A framework for generalized conjugate gradient methods-with special emphasis on contributions by Rüdiger Weiss

Authors:
Martin H. Gutknecht;Miroslav Rozlozník
Affiliations:
Seminar for Applied Mathematics, ETH-Zentrum HG, CH-8092 Zürich, Switzerland;Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodárenskou vezí 2, CZ-18207 Prague 8, Czech Republic
Venue:
Applied Numerical Mathematics - Developments and trends in iterative methods for large systems of equations—in memoriam Rüdiger Weiss
Year:
2002

Citing 14
Cited 1

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On the stable implementation of the generalized minimal error method

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Conjugate gradient methods for partial differential equations.

Conjugate gradient methods for partial differential equations.
Iterative methods for large, sparse, nonsymmetric systems of linear equations

Iterative methods for large, sparse, nonsymmetric systems of linear equations

Lanczos-type variants of the COCR method for complex nonsymmetric linear systems

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Abstract

We discuss a general framework for generalized conjugate gradient methods, that is, iterative methods (for solving linear systems) that are based on generating residuals that are formally orthogonal to each other with respect to some true or formal inner product. This includes methods that generate residuals that are minimal with respect to some norm based on an inner product. A similar, even more general framework was introduced and extensively discussed by Weiss in his work.Weiss also emphasized in his work the relationship between certain pairs of orthogonal and minimal residual methods, where the results of the second method can be generated from those of the first method by applying the minimal residual smoothing process. Examples of such pairs are the conjugate gradient (CG) and the conjugate residual (CR) methods, the full orthogonalization method (FOM) and the generalized minimal residual (GMRES) method, as well as the CGNE and CGNR versions of applying CG to the normal equations.