DESCENT DIRECTIONS OF QUASI-NEWTON METHODS FOR SYMMETRIC NONLINEAR EQUATIONS∗

Abstract. In general, when a quasi-Newton method is applied to solve a system of nonlinear equations, the quasi-Newton direction is not necessarily a descent direction for the norm function. In this paper, we show that when applied to solve symmetric nonlinear equations, a quasi-Newton method with positive definite iterative matrices may generate descent directions for the norm func-tion. On the basis of a Gauss–Newton based BFGS method [D. H. Li and M. Fukushima, SIAM J. Numer. Anal., 37 (1999), pp. 152–172], we develop a norm descent BFGS method for solving symmetric nonlinear equations. Under mild conditions, we establish the global and superlinear con-vergence of the method. The proposed method shares some favorable properties of the BFGS method for solving unconstrained optimization problems: (a) the generated sequence of the quasi-Newton matrices is positive definite; (b) the generated sequence of iterates is norm descent; (c) a global convergence theorem is established without nonsingularity assumption on the Jacobian. Preliminary numerical results are reported, which positively support the method. Key words. BFGS method, norm descent direction, global convergence, superlinear convergence AMS subject classifications. 65H10, 90C53 PII. S003614290139742