# GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems

**Youcef Saad****Martin H. Schultz**

CiteWeb id: 19860000022

CiteWeb score: 9587

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 /2-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. For solving indefinite symmetric systems, Paige and Saunders (10) proposed an approach which exploits the relationship between the conjugate gradient method and the Lanczos method. In particular, it is known that the Lanczos method for solving the eigenvalue problem for an N xN matrix A is a Galerkin method onto the Krylov subspace Kk =- span{v1, AVl,"" ", Ak-lvl}, while the conjugate gradient method is a Galerkin method for solving the linear system Ax =f, onto the Krylov subspace Kk with v to/II roll. Thus, the Lanczos method computes the matrix representation Tk of the linear operator PkAII, the restriction of PkA to Kk, where Pk is the 12-0rthogonal projector onto Kk. The Galerkin method for Ax=f in Kk leads to solving a linear system with the matrix Tk which is tridiagonal if A is symmetric. In general, Tk is indefinite when A is and some stable direct method must be used to solve the corresponding tridiagonal Galerkin system. The basis ofPaige and Saunders'SYMMLQ algorithm is to use the stable LQ factorization of T. Paige and Saunders also showed that it is possible to formulate an algorithm called MINRES using the Lanczos basis to compute an approximate solution Xk which minimizes the residual norm over the Krylov subspace K. In the present paper we introduce and analyse a generalization of the MINRES algorithm for solving nonsymmetric linear systems. This generalization is based on the Arnoldi process 1 ), 12) which is an analogue of the Lanczos algorithm for nonsym-

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Youcef Saad, Martin H. Schultz,

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