4.1 Bisection

Let's compute ?2. We will use interval bisection, which is a kind of systematic trial and error. We know that ?2 is between 1 and 2. Try . Because x ² is greater than 2, this x is too big. Try . Because x ² is less than 2, this x is too small. Continuing in this way, our approximations to ?2 are

Here is a MATLAB program, including a step counter.

     M = 2     a = 1     b = 2     k = 0;     while b-a > eps        x = (a + b)/2;        if x^2 > M           b = x        else           a = x        end        k = k + 1;     end

We are sure that ?2 is in the initial interval [ a,b]. This interval is repeatedly cut in half...