On Local and Superlinear Convergence of Secant Methods for Nonlinear Equations

Abstract

This paper considers local convergence of secant methods for a non-linear system of equations. The well-known local convergence theory has been developed by Broyden, Dennis and Moré (1973). They used a norm inequality such that the difference between two vectors transformed by some matrix is bounded above by an order of one of the two. Instead, in the present paper, we use an inequality that bounds the angle between the vectors. This inequality has a merit of scale invariance whereas the norm inequality does not.