Descent directions of quasi-Newton methods for symmetric nonlinear equations. (English) Zbl 1047.65032

Gu, Guang-Ze; Li, Dong-Hui; Qi, Liqun; Zhou, Shu-Zi

SIAM J. Numer. Anal. 40, No. 5, 1763-1774 (2002).

Quasi-Newton methods for solving systems of nonlinear equations \(F(x) = 0\) with symmetric Jacobian are considered. The authors develop a modification of the Gauss-Newton BFGS method proposed by D. Li and M. Fukushima [SIAM J. Numer. Anal. 37, 152–172 (1999; Zbl 0946.65031)]. Contrary to the method proposed by Li and Fukushima the new method is norm descent: The step length and the search direction are adapted simultaneously in such a way that the search direction becomes a descent direction of the norm function \(\| F(x)\| \). Under mild conditions, global and superlinear convergence of the method are shown. Numerical results are reported, where the new approach is compared to the method proposed by Li and Fukushima.

Reviewer: Walter Zulehner (Linz)

MSC:

65H10	Numerical computation of solutions to systems of equations
90C53	Methods of quasi-Newton type
90C30	Nonlinear programming

Keywords:

symmetric nonlinear equations; norm descent direction; global convergence; superlinear convergence; comparison of methods; quasi-Newton methods; systems of nonlinear equations; Gauss-Newton BFGS method; numerical results

Citations:

Zbl 0946.65031