A Generalized Multivariable Newton Method

TL;DR

This paper introduces a family of generalized multivariable Newton methods that expand the convergence region and improve properties near solutions, supported by theoretical proofs and extensive numerical testing.

Contribution

The paper proposes new variants of the Newton method with larger convergence regions and quadratic convergence, backed by theoretical analysis and practical testing.

Findings

Quadratic convergence of the new methods is proven.

The new methods outperform traditional Newton in larger convergence regions.

Numerical experiments identify the most effective method for different scenarios.

Abstract

It is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a larger convergence region as well as more desirable properties near a solution. We prove quadratic convergence of the new family, and provide specific bounds for the asymptotic error constant. We illustrate the advantages of the new methods by means of test problems, including two and six variable polynomial systems, as well as a challenging signal processing example. We present a numerical experimental methodology which uses a large number of randomized initial guesses for a number of methods from the new family, in turn providing advice as to which of the methods employed is preferable to use in a particular search domain.