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 \mbox{%DIFAUXCMD MATLAB,More1978 ) and the classical homotopy continuation methods (HOMPACK90 \mbox{%DIFAUXCMD \cite{WSMMW1997 }\hspace{0pt}%DIFAUXCMD and the built-in subroutines psolve. m for polynomial systems, GaussNewton.m for non-polynomial systems of the NAClab environment \mbox{%DIFAUXCMD \cite{ZL2013}}\hspace{0pt}%DIFAUXCMD )}  .
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 consumed time is heavy since they need to solve many auxiliary nonlinear systems during the intermediate continuation process . In order to overcome this shortcoming, we consider the special explicit continuation Newton method with the new time-stepping scheme for nonlinear equations. According to our numerical experiments, it 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 }\hspace{0pt}%DIFAUXCMD and the built-in subroutines psolve. m for polynomial systems, GaussNewton.m for non-polynomial systems of the NAClab environment \mbox{%DIFAUXCMD \cite{ZL2013}}\hspace{0pt}%DIFAUXCMD )} and NAClab). Furthermore, we analyze the global convergence and the local superlinear convergence of the new method .
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the problem of
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fairly
quite
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systems of nonlinear equations
nonlinear systems
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processes
process
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continuation method based on the Newton flow and follow its trajectory
explicit continuation Newton method
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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
for nonlinear equations. According to our numerical experiments, it is more robust and faster to find the required solution of the
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problems are also reported, with comparison to the traditional trust-region
problem than the traditional optimization
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\mbox{%DIFAUXCMD MATLAB,More1978
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classical
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\mbox{%DIFAUXCMD \cite{WSMMW1997
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and NAClab). Furthermore, we analyze the global convergence and the local superlinear convergence of the new method