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

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.


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


Zbl 0946.65031
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