A modified quasi-Newton method for nonlinear equations

Abstract

In this paper, a modified quasi-Newton method is proposed for solving the nonlinear equation F(x)=0, which is based on a new quasi-Newton approach. The usual quasi-Newton equation is Bk+1sk=yk, where sk=xk+1−xk, yk=F(xk+1)−F(xk). The new quasi-Newton equation is Bk+1s̃k=ỹk, in which s̃k is based on the iterates xk+1,xk,xk−1 and ỹk is based on the function values F(xk+1),F(xk),F(xk−1). The new quasi-Newton equation exploits additional information by assuming a quadratic relationship between the information from the last three iterates. The modified quasi-Newton method is based on the new quasi-Newton equation, and possess local superlinear convergence properties. Numerical experiments show that the modified quasi-Newton method is promising.