Existence of Solutions and Iterations for Nonlinear Equations

Abstract

This chapter discusses the iterations for nonlinear equations and the use of interval analysis to prove the existence of solutions of an equation. It is well-known that if for a real continuous function f: ℝ → ℝ, there exist reals a and b where a < b such that f(a) f(b) ≤ 0, then there exists an x* ∈ [a, b] such that f (x*)= 0. Furthermore, x* can be computed by the well-known bisection process. If, however, the condition f (a) f (b) ≤ 0 does not hold, then there is no statement possible whether there is a zero in [a, b] or not. Under practical aspects, the requirement that the interval arithmetic evaluation exists is not restrictive. Most mappings that appear in numerical computation are composed of the four algebraic operations and of the elementary functions for which interval arithmetic evaluations can be defined in a natural manner.