Optimal secant-updates of rank 1

Abstract

We consider a class of secant update iterative methods for solving nonlinear operator equations in Hilbert spaces. Our goal is to characterize optimal inverse secant-updates of rank 1. As the optimality criterion we use the factor, by which one iteration of a secant update iterative method reduces the entropy of solution’s position within the set of its guaranteed existence and uniqueness. Having got the optimality condition, we select among optimal updates of rank 1 one that minimizes the condition number of the updated operator. The resulting update determines a new secant update iterative method.