Question

Let the mathematical function f(x) be defined as: f(x) = exp(-0.5x) cos(5x)-0.5 , x 〉 0 Write a Matlab function called Newton1 that would find the zero based on a passing initial guess as an input argument x0. The function returns the estimated zero location x, the function value at the zero location (f) and the number of iteration k. The iteration function converges if f(%) 10000. When it diverges, it should state so using disp() command. The function should starts as: function [x,fs,k]-Newton1(xo) % enter your code here end Test the algorithm to find the zero as follows. >> [x,f,k-Newton! (O) → x-1.1641, f=-1.6653e-16, k = 7 >> [x,f,k-Newton! (0.1) → x = 0.1972, f= 1.1102e-16, k :5 >> [x,f,k]-Newton1(1.5) → x= 1.3111, f=-1.1102e-16, k = 6 >> [x,f,k-Newton! (2) → Warning iteration diverged, x-2.8808, f=-0.5624, k = 10001

Answer 1

MATLAB CODE;

clc;

clear all;

close all;

function [ xz fs k ]=Newton1(x0)
format long
tol=5*eps;
c(1)=x0;
syms x;
f=exp(-0.5*x)*cos(5*x)-0.5 ;
for k=1:1000
l1=subs(f,c(k));
l2=subs(diff(f),c(k));
c(k+1)=c(k)-l1/l2;
l3=(subs(f,c(k+1)));

fs=l3;
xz=c(k+1);
if (abs(l3) < tol)

break;
end
end
end

%%%%%%% TEST %%%% %%%

[x fs k]=Newton1(0);
fprintf(' x= %f \n',x);
fprintf(' fx= %1.32f \n',fs);
fprintf(' k= %f \n',k);
if k>1000
fprintf('Iteration diverged');
end

MATLAB O/P::

x= 1.164097
fx= -0.00000000000000006861264023336157
k= 7.000000
>>

%%%%%%%%TEST%%%%%%

clc;
close all;
clear all;
format long;

[x fs k]=Newton1(1.5);
fprintf(' x= %f \n',x);
fprintf(' fx= %1.32f \n',fs);
fprintf(' k= %f \n',k);
if k>1000
fprintf('Iteration diverged');
end

MATLAB O/P:

x= 1.311127
fx= -0.00000000000000002916434802119459
k= 6.000000

%%%%%%%%%%%%%TEST%%%%%

clc;
close all;
clear all;
format long;

[x fs k]=Newton1(2);
fprintf(' x= %f \n',x);
fprintf(' fx= %1.32f \n',fs);
fprintf(' k= %f \n',k);
if k>1000
fprintf('Iteration diverged');
end

MATLAB O/P:

x= 2.399596
fx= -0.24611144691801684070142641758139
k= 1000.000000
>>>>>>>>>>>>>Iteration diverged