Descent directions of quasi-Newton methods for symmetric nonlinear equations

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

Research output: Journal article publicationJournal articleAcademic researchpeer-review

58 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)1763-1774
Number of pages12
JournalSIAM Journal on Numerical Analysis
Issue number5
Publication statusPublished - 1 Oct 2002


  • BFGS method
  • Global convergence
  • Norm descent direction
  • Superlinear convergence

ASJC Scopus subject areas

  • Numerical Analysis


Dive into the research topics of 'Descent directions of quasi-Newton methods for symmetric nonlinear equations'. Together they form a unique fingerprint.

Cite this