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

Authors

Youcef Saad, Martin H. Schultz

Year

1986

Journal Paper

Algorithm

GMRES

Source

SIAM Journal on Scientific and Statistical Computing

Identifiers

DOI 10.1137/0907058

Abstract¶

We present an iterative method for solving linear systems, which has the property of minimizing 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.

Summary¶

Saad and Schultz introduce GMRES, a Krylov-subspace method for nonsymmetric linear systems that minimizes the residual norm over the Arnoldi basis at each iteration. The paper is foundational for modern iterative linear solvers, especially large sparse nonsymmetric systems where conjugate gradients does not apply.