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.

翻译：本文致力于解决具有Hölder连续雅可比矩阵的大规模非线性方程组的求解难题。我们提出了一种具有显式超线性收敛速率的新型增量高斯-牛顿方法，其性能优于仅能达到线性收敛速率的现有方法。具体而言，我们将问题建模为具有有限和结构的非线性最小二乘问题，所提方法在每轮迭代中仅利用单个分量的信息进行增量更新。我们还为该方法提供了小批量扩展版本，该版本可获得更快的超线性收敛速率。此外，我们通过数值实验验证了所提方法的优越性。