Question

7. Finding Roots Using the Bisection Method

Write a function that implements the "bisection method" for finding the roots of function. The signature of your function should look like

def find_root(f,a,b,n):

where n is the maximum number of iterations of to search for the root.

The code should follow this algorithm:

• We are given a continuous function f and numbers a and b and with a<b with f(a)<0<f(b). From the intermediate value theorem we know that there exists a c with a<c<b with f(c)=0. We want to find c.

• Set a1=a and b1=b and m=12(a+b).

• For i=1 to n do

  • If f(m)=0, then c=m so break and return m.
  • Else If f(m)>0 then set ai+1=ai and bi+1=m.
  • Else If f(m)<0, then set ai+1=mi and bi+1=bi.

Answer 1

%%Matlab code for finding root using Bisection method
clear all
close all

%Function for which root have to find
fun=@(x) sin(x)-1/2;

%displaying the function
fprintf('\tFor the function\n')
disp(fun)

%Root using Bisection method
x0=0; x1=1; %Initial guess
n=100; %maximum iteration
[root]=find_root(fun,x0,x1,n);
fprintf('Root using Bisection method for initial guess[%f,%f] is %2.15f with iteration count %d.\n\n',x0,x1,root,n);

%Matlab function for Bisection Method
function [root]=find_root(f,a,b,n)
if f(a)<=0
    t=a;
    a=b;
    b=t;
end
%f(x1) should be positive
%f(x0) should be negative
for i=1:n
    xx(i)=(a+b)/2;
    mm=double(f(xx(i)));
    if mm>=0
        a=xx(i);
    else
        b=xx(i);
    end
end
root=xx(end);
end

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