Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/5955
Title: Descent directions of quasi-Newton methods for symmetric nonlinear equations
Authors: Gu, GZ
Li, DH
Qi, L 
Zhou, SZ
Keywords: BFGS method
Norm descent direction
Global convergence
Superlinear convergence
Issue Date: 2002
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on numerical analysis, 2002, v. 40, no. 5, p. 1763–1774 How to cite?
Journal: SIAM Journal on numerical analysis 
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 function. 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 convergence 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.
URI: http://hdl.handle.net/10397/5955
ISSN: 0036-1429 (print)
1095-7170 (online)
DOI: 10.1137/S0036142901397423
Rights: © 2002 Society for Industrial and Applied Mathematics
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