On measures of ill-conditioning for nonlinear equations

Abstract

Let x ∗ {x^\ast } be a solution of the nonlinear equation F x = b Fx = b on a normed linear space and y ∗ {y^\ast } that of a perturbed equation G x = c Gx = c . Estimates for the relativized error between x ∗ {x^\ast } and y ∗ {y^\ast } are derived which extend a known estimate for the corresponding matrix case. The condition number of F depends now also on the domain, and special considerations are needed to determine the existence of the solution of the perturbed equation. For differentiable F , when the domain shrinks to a point, the condition number of F is shown to reduce to that of the derivative at that point.