On a non-archimedean broyden method

TL;DR

This paper adapts Broyden's quasi-Newton method to non-archimedean settings, demonstrating convergence properties and providing numerical evidence for its effectiveness.

Contribution

It introduces a non-archimedean version of Broyden's method, analyzing its convergence without relying on inner products.

Findings

Converges at least Q-linearly

Achieves R-superlinear convergence with order 2^{1/(2m)}

Numerical experiments support theoretical results

Abstract

Newton's method is an ubiquitous tool to solve equations, both in the archimedean and non-archimedean settings -- for which it does not really differ. Broyden was the instigator of what is called "quasi-Newton methods". These methods use an iteration step where one does not need to compute a complete Jacobian matrix nor its inverse. We provide an adaptation of Broyden's method in a general non-archimedean setting, compatible with the lack of inner product, and study its Q and R convergence. We prove that our adapted method converges at least Q-linearly and R-superlinearly with R-order $2^{\frac{1}{2m}}$ in dimension m. Numerical data are provided.