Use the Bisection method to find the roots of fx 2x x cosx

Use the Bisection method to find the roots of f(x) = (2/x - x) cosx - 1 on the interval [0.5, 5]. First write the function f(x) as myeq.m in MATLAB; use the brack Plot.m function to plot the subintervals that may contain potential roots, for [a, b] = [0.5, 5]. Write a MATLAB script file (NOT a function file) that applies the bisect function on each of the brackets you have found from (1) and produces all the numerical roots of f(x) = 0.

Solution

(1)

function f=myeq(a,b)

x=a:0.01:b;

f=(2./x-x).*cos(x)-1;

end

--------------------brackplot.m------------

clc;
clear all;
close all;

a=0.5;
b=5;
f=myeq(a,b);
plot(f)

(2)

clc;

clear all;

close all;

x1=0.5;

x2=5;

y1=(2/x1 -x1)*cos(x1)-1;

if(y1<0)

x1=x1;

x2=x2;

else

x=x2;

x2=x1;

x1=x;

end

n=100;

i=1;

while(n>0)

y1=(2/x1-x1)*cos(x1)-1;

y2=(2/x2-x2)*cos(x2)-1;

y=y1*y2;

if (y<0)

x=(x1+x2)/2

y=(2/x-x)*cos(x)-1;

if(y<0)

x1=x;

else

x2=x;

end

end

tol(i)=abs(y);

if(tol(i)<(0.01))

n=-1;

end

  

i=i+1;

n=n-1;

end

fprintf(\'Rooots of given problem by using Bi-Section Method : %d and %d(Iterations)\ \',x,i-1);

OUTPUT:

x =
2.7500
x =
3.8750
x =
4.4375
x =
4.7188
x =
4.5781
x =
4.5078
x =
4.4727
x =
4.4551
x =
4.4639
x =
4.4595
Rooots of given problem by using Bi-Section Method : 4.459473e+00 and 10(Iterations)
>>