8 - NONLINEAR EQUATIONS

Abstract

This chapter discusses the methods and software that help solve the general nonlinear equations problem. The key to putting problems into the form f(x) = 0 or f(x) = 0 is to realize that, if there are conditions to be satisfied and if they can be calculated, then a subprogram can be written whose output is the amounts by which the conditions fail to be satisfied. Most methods for systems of nonlinear equations are extensions of methods for one equation, and so it is important to systematically survey most of the possibilities in the simpler case of one equation. Iterations for nonlinear equations usually converge fast, and the convergence test is only occasionally a troublesome part of the method. Convergence tests appear throughout numerical software. Moreover, there are some special situations that must be handled to obtain a reliable program for solving f(x) = 0. If the function f(x) is a polynomial then this fact can be exploited to improve upon the general methods and to devise special methods. The software for nonlinear equations is divided into five categories: (1) simple implementation of basic algorithms, (2) software for a single nonlinear equation, (3) software for systems of nonlinear equations, (4) polynomial zero finders, and (5) special procedures.