Continuation Newton methods with the residual trust-region time-stepping scheme for nonlinear equations

TL;DR

This paper introduces a new explicit continuation Newton method with a residual trust-region time-stepping scheme that improves robustness and speed in solving nonlinear equations, outperforming traditional methods and homotopy continuation techniques.

Contribution

The paper presents a novel explicit continuation Newton method with a residual trust-region scheme, offering enhanced efficiency and convergence properties for nonlinear equations.

Findings

The new method is more robust than traditional solvers.

It converges faster than homotopy continuation methods.

Numerical experiments confirm improved performance.

Abstract

For nonlinear equations, the homotopy methods (continuation methods) are popular in engineering fields since their convergence regions are large and they are quite reliable to find a solution. The disadvantage of the classical homotopy methods is that their computational time is heavy since they need to solve many auxiliary nonlinear systems during the intermediate continuation processes. In order to overcome this shortcoming, we consider the special explicit continuation Newton method with the residual trust-region time-stepping scheme for this problem. According to our numerical experiments, the new method is more robust and faster to find the required solution of the real-world problem than the traditional optimization method (the built-in subroutine fsolve.m of the MATLAB environment) and the homotopy continuation methods(HOMPACK90 and NAClab). Furthermore, we analyze the global convergence and the local superlinear convergence of the new method.