On a non-archimedean broyden method

Xavier Dahan (1) , Tristan Vaccon (2)

1 Tohoku University [Sendai]
2 XLIM - XLIM

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.

Mots clés: System of equations; Broyden's method; Quasi-Newton; p-adic ap- proximation; Power series; Symbolic-numeric; p-adic algorithm

Domaines: Calcul formel [cs.SC]; Analyse numérique [math.NA]; Théorie des nombres [math.NT]

Xavier Dahan, Tristan Vaccon. On a non-archimedean broyden method. ISSAC '20: International Symposium on Symbolic and Algebraic Computation, Jul 2020, Kalamata Greece, France. pp.114-121, ⟨10.1145/3373207.3404045⟩. ⟨hal-02928697⟩