Question

PLEASE USE MATLAB

this problem you are trying to find an approximation to the periodic function /(t) esint-1) over one period, o <t < 2π. In t-linspace(0,2*pi,200)' and let b be a column vector of evaluations of f at those points. (a) Find the coefficients of the least squares fit In MATLAB, let (b) Find the coefficients of the least squares fit f(t)ndy+d2 cos(t)+ d, sin(t)+d, cos(2t)+dy sin(2t). (c) Plot the original function f(t)and the two approximations from (a) and (b) together on a well-labeled graph.

Answer 1

MATLAB Code:

close all
clear
clc

t = linspace(0, 2*pi, 200)';
b = exp(sin(t-1));

fprintf('Part (a)\n-----------------------------------------------\n')
T = [ones(size(t)) t t.^2 t.^3 t.^4 t.^5 t.^6 t.^7];
CA = T\b;
fprintf('f(t) ~ %f + %f*t + %f*t^2 + %f*t^3 + %f*t^4 + %f*t^5 + %f*t^6 + %f*t^7\n', CA)

fprintf('\nPart (b)\n-----------------------------------------------\n')
T = [ones(size(t)) cos(t) sin(t) cos(2*t) sin(2*t)];
CB = T\b;
fprintf('f(t) ~ %f + %f*cos(t) + %f*sin(t) + %f*cos(2*t) + %f*sin(2*t)\n', CB)

plot(t, b), hold on
plot(t, CA(1) + CA(2)*t + CA(3)*t.^2 + CA(4)*t.^3 + CA(5)*t.^4 + CA(6)*t.^5 + CA(7)*t.^6 + CA(8)*t.^7)
plot(t, CB(1) + CB(2)*cos(t) + CB(3)*sin(t) + CB(4)*cos(2*t) + CB(5)*sin(2*t))
xlabel('t'), ylabel('f(t)')
legend('Original Function', 'Part (a)', 'Part (b)')

Output:

Part (a)
-----------------------------------------------
f(t) ~ 0.326349 + 1.644306*t + -3.880251*t^2 + 4.701695*t^3 + -2.320595*t^4 + 0.540292*t^5 + -0.060086*t^6 + 0.002586*t^7

Part (b)
-----------------------------------------------
f(t) ~ 1.266081 + -0.951099*cos(t) + 0.610714*sin(t) + 0.113013*cos(2*t) + -0.246870*sin(2*t)

Plot:

Original Function -Part (a) Part (b) 2.5 2 ︵ 1.5 0.5 7 6 5 4 2 0