Extending the applicability of Newton's and Secant methods under regular smoothness

Ioannis K Argyros,Santhosh George,Shobha M Erappa

doi:10.5269/bspm.42132

Abstract

The concept of regular smoothness has been shown to be an appropriate and powerfull tool for the convergence of iterative procedures converging to a locally unique solution of an operator equation in a Banach space setting. Motivated by earlier works, and optimization considerations, we present a tighter semi-local convergence analysis using our new idea of restricted convergence domains. Numerical examples complete this study.

Highlights

Semilocal convergence analysis for Newton’s methodLet T denote the class of nondecreasing continuous functions v : [0, ∞) −→ [0, ∞), that have convex subgraphs {(s, t) : s ≥ 0 and t ≤ v(s)}, and vanish at zero, i.e., they are concave [3], [6], [7]
The Secant method has some attractive properties: it is self–correcting, it exhibits superlinear convergence, and no knowledge of the derivatives of the operators involved is required
In the excellent works by Galperin [6], [7], the concept of regular smoothness was introduced, which became a viable framework for the study of the convergence of iterative procedures such as Newton’s method, and Secant method

Summary

Semilocal convergence analysis for Newton’s method

Let T denote the class of nondecreasing continuous functions v : [0, ∞) −→ [0, ∞), that have convex subgraphs {(s, t) : s ≥ 0 and t ≤ v(s)}, and vanish at zero, i.e., they are concave [3], [6], [7]. The creation of function ω was not possible before in the studies using only function ω1 [1], [2], [5]– [18] In these studies ω can replace ω1 leading to the advantages as stated previously, when strict inequality holds in (2.1). These advantages are obtained under the same computational cost, since in practice the computation of function ω1 requires the computation of functions ω0 and ω1 as special cases. The sequence {xn} (n ≥ 0), generated by Newton’s method (1.2) is well defined, remains in U (x0, t∞) for all n ≥ 0, and converges to a solution x∞ of equation F (x) = 0. Note that in Application 3.6, we show how to replace delicate condition (2.24)

Semilocal convergence of the Secant method

Conclusion