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.

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关于非阿基米德布罗伊登方法

牛顿法是一种在阿基米德和非阿基米德环境中求解方程的普遍工具——在这方面它并没有真正的区别。布罗伊登是所谓的“准牛顿方法”的发起者。这些方法使用迭代步骤，其中不需要计算完整的雅可比矩阵或其逆矩阵。我们在一般非阿基米德设置中提供了对 Broyden 方法的改编，与内积的缺乏兼容，并研究其 Q 和 R 收敛。我们证明我们的适应方法至少在 Q 线性和 R 超线性收敛，R 阶 $2^{\frac{1}{2m}}$ 在维度 m。提供了数值数据。