Abstract
In the present paper, we are concerned by weighted Arnoldi like methods for solving large and sparse linear systems that have different right-hand sides but have the same coefficient matrix. We first give detailed descriptions of the weighted Gram-Schmidt process and of a Ruhe variant of the weighted block Arnoldi algorithm. We also establish some theoretical results that links the iterates of the weighted block Arnoldi process to those of the non weighted one. Then, to accelerate the convergence of the classical restarted block and seed GMRES methods, we introduce the weighted restarted block and seed GMRES methods. Numerical experiments that are done with different matrices coming from the Matrix Market repository or from the university of Florida sparse matrix collection are reported at the end of this work in order to compare the performance and show the effectiveness of the proposed methods.
Highlights
IntroductionWe are interested in solving multiple linear systems that have the same coefficient matrix and which have the form
In this paper, we are interested in solving multiple linear systems that have the same coefficient matrix and which have the form A X = B, (1.1)where A is a non-symmetric real square matrix of size n and B := [b(1), . . . , b(s)], X := [x(1), . . . , x(s)] are n × s real rectangular matrices, such that the block righthand side B is full rank and s ≪ n.Block linear systems of kind (1.1) are encountered in many problems of scientific computing and engineering applications
To compare the performance of the proposed methods when they are combined with a preconditioning strategy, we report in Table 3 the results obtained when comparing the preconditioned restarted block GMRES (denoted here by PBl-GMRES(m)), the preconditioned WBl-GMRES method (denoted here by PWBl-GMRES(m)) and the preconditioned WSGMRES method (denoted here by PWSGMRES(m))
Summary
We are interested in solving multiple linear systems that have the same coefficient matrix and which have the form. Block Krylov methods are more efficient when they are applied to relatively dense linear systems and combined with some preconditioning techniques [9]. The numerical experiments we have conducted show that the convergence of the weighted Bl-GMRES method may be affected by the occurrence of a linear dependency between the columns of the Krylov matrix This led us to apply the weighting technique to the seed GMRES algorithm and to propose the weighted seed GMRES method as an improvement of the classical seed GMRES algorithm. These results generalize to the block case the results obtained in [3] for the case of a single linear system. Tr(XT D X), which is associated to the D-inner product < X, Y >D
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