Finding roots of equations in given neighborhoods

摘要

This paper discusses a method of finding the roots of a continuously differentiable mapping f:Rn → Rn once they have been located in a bounded, open convex region of Rn. The method consists in showing that for suitable restrictions inposd on f, comparable but weaker than the standard restrictions placed on f so that Newton's method maybe employed, one is able to solve a functional equation by solving a differential equation whose solutions satisfy the functional equation and lead to the roots in question. Examples where Newton's method fails but the considered method succeeds are given. Rates of convergence to a root and other matters are also considered.