1}(\\cos\\theta) + T_{n-1}(\\cos\\theta) = 2 \\cos \\theta \\cos n\\theta = 2 \\cos(\\theta) T_n(\\cos\\theta)$ and thus the recursion relation follows.\n", "\n", "One can therefore show inductively that $T_n$ is a polynomial of degree $n$. Using the recursion, the coeficient of $x^{n+1}$ in $T_{n+1}$ is $2$ times the coefficient of $x^n$ in $T_{n}$. You can therefore argue inductively that the coeficient of $x^n$ in $T_{n}$ is $2^{n-1}$.\n", "\n", "We have $|T_n(\\cos\\theta)| = |\\cos n\\theta| \\leq 1$ and this maximum is achieved when $n\\theta = j\\pi$ for some $j \\in \\mathbb Z$. Restricting this to $\\theta = \\frac{j\\pi}{n} \\in [0,\\pi]$ gives $j = 0,\\dots,n$. At these values, we have $T_n( \\cos\\tfrac{j\\pi}{n} ) = \\cos j\\pi = (-1)^j$.\n", "\n", "$T_{n+1}$ is a polynomial of degree $n+1$ and thus has at most $n+1$ zeros in the interval $[-1,1]$. We have $T_{n+1}(\\cos\\theta) = \\cos (n+1)\\theta = 0$ when $(n+1)\\theta = \\frac{\\pi}{2} + j \\pi$ for $j \\in \\mathbb Z$ such that $\\theta = \\frac{ (2j+1) \\pi }{ 2(n+1) } \\in [0,\\pi]$. That is, for $j = 0,\\dots,n$.\n", "\n", "
\n", "\n", "Here, we plot the first $5$ Chebyshev polynomials on $[-1,1]$ together with $(x, T_n(x))$ for $x \\in X_{\\mathrm{II}}$ with $n=5$" ] }, { "cell_type": "code", "execution_count": 28, "id": "53454b29", "metadata": {}, "outputs": [ { "data": { "image/png": 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= 3)\n", "\n", "scatter!( @. cos( pi*(0:5)/5 ), @. p( cos( pi* (0:5)/5 ) ) )" ] }, { "cell_type": "markdown", "id": "077b4b28", "metadata": {}, "source": [ "We define the *monic* (with leading coefficient equal to $1$) Chebyshev polynomials as $t_0(x) = 1$ and $t_n(x) := 2^{1-n} T_n(x)$ for $n \\geq 1$. We are ready for the main result of this section (\"Which nodes to choose?\"):\n", "\n", "
Theorem. \n", "\n", "The monic Chebyshev polynomials solve the Chebyshev problem (on $[-1,1]$). That is, \n", "\n", "\\begin{align}\n", " \\|t_{n+1} \\|_{L^\\infty([-1,1])} \\leq \\min_{p = x^{n+1} + q : \\,\\, q \\in \\mathcal P_n } \\big\\| p \\big\\|_{L^\\infty([-1,1])}.\n", "\\end{align}\n", "\n", "
\n", "\n", "
Proof. \n", "\n", "Here, we just write $\\|\\,\\cdot\\,\\|_{L^\\infty}$ for $\\|\\,\\cdot\\,\\|_{L^\\infty([-1,1])}$.\n", "\n", "Suppose that $p$ minimises the right hand side of (1). Then, because $t_{n+1}$ is a monic polynomial of degree $n+1$, we must have \n", "\n", "\\begin{align*}\n", " \\big\\|p \\big\\| \\leq \\| t_{n+1} \\|_{L^\\infty} = 2^{1-(n+1)} \\| T_{n+1} \\|_{L^\\infty} = 2^{-n}\n", "\\end{align*}\n", "\n", "Define $r(x) = t_{n+1}(x) - p(x)$ and note that \n", "\n", "\\begin{align}\n", " r( z_j ) = (-1)^j 2^{-n} - p( z_j ), \\qquad \\text{for } z_j := \\cos \\frac{j\\pi}{n+1}\n", "\\end{align}\n", "\n", "and $j = 0,1,\\dots,n+1$ (here, we have used the fact that $z_j$ are the extreme values of $T_{n+1}$ for $j=0,\\dots,n+1$). \n", "\n", "Therefore, since $|p(x)| \\leq 2^{-n}$ for all $x \\in [-1,1]$, we have\n", "\n", "\\begin{align}\n", " r( z_j ) &= 2^{-n} - p( z_j ) \\geq 0 \\qquad &\\text{for even } j \\\\\n", " r( z_j ) &= (-1) 2^{-n} - p( z_j ) \\leq 0 \\qquad &\\text{for odd } j.\n", "\\end{align}\n", "\n", "By the change of sign theorem, there exists $x_1,\\dots,x_{n+1}$ such that $z_j \\leq x_{j+1} \\leq z_{j+1}$ and $r(x_j) = 0$. As a result, $r$ is a polynomial of degree at most $n$ with $n+1$ zeros and thus $r = 0$ and $p = t_{n+1}$. \n", "\n", "
\n", "\n", "We have therefore shown that, choosing $X =X_{\\mathrm{I}}= \\{ \\text{zeros of }t_{n+1} \\}$ minimises the node polynomial in the following sense\n", "\n", "\\begin{align}\n", " \\min_{ y_0 < \\dots < y_n } \\| \\ell_{Y} \\|_{L^\\infty([-1,1])} \\geq \\| \\ell_X \\|_{L^\\infty([-1,1])} = 2^{-n}\n", "\\end{align} \n", "\n", "It turns out that $\\ell_{X_{\\mathrm{II}}}(x) = 2^{-n} \\big( T_{n+1}(x) - T_{n-1}(x) \\big)$ and so $\\|\\ell_{X_{\\mathrm{II}}}\\|_{L^\\infty([-1,1])} \\leq 2^{1-n}$ (here, we used $|T_{n}|\\leq 1$) and so, in practice, one expects the same approximation rates when using $X = X_{\\mathrm{II}} = \\{\\text{extreme points of } T_n \\}$. " ] }, { "cell_type": "markdown", "id": "1c180a4a", "metadata": {}, "source": [ "## Barycentric formula for Chebyshev\n", "\n", "Using the standard Lagrange formulation $p(x) = \\sum_{j=0}^n \\ell_j(x) f(x_j)$ is slow and unstable! Computing $\\ell_j(x)$ for a single $x$ requires $O(n)$ operations and thus evaluating $p(x)$ requires $O(n^2)$. Moreover, due to subtractive cancellation, this method is unstable." ] }, { "cell_type": "code", "execution_count": 35, "id": "81c6786c", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " 5.944383 seconds (25.78 M allocations: 15.129 GiB, 22.59% gc time, 2.50% compilation time)\n" ] }, { "data": { "image/png": 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"\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "n = 1000\n", "\n", "x = @. cos( π*(0:n)/n )\n", "y = @. f₃( x )\n", "\n", "function L( j, t ) \n", " if j == 0\n", " return prod( @. t - x[2:end] ) / prod( @. x[1] - x[2:end] )\n", " elseif j == length(x)-1 \n", " return prod( @. t - x[1:end-1] ) / prod( @. x[end] - x[1:end-1] )\n", " else\n", " return prod( @. t - x[1:j] ) * prod( @. t - x[(j+2):end] ) / ( prod( @. x[j+1] - x[1:j] ) * prod( @. x[j+1] - x[(j+2):end] ) )\n", " end\n", "end \n", "\n", "function p_naive( t ) \n", " r = 0\n", " for j in 0:(length(x)-1)\n", " r = r + L(j, t ) * y[j+1] \n", " end\n", " return r\n", "end \n", "\n", "plot(f₃, -1, 1, label=L\"y = f(x)\", lw = 3, linestyle = :dash, legend=false)\n", "@time plot!( t -> p_naive( t ), -1, 1, title=\"Naive implementation: slow/unstable\", lw = 3 )\n", "scatter!( [x], [f₃.(x)])" ] }, { "cell_type": "markdown", "id": "d8bf0f17", "metadata": {}, "source": [ "Instead, we may use the Barycentric formula: First notice that \n", "\n", "\\begin{align}\n", " \\ell_j(x) = \\prod_{k \\not= j} \\frac{x - x_k}{x_j - x_k} = \\frac{\\ell_X(x)}{x- x_j} \\frac{1}{ \\prod_{k \\not= j} (x_j - x_k) }\n", "\\end{align}\n", "\n", "Therefore on defining $\\lambda_j := \\left( \\prod_{k \\not= j} (x_j - x_k) \\right)^{-1}$ and dividing by $\\sum_{j=0}^n \\ell_j(x) = 1$, we obtain\n", "\n", "\\begin{align}\n", " p(x) &= \\left. \\sum_{j=0}^n \\ell_j(x) f(x_j) \\middle/ \\sum_{j=0}^n \\ell_j(x) \\right. \\nonumber\\\\\n", " &= \\left. \\ell_X(x) \\sum_{j=0}^n \\frac{\\lambda_j f(x_j)}{x - x_j} \\middle/ \\ell_X(x) \\sum_{j=0}^n \\frac{\\lambda_j}{x - x_j} \\right. \\nonumber\\\\\n", " &= \\left. \\sum_{j=0}^n \\frac{\\lambda_j f(x_j)}{x - x_j} \\middle/ \\sum_{j=0}^n \\frac{\\lambda_j}{x - x_j} \\right.\n", "\\end{align}\n", "\n", "which is known as the Barycentric Formula. \n", "\n", "For a fixed interpolation set, one may compute $\\lambda_j$ once and then evaluating $p(x)$ requires only $O(n)$ computations. \n", "\n", "In fact, for the Chebyshev nodes $X_{\\mathrm{II}} = \\big\\{ \\cos \\frac{j\\pi}{n} \\big\\}_{j=0}^n$, we have a simple formula for the $\\lambda_j$:\n", "\n", "\\begin{align}\n", " \\lambda_j = \\begin{cases}\n", " \\frac{2^{n}}{4n} & \\text{if } j = 0\\nonumber\\\\\n", " (-1)^j \\frac{2^{n}}{2n} & \\text{if } j = 1,\\dots,n-1 \\nonumber\\\\\n", " (-1)^n \\frac{2^{n}}{4n} & \\text{if } j = n\n", " \\end{cases}\n", "\\end{align}\n", "\n", "Moreover, this formula has been proved to be numerically stable which we demonstrate in the following:" ] }, { "cell_type": "code", "execution_count": 42, "id": "23b11076", "metadata": {}, "outputs": [ { "data": { "image/png": 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"\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "n = 1000\n", "x = @. cos( π*(0:n)/n )\n", "y = @. f₃( x )\n", "\n", "λ = zeros(n+1)\n", "λ[1] = 1/2\n", "λ[n+1] = (-1)^n/2\n", "for j in 1:(n-1)\n", " λ[j+1] = (-1)^j\n", "end\n", "\n", "function p_bary( t )\n", " return sum( @. λ * y / (t - x) ) / sum( @. λ / (t - x) )\n", "end\n", "\n", "plot(f₃, -1, 1, label=L\"y = f(x)\", lw = 3, linestyle = :dash, title = \"Barycentric formula\")\n", "plot!(p_bary, -1, 1, label=L\"y = p(x)\", lw = 3 )\n", "scatter!( [x], [f₃.(x)], primary = false)" ] }, { "cell_type": "markdown", "id": "be1d7631", "metadata": {}, "source": [ "## Newton Form & Divided Differences\n", "\n", "Next time..." ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.11.6", "language": "julia", "name": "julia-1.11" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.11.6" } }, "nbformat": 4, "nbformat_minor": 5 }