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GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems

Youcef Saad; Martin H. Schultz · SIAM Journal on Scientific and Statistical Computing · 1986

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Author / responsible party
Youcef Saad; Martin H. Schultz
Publisher
SIAM Journal on Scientific and Statistical Computing
Year
1986
Language
English

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We present an iterative method for solving linear systems, which has the property ofminimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an l2-orthogonal basis of Krylov subspaces. It can be considered as a generalization of Paige and Saunders’ MINRES algorithm and is theoretically equivalent to the Generalized Conjugate Residual (GCR) method and to ORTHODIR. The new algorithm presents several advantages over GCR and ORTHODIR.

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APA 7

Saad, Y. & Schultz, M. H. (1986). GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. https://doi.org/10.1137/0907058

MLA

Saad, Youcef, and Martin H. Schultz. "GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems." 1986. https://doi.org/10.1137/0907058.

Chicago

Saad, Youcef and Martin H. Schultz. 1986. "GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems.". https://doi.org/10.1137/0907058.

Harvard

Saad, Y. and Schultz, M. H. 1986, GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM Journal on Scientific and Statistical Computing, available at: https://doi.org/10.1137/0907058 [Accessed 7 Aug. 2026].

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Title
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
Author / contributors
Youcef Saad; Martin H. Schultz
Publisher
SIAM Journal on Scientific and Statistical Computing
Publication year
1986
Language
English

Subjects

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Generalized minimal residual method; Krylov subspace; Residual; Mathematics; Conjugate residual method; Conjugate gradient method; Linear subspace; Linear system; Algorithm; Applied mathematics; Orthogonalization; Iterative method
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