Singularity-Avoiding Multi-Dimensional Root-Finder

TL;DR

The paper introduces the W4 method, a novel extension of Newton-Raphson, capable of solving nonlinear systems with singular Jacobians, overcoming limitations of previous methods and successfully solving previously unsolvable problems.

Contribution

It presents the W4 method, which extends Newton-Raphson to handle singular Jacobians using singular value decomposition, enabling solutions to previously unsolvable problems.

Findings

The W4 method defines a non-singular iteration map for singular Jacobian problems.

It converges to the correct solution under certain conditions.

Successfully solves all standard 2D problems previously unsolvable by existing methods.

Abstract

We proposed in this paper a new method, which we named the W4 method, to solve nonlinear equation systems. It may be regarded as an extension of the Newton-Raphson~(NR) method to be used when the method fails. Indeed our method can be applied not only to ordinary problems with non-singular Jacobian matrices but also to problems with singular Jacobians, which essentially all previous methods that employ the inversion of the Jacobian matrix have failed to solve. In this article, we demonstrate that (i) our new scheme can define a non-singular iteration map even for those problems by utilizing the singular value decomposition, (ii) a series of vectors in the new iteration map converges to the right solution under a certain condition, (iii) the standard two-dimensional problems in the literature that no single method proposed so far has been able to solve completely are all solved by our new method.