This paper addresses the challenge of solving large-scale nonlinear equations with H\"older continuous Jacobians. We introduce a novel Incremental Gauss--Newton (IGN) method within explicit superlinear convergence rate, which outperforms existing methods that only achieve linear convergence rate. In particular, we formulate our problem by the nonlinear least squares with finite-sum structure, and our method incrementally iterates with the information of one component in each round. We also provide a mini-batch extension to our IGN method that obtains an even faster superlinear convergence rate. Furthermore, we conduct numerical experiments to show the advantages of the proposed methods.

本文针对具有 H"older 连续雅可比矩阵的大规模非线性方程的求解挑战，引入了一种新颖的具有显式超线性收敛速率的增量高斯-牛顿（IGN）方法，其性能优于仅实现线性收敛速率的现有方法。具体而言，我们通过具有有限和结构的非线性最小二乘问题来表述我们的问题，并且我们的方法在每一轮中逐渐迭代一个组件的信息。我们还为我们的IGN方法提供了小批量扩展，可以获得更快的超线性收敛速率。此外，我们进行了数值实验以展示所提方法的优势。

具有超线性收敛速度的增量高斯-牛顿方法