Overview

In this chapter we shall build on the foundations so far laid, and illustrate their power, by developing some routines for the extraction of roots of non-linear equations in one variable with real coefficients, of the form f ( x) = 0, where f ( x) is a continuous and differentiable function. Firstly, we shall consider methods which assume no knowledge of the derivative of the function. Next, we shall consider fixed point methods, where the equation f ( x) = 0 is transformed into x = g( x). Lastly, we return to Newton's method, which uses the derivative explicitly, and shall in particular show how to calculate the roots of a polynomial.

In each method, we begin by assuming an interval ( a, b) containing a single root of f ( x) = 0. If f ( a) and f ( b) are of opposite sign, then by the intermediate value theorem there must be a number c ? ( a, b), such that f ( c) = 0. It is this number that we seek. We shall illustrate each method by applying it to the function F ( x) = x ? e ^{? x}. Since F(0) = ?1, and F(1) ? 0.63, there is a root (the only real root) in between; it is in fact...