ADTs, Data Structures, and Problem Solving with C++ 2nd Edition by Larry Nyhoff

Exercise 8

A root of a polynomial P ( x ) is a number c for which P ( c ) = 0. The bisection method is one scheme that can be used to find an approximate root of P ( x ) in some given interval [ a , b ], where P ( a ) and P ( b ) have opposite signs (thus guaranteeing that P ( x ) has a root in [ a , b ]). In this method, we begin by bisecting the interval [ a , b ] and determining in which half P ( x ) changes sign, because P must have a root in that half of the interval. Now bisect this subinterval and determine in which half of this subinterval P ( x ) changes sign. Repeating this process gives a sequence of smaller and smaller subintervals, each of which contains a root of P ( x ), as pictured in the following diagram. The process can be terminated when a small subinterval-say, of length less than 0.0001-is obtained or when P ( x ) has the value 0 at one of the endpoints:

Add and test a member function root() to the Polynomial class template so that, for a Polynomial object p,p.root (a,b) returns an approximate root of the polynomial in the interval [a, b] (if there is one), using the bisection method.