Taylor series expansion MATLAB Example

Introduction

In this post, we will learn MATLAB program to compute the Taylor series approximation of a simple function like sin(x). In fact, we can use this approximations to other functions and polynomials.

Taylor Series

Taylor series of a function f(x) is an infinite series at a point a and can be written as :

$\LARGE f(x) = \sum_{n = 0}^{\infty} \frac{ f^{n}(a) (x-a)^{n}}{n!}$

where

$\LARGE f^{n}(a)$ is the nth derivative of the function at point a. The series is also called Maclaurin series when a = 0.

For more information : https://en.wikipedia.org/wiki/Taylor_series

Example

Let’s compute the Taylor series for sin(x) at point a = 0.

$\LARGE f(x) = sin(x)$

Apply the Taylor series expansion formula:

For better understanding of the series lets calculate each term individually for first few terms

The first tern would be = $\LARGE \frac{f^{0}(0)(x - 0)^{0}}{0!} = \frac{sin(0)(x)^{0}}{0!}$

Second term in the series = $\LARGE \frac{f^{1}(0)(x - 0)^{1}}{1!} = \frac{cos(0)(x)^{1}}{1!}$

The derivative of sin(x) = cos(x)

The third term in the series = $\LARGE \frac{f^{2}(0)(x - 0)^{2}}{2!} = \frac{- sin(0)(x)^{2}}{2!}$

The second derivative of sin(x) = -sin(x)

The fourth term in the series = $\LARGE \frac{f^{3}(0)(x - 0)^{3}}{3!} = \frac{-cos(0)(x)^{3}}{3!}$

We can ignore the even terms that contain the sin(0) in the series as sin(0) = 0.

We can generalize the series as :

Note that: cos(0) = 1

$\LARGE sin(x) = \sum_{n = 0}^{\infty } \frac{(-1)^{n}x^{2n+1}}{(2n+1)!}$

MATLAB Programs

Function

function [out] = sinTaylorApprox(x, N)
% Input = x, N
% out vector : Taylor series approx of sin(x)
out=zeros(1,numel(x));

for i = 1 : numel(x)
    for k = 0: N
        out(i) = out(i)+ ( (-1)^k * (x(i))^(2*k + 1) ) / factorial(2*k + 1);
    end
end
end

Test Driver Script

Using the defined function to compute the Taylor series approximation for the sin(x), we will create a driver script to plot the graphs for 3rd and 5th order approximations.

The driver script invokes the function twice with input arguments.

 
% Test Driver Script for plotting Taylor Series Approximation
% 2nd order and 4th Order using the function.
% MATLAB Tutorials. - www.TestingDocs.com
clear
close all

x = linspace(-2*pi, 2*pi, 100);
f = @(x) sin(x);

plot(x,f(x),'--rs','LineWidth',1.5,...
    'MarkerEdgeColor','k',...
    'MarkerFaceColor','g',...
    'MarkerSize',9)
hold on;
grid on;
plot(x,sinTaylorApprox(x, 1),'--rs','LineWidth',1.5,...
    'MarkerEdgeColor','c',...
    'MarkerFaceColor','r',...
    'MarkerSize',9)
plot(x,sinTaylorApprox(x, 2),'--rs','LineWidth',1.5,...
    'MarkerEdgeColor','m',...
    'MarkerFaceColor','b',...
    'MarkerSize',9)
xlabel('x - axis');
ylabel('y - axis');
axis([-6 6 -1 1])
legend('Sin(x) Function', '3rd Order Taylor Approx','5th Order Taylor Approx');
title('Taylor Series Approximation Plot - www.TestingDocs.com');
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