Topic 10.1: Bisection Method (Matlab)

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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