Example 1.  Find all the real solutions to the cubic equation  .  

Solution 1.

Plot the function.

There appears to be only one real root which lies in the interval [1,2].

Call the Bisection subroutine on the interval [1,2] using 10 iterations

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After 10 iterations, the interval has been reduced to [a,b] where

The root lies somewhere in the interval [a,b] the width of which is

The reported root is alleged to be

The accuracy we can guarantee is one half of the interval width.

Is this the desired accuracy you want ?  If not, more iterations are required.

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Compare our result with Mathematica's built in root finder.

Question.  Why is Mathematica's answer different ?
How many bisections would it take to reduce the interval width to ?

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Remember. The bisection method can only be used to find a real root in an interval [a,b] in which f[x] changes sign.

(c) John H. Mathews 2004