On a non-archimedean broyden method
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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 2^1/2m in dimension m. Numerical data are provided.
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