On measures of ill-conditioning for nonlinear equations

Abstract:

Let ${x^\ast }$ be a solution of the nonlinear equation $Fx = b$ on a normed linear space and ${y^\ast }$ that of a perturbed equation $Gx = c$. Estimates for the relativized error between ${x^\ast }$ and ${y^\ast }$ 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.