Write two codes Newton Method and Bisection Method to find the roots. (a) Using

Question

Write two codes Newton Method and Bisection Method to find the roots. (a) Using bisection method: on the interval xL = 0, xR = pi . Do you encounter any problems?
Is there a way the code can be improved (again, in the sense of making it more `reliable' as
in the first homework)? Explain any changes you make to your code.

(b) Using Newton's method: with the following initial guesses:
x0 = pi/2
x0 = 5pi
x0 = 10pi
Use an accuracy of a = 10-5, and a maximum number of 20000 iterations. Explain the
different behavior for the three starting values?

For each part, submit the equivalent files: a function file, script file, and
output file. Your output must also meet the same requirements: the value of the root, function
value at the root, and number of iterations performed.

Explanation / Answer

a) %%% Bisection method

at both point x=0 and x=pi f(x) is positive

Thus there need to change a initial point

lets Xr=pi/2

%%% Matlab code

clc;
clear all;
close all;
format long
format compact

syms x
f=1/2+x^2/4-x*sin(x)-cos(2*x)/4;
tol=0.00001;

% % % bisection method
disp('Bisection method :');
a=pi/2;
b=0;
for n=1:100
l1=subs(f,a);
l2=subs(f,b);
  
c(n)=(a+b)/2;
l3=subs(f,c(n));
if (n>2)
if (abs( c(n)-c(n-1))< tol)
break;
end
end
  
if ( l3 < 0 )
a=c(n);
else
b=c(n);
end

end
fprintf('roots of eqution is : %f ', c(end) );

OUTPUT:

Bisection method :
roots of eqution is : 1.029271

b) Newton raphson method

syms x
f=1/2+x^2/4-x*sin(x)-cos(2*x)/4;
tol=0.00001;
x0=[pi/2 5*pi 10*pi];
% % % % N-R Method
% disp('Newton Raphson method');
fprintf('Initial guess Root value at root Number of iteration ');
for k=1:length(x0)
s(1)=x0(k);
for n=1:2000
l1=subs(f,s(n));
l2=subs(diff(f),s(n));
s(n+1)=s(n)-l1/l2;
e=abs(s(n+1)-s(n));
if (e < tol)
break;
end
end
v=subs(f,s(n+1));
fprintf(' %f %f %1.8f %d ',x0(k),s(n+1),v,n);
end

OUTPUT:

Initial guess Root value at root Number of iteration
1.570796 1.029276 0.00000000 5  
15.707963 2.192591 0.00000000 33  
31.415927 -1.029276 0.00000000 87  

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