ADAPTIVELY PRECONDITIONED GMRES ALGORITHMS

The restarted GMRES algorithm proposed by Saad and Schultz [SIAM J. Sc i. Statist. Comput., 7 (1986), pp. 856-869] is one of the most popular iterative methods for the solution of large linear systems of equatio ns Ax = b with a nonsymmetric and sparse matrix. This algorithm is par ticularly attractive when a good preconditioner is available. The pres ent paper describes two new methods for determining preconditioners fr om spectral information gathered by the Arnoldi process during iterati ons by the restarted GMRES algorithm. These methods seek to determine an invariant subspace of the matrix A associated with eigenvalues clos e to the origin and to move these eigenvalues so that a higher rate of convergence of the iterative methods is achieved.