Continuation Newton method with the trust-region time-stepping scheme
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For the problem of nonlinear equations, the homotopy methods (continuation methods) are popular in engineering fields since their convergence regions are large and they are fairly reliable to find a solution. The disadvantage of the classical homotopy methods is that their consumed time is heavy since they need to solve many auxiliary systems of nonlinear equations during the intermediate continuation processes. In order to overcome this shortcoming, we consider the special continuation method based on the Newton flow and follow its trajectory with the new time-stepping scheme based on the trust-region technique. Furthermore, we analyze the global convergence and local superlinear convergence of the new method. Finally, the promising numerical results of the new method for some real-world problems are also reported, with comparison to the traditional trust-region method (the built-in subroutine fsolve.m of the MATLAB environment <cit.>) and the classical homotopy continuation methods (HOMPACK90 <cit.> and the built-in subroutines psolve.m for polynomial systems, GaussNewton.m for non-polynomial systems of the NAClab environment <cit.>).
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