A derivative-free line search and global convergence of Broyden-like method for nonlinear equations

Abstract

Abstract In this paper, by using derivative-free line search, we propose a quasi-Newton method for smooth nonlinear equations. Under appropriate conditions, we show that the proposed quasi-Newton method converges globally and superlinearly. For nonlinear equations involving a mapping with positive definite Jacobian matrices, we also propose a norm descent quasi-Newton method and establish its global and superhnear convergence. Finally, we report results of preliminary numerical experiments Keywords: Derivative-free Line SearchQuasi-Newton MethodGlobal ConvergenceSuperliear Convergence ∗The work of the first was done while he was visiting Kyoto University ∗The work of the first was done while he was visiting Kyoto University Notes ∗The work of the first was done while he was visiting Kyoto University