{ "cells": [ {"cell_type":"markdown","source":"

Questions to be handed in for Taylor Polynomials

","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

We begin by loading packagse for plotting, automatic differentiation, and symbolic manipulations:

","metadata":{}}, {"outputs":[],"cell_type":"code","source":["using Plots\nusing Roots\nusing ForwardDiff\nD(f, k=1) = k > 1 ? D(D(f),k-1) : x -> ForwardDiff.derivative(f, float(x))\nusing QuadGK\nusing SymPy"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Quick background

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

A Taylor polynomial of a function $f(x)$ about $x=c$ of degree $n$ is formally defined by

","metadata":{}}, {"cell_type":"markdown","source":"\n$$\nT_n(x) = f(c) + \\frac{f'(c)}{1!}(x-c) \\frac{f''(c)}{2!}(x-c)^2 + \\cdots + \\frac{f^{(n)}(c)}{n!}(x-c)^n.\n$$\n","metadata":{}}, {"cell_type":"markdown","source":"

When $n=1$ this is the familiar tangent line approximation. Higher orders, generally yield better approximations.

","metadata":{}}, {"cell_type":"markdown","source":"

In julia we can create this series numerically using the D function. It is useful to write our function so that it returns a function, as is done with the following. We use a quadratic approximation for the default order and expand around $0$ as the default (the Maclaurin polynomial).

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["taylor (generic function with 2 methods)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["function taylor(f, n=2; c::Real=0)\n\t x -> f(c) + sum([D(f, k)(c)/factorial(k)*(x-c)^k for k in 1:n])\nend"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

We can see graphically that the Taylor polynomial approximates the function $f(x)$ around the $x=c$. For example, we approximate $\\cos(x)$ at $x=0$ and $x=1$:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["f(x) = cos(x)\nplot([f, taylor(f, 2, c=0), taylor(f, 2, c=pi/4)], -pi/2, pi/2)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

We can see that taking higher orders can lead to better approximations. In the following we use a comprehension and splatting to avoid having to type plot([f, taylor(f,1), taylor(f,2), taylor(f,3), ..., taylor(f,6) ], 0, 4):

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["f(x) = exp(x)\nfs = [f, [taylor(f,n) for n in 1:6]...]\nplot(fs, 0, 4) # 6 polynomial approximations"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

You should be able to confirm that the higher-order approximations do a better job.

","metadata":{}}, {"cell_type":"markdown","source":"

Mathematically, the above picture is governed by the following error bound for Taylor polynomials:

","metadata":{}}, {"cell_type":"markdown","source":"\n$$\n| f(x) - T_n(x) | \\leq | \\frac{f^{(n+1)}(z^*)}{(n+1)!} \\cdot (x-c)^{n+1} |\n$$\n","metadata":{}}, {"cell_type":"markdown","source":"

where $z^*$ is chosen to maximize the above expression between $x$ and $c$. That is, the largest possible value (in absolute value) for the next term in the Taylor polynomial provides a bound for the error.

","metadata":{}}, {"cell_type":"markdown","source":"
","metadata":{}}, {"cell_type":"markdown","source":"

Questions

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"

Maclaurin Polynomials. Useful approximations

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Maclaurin Polynomials

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Maclaurin Polynomials better for bigger $k$?

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Is 4th better then 3rd?

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Approximate integration by approximating the function – not the area.

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["(0.46128100641279246, 0.0)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["f(x) = exp(-x^2)\nquadgk(f, 0, 0.5)"],"metadata":{},"execution_count":1}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Verifying the error bound

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

A physics application

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"","metadata":{}}, {"cell_type":"markdown","source":"\n$$\nk_1(v) = \\frac{1}{2}v^2, \\quad k_2(v,c) =c^2((1-(v/c)^2)^{-1/2} - 1).\n$$\n","metadata":{}}, {"cell_type":"markdown","source":"

Which we can express via:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["k2 (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["k1(v) = v^2 / 2\nk2(v; c=10) = c^2 * (1/sqrt(1 - (v/c)^2) - 1)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

Both describe the kinetic energy, but one uses Einstein's theory of relativity. For any value of $c > 0$, show that the two agree up to second order approximations by Maclaurin polynomials.

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

A different motivation for the polynomials

","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}}, {"cell_type":"markdown","source":"","metadata":{}}, {"cell_type":"markdown","source":"

For concreteness let $f(x) = e^x$. For any $b > 0$, there is just one fifth degree polynomial, $p(x)$, satisfying $p(i\\cdot b) = f(i\\cdot b)$ for $i=0,1,2,3,4,5$. This is polynomial interpolation at 6 points.

","metadata":{}}, {"cell_type":"markdown","source":"

This particular solution can be found as follows using SymPy:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["\\begin{equation*}\\begin{cases}a_{2} & \\text{=>} &\\frac{- 10 e^{5 b} + 61 e^{4 b} - 156 e^{3 b} + 214 e^{2 b} - 154 e^{b} + 45}{24 b^{2}}\\\\a_{4} & \\text{=>} &\\frac{- 2 e^{5 b} + 11 e^{4 b} - 24 e^{3 b} + 26 e^{2 b} - 14 e^{b} + 3}{24 b^{4}}\\\\a_{0} & \\text{=>} &1\\\\a_{5} & \\text{=>} &\\frac{e^{5 b} - 5 e^{4 b} + 10 e^{3 b} - 10 e^{2 b} + 5 e^{b} - 1}{120 b^{5}}\\\\a_{3} & \\text{=>} &\\frac{7 e^{5 b} - 41 e^{4 b} + 98 e^{3 b} - 118 e^{2 b} + 71 e^{b} - 17}{24 b^{3}}\\\\a_{1} & \\text{=>} &\\frac{12 e^{5 b} - 75 e^{4 b} + 200 e^{3 b} - 300 e^{2 b} + 300 e^{b} - 137}{60 b}\\\\\\end{cases}\\end{equation*}"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["@vars a0 a1 a2 a3 a4 a5 b\nf(x) = exp(x)\ng(x) = a0 + a1*x + a2*x^(2//1) + a3*x^(3//1) + a4*x^(4//1) + a5*x^(5//1) - f(x)\ndi = solve(Sym[g(i*b) for i in 0:5], [a0, a1,a2,a3,a4,a5])"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

At first glance, this seems to have nothing to do with the Maclaurin polynomial for $f(x)$, $T_5(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120$. But wait, let's take the limit as $b$ goes to zero of each value in di. For example, the fifth term:

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":["$$\\frac{1}{120}$$"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["limit(di[a5], b, 0)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

The answer is the a5 coefficient of $T_5(x)$. Verify that this is the case for each of a0, ..., a4.

","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"

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