Some Variants of the GMRES Iterative Methods and Other Krylov Subspace Iterative Methods
by
David R. Kincaid
Center for Numerical Analysis, Texas Institute for Computational and Appled Mathematics, The University of Texas at Austin, Texas 78712, U.S.A.

We are interested in iterative methods for solving systems of linear equations of the form Au=b, where A is a large sparse nonsingular matrix. When A is symmetric positive definite, conjugate-gradient-type methods are often used and are fairly well understood. On the other hand, when A is nonsymmetric, the choice of iterative method is much more difficult.

We discuss variants of the Generalized Minimal Residual (GMRES) method such as the GGMRES method, which is a slight generalization of the GMRES method, and the MGMRES method, which is a modifications of the GMRES method, as well as another method-LAN/MGMRES. Instead of using a minimization process as in GGMRES, we use a Galerkin condition to derive the MGMRES method. The LAN/MGMRES method is designed to combine the reliability of GMRES with the reduced work of a Lanczos-type method. [J-Y. Chen, D. R. Kincaid, and D. M. Young, Generalizations and modifications of the GMRES iterative method, Numerical Algorithms 1999, 21:119-146.]

We also discuss several other Krylov subspace iterative methods for the approximation of the solution of general nonsymmetric indefinite linear systems such as a restarted version and a truncated version of the ODIR method. [A. T. Chronopoulos and D. R. Kincaid, On the Odir iterative method for non-symmetric indefinite linear systems, Numerical Linear Algebra with Applications 2001, 8:71-82.]

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Date received: March 21, 2001