The bisection method in Matlab is quite straight-forward. Assume a file `f.m` with
contents

function y = f(x) y = x.^3 - 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 >> [a b] ans = 1.259918212890625 1.259925842285156 >> abs(f(a)) ans = 0.0000135103601622 >> abs(f(b)) ans = 0.0000228224229404

Thus, we would choose 1.259918212890625 as our approximation to the cube-root of 2, which has an actual value (to 16 digits) of 1.259921049894873.

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

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.