Generalized least-squares polynomial preconditioners for symmetric indefinite linear equations

Abstract

Polynomial preconditioners are frequently used in a parallel environment for the computation of the solution of large-scale sparse linear equations ( Ax= b) because of their easy implementation and trivial parallelization. With respect to symmetrical indefinite (SID) linear systems, the use of generalized least-squares (GLS) polynomial preconditioning is preferable to other polynomial preconditioning methods because of the ability to use a three-term recurrence relationship and the low implementation costs. The GLS preconditioning polynomial and its influence on the flexible generalized minimized residual (FGMRES) solver are discussed in this paper. The orthogonal polynomials required in the solution of the least-squares approximation problem are constructed using the Stieltjes procedure in multiple disjoint intervals which exclude the origin. The time-consuming numerical integration associated with this procedure is computed efficiently using Chebyshev polynomials of the first kind and the GLS polynomial reconditioned FGMRES algorithm is implemented using MPI in a highly parallel IBM SP2 environment. Experimental results using classical benchmark systems are presented and compared with those obtained using the recently developed SPAI preconditioned Bi-CGSTAB iterative method. The performance of the GLS preconditioned FGMRES solver is critically accessed.