Using the pseudocode for the Bisection Method in the class n

Using the pseudocode for the Bisection Method in the class note write a MATLAB code to approximate the root of f(x) = e^x - x - 2 on [0, 2] with accuracy within 10^-8. Use maximum 100 iterations. Explain steps by commenting on them.

Solution

I wrote the bisection method in matlab code, it is quite straight-forward.

Assume a file f.m with contents

function y = f(x)
y = e^x-x-2;
exists. Then:

>> format long
>> eps_abs = 1e-5;
>> eps_step = 1e-5;
>> a = 0.0;
>> b = 2.0;
>> while (b - a >= eps_step || ( abs( f(a) ) >= eps_abs && abs( f(b) ) >= eps_abs ) )
c = (a + b)/2;
if ( f(c) == 0 )
break;
elseif ( f(a)*f(c) < 0 )
b = c;
else
a = c;
end
end
- An implementation of a function would be -

function [ r ] = bisection( f, a, b, N, eps_step, eps_abs )
% Check that that neither end-point is a root
% and if f(a) and f(b) have the same sign, throw an exception.

if ( f(a) == 0 )
   r = a;
   return;
elseif ( f(b) == 0 )
   r = b;
   return;
elseif ( f(a) * f(b) > 0 )
error( \'f(a) and f(b) do not have opposite signs\' );
end

% We will iterate N times and if a root was not
% found after N iterations, an exception will be thrown.

for k = 1:N
% Find the mid-point
c = (a + b)/2;

% Check if we found a root or whether or not
% we should continue with:
% [a, c] if f(a) and f(c) have opposite signs, or
% [c, b] if f(c) and f(b) have opposite signs.

if ( f(c) == 0 )
r = c;
return;
elseif ( f(c)*f(a) < 0 )
b = c;
else
a = c;
end
% If |b - a| < eps_step, check whether or not
% |f(a)| < |f(b)| and |f(a)| < eps_abs and return \'a\', or
% |f(b)| < eps_abs and return \'b\'.

if ( b - a < eps_step )
if ( abs( f(a) ) < abs( f(b) ) && abs( f(a) ) < eps_abs )
r = a;
return;
elseif ( abs( f(b) ) < eps_abs )
r = b;
return;
end
end
end

error( \'the method did not converge\' );
end