This paper develops a gradient descent (GD) method for solving a system of nonlinear equations with an explicit formulation. We theoretically prove that the GD method has linear convergence in general and, under certain conditions, is equivalent to Newton’s method locally with quadratic convergence. A stochastic version of the gradient descent is also proposed for solving large-scale systems of nonlinear equations. Finally, several benchmark numerical examples are used to demonstrate the feasibility and efficiency compared to Newton’s method.

中文翻译：

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求解非线性方程组的梯度下降方法

本文开发了一种梯度下降（GD）方法，用于求解带有显式公式的非线性方程组。我们从理论上证明了GD方法通常具有线性收敛性，并且在一定条件下与局部具有二次收敛性的牛顿法等效。还提出了梯度下降的随机形式，用于求解非线性方程的大规模系统。最后，使用几个基准数值示例来证明与牛顿法相比的可行性和效率。