PSB Method For Symmetric Nonlinear Equations

Quasi-Newton methods are efficient methods for solving nonlinear equations and optimization problems. Under appropriate conditions, these methods possess local superlinear convergence property. Moreover, when applied to solving opti-mization problems, most quasi-Newton methods are globally convergent if some line search technique is used. However, when applied to solve a system of nonlin-ear equations, the global convergence of most qusi-Newton methods is not known because the quasi-Newton direction is generally not a descent direction of the norm function. Therefore, to develop a globally convergent quasi-Newton method, derivative-free line search is necessary. The earliest global convergence result is due to Griewank(1986), where a derivative-free line search is proposed. By the' use of this monotone line search, it was shown that Broyden rank one method is globally convergent when they are applied to solve a system of nonlinear equa-tions. But there is a certain difficulty with the line search. Li-Fukushima(1999) developed a derivative-free line search that do not suffer from such difficulty, and the line search posses approximately norm descent property. Li-Fukushima proved that Broyden-like methods with that line search is globally and superlinearly con-vergent. For symmetric nonlinear equations, Li-Fukushima (1999) introduced a DFP method with another derivative-free line search and established its global convergence.In this paper, we study the global congvergence of the PSB method with the so-called approximately norm descent derivative-free line search developed by Li and Fukushima.We show that if the Jacobi of the symmetric equation Lipschitz continuous and nonsingular everywhere, then the PSB method with that line search is globally and superlinearly convergent. We also do some numerical experiments. The results show that the PSB method is an effective method for solving nonlinear equations.