This paper considers the generalized continuation Newton method and the trust-region updating strategy for the underdetermined system of nonlinear equations. Moreover, in order to improve its computational efficiency, the new method will not update the Jacobian matrix when the current Jacobian matrix performs well. The numerical results show that the new method is more robust and faster than the traditional optimization method such as the Levenberg–Marquardt method (a variant of trust-region methods, the built-in subroutine fsolve.m of the MATLAB R2020a environment). The computational time of the new method is about 1/8 to 1/50 of that of fsolve. Furthermore, it also proves the global convergence and the local superlinear convergence of the new method under some standard assumptions.

本文考虑了广义延拓牛顿法和非线性方程组欠定系统的信任域更新策略。而且，为了提高其计算效率，新方法在当前雅可比矩阵表现良好时不会更新雅可比矩阵。数值结果表明，新方法比传统的优化方法如Levenberg-Marquardt方法（信任域方法的变体，MATLAB R2020a环境的内置子程序fsolve.m）更鲁棒和更快。新方法的计算时间约为 fsolve 的 1/8 到 1/50。此外，它还证明了新方法在一些标准假设下的全局收敛性和局部超线性收敛性。