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Fast Secant Methods for the Iterative Solution of Large Nonsymmetric Linear Systems.

Please always quote using this URN: urn:nbn:de:0297-zib-330
  • A family of secant methods based on general rank-1 updates has been revisited in view of the construction of iterative solvers for large non- Hermitian linear systems. As it turns out, both Broydens "good" and "bad" update techniques play a special role - but should be associated with two different line search principles. For Broydens "bad" update technique, a minimum residual principle is natural - thus making it theorectically comparable with a series of well-known algorithms like GMRES. Broydens "good" update technique, however, is shown to be naturally linked with a minimum "next correction" principle - which asymptotically mimics a minimum error principle. The two minimization principles differ significantly for sufficiently large system dimension. Numerical experiments on discretized PDE's of convection diffusion type in 2-D with internal layers give a first impression of the possible power of the derived "good" Broyden variant. {\bf Key Words:} nonsymmetric linear system, secant method, rank-1 update, Broydens method, line search, GMRES. AMS(MOS) {\bf Subject Classifications:} 65F10, 65N20.

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Metadaten
Author:Peter Deuflhard, R. Freund, Artur Walter
Document Type:ZIB-Report
Tag:Broyden's method; GMRES; line search; nonsymmetric linear systems; rank-1 update; secant method
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F10 Iterative methods for linear systems [See also 65N22]
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N20 Ill-posed problems
Date of first Publication:1990/06/07
Series (Serial Number):ZIB-Report (SC-90-05)
ZIB-Reportnumber:SC-90-05
Published in:Appeared in: IMPACT Comp. Sci. Eng. 2, pp. 244-276 (1990)
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