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

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 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.