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

1986/07/01 by Youcef Saad, Martin H. Schultz · 20 citations
Computer Science · Mathematics · Physics and Astronomy · #Electromagnetic Scattering and Analysis #Matrix Theory and Algorithms #Statistical and numerical algorithms

paper · doi:10.1137/0907058

openalex publication_date 1986/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

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.

Citations

• Solution of Sparse Indefinite Systems of Linear Equations

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