On Computational Order of Convergence of some Multi-Precision Solvers of Nonlinear Systems of Equations

TL;DR

This paper revisits the local order of convergence in iterative methods for nonlinear systems, introduces adaptive multi-precision arithmetic, and generalizes convergence order concepts to multiple dimensions with illustrative examples.

Contribution

It provides shorter analytic proofs of convergence order and extends the concept to multi-dimensional cases with adaptive precision techniques.

Findings

Adaptive multi-precision arithmetic improves convergence analysis.

Generalized convergence order definitions for multi-dimensional systems.

Examples demonstrate the effectiveness of the proposed methods.

Abstract

In this paper the local order of convergence used in iterative methods to solve nonlinear systems of equations is revisited, where shorter alternative analytic proofs of the order based on developments of multilineal functions are shown. Most important, an adaptive multi-precision arithmetics is used hereof, where in each step the length of the mantissa is defined independently of the knowledge of the root. Furthermore, generalizations of the one dimensional case to m-dimensions of three approximations of computational order of convergence are defined. Examples illustrating the previous results are given.