This article is about numerical methods for the solution of nonlinear equations. We consider both the fixed-point form $\mathbf{x}=\mathbf{G}(\mathbf{x})$ and the equations form $\mathbf{F}(\mathbf{x})=0$ and explain why both versions are necessary to understand the solvers. We include the classical methods to make the presentation complete and discuss less familiar topics such as Anderson acceleration, semi-smooth Newton’s method, and pseudo-arclength and pseudo-transient continuation methods.

中文翻译：

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非线性方程的数值方法

本文是关于求解非线性方程的数值方法。我们同时考虑定点形式$\mathbf{x}=\mathbf{G}(\mathbf{x})$和方程形式$\mathbf{F}(\mathbf{x})=0$并解释为什么两个版本都是理解求解器所必需的。我们包括使演示完整的经典方法，并讨论不太熟悉的主题，例如安德森加速、半光滑牛顿法以及伪弧长和伪瞬态延续方法。