A modified quasi-Newton method for nonlinear equations

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 = y(k), where s(k) = x(k+1) - x(k), y(k) = F(x(k+1)) - F(x(k)). The new quasi-Newton equation is Bk+1(s) over tilde (k) = (y) over tilde (k), in which (s) over tilde (k) is based on the iterates x(k+1), x(k), x(k-1) and (y) over tilde (k) is based on the function values F(x(k+1)), F(x(k)), F(x(k-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. (C) 2017 Elsevier B.V. All rights reserved.