Given an n × n symmetric positive definite (SPD) matrix A and an SPD preconditioner P, we propose a new class of generalized block tuned (GBT) preconditioners. These are defined as a p-rank correction of P with the property that arbitrary (positive) parameters γ1, …, γp are eigenvalues of the preconditioned matrix. We propose to employ these GBT preconditioners to accelerate the iterative solution of linear systems like (A − θI)s = r in the framework of iterative eigensolvers. We give theoretical evidence that a suitable, and effective, choice of the scalars γj is able to shift p eigenvalues of P(A − θI) very close to one. Numerical experiments on various matrices of very large size show that the proposed preconditioner is able to yield an almost constant number of iterations, for different eigenpairs, irrespective of the relative separation between consecutive eigenvalues. We also give numerical evidence that the GBT preconditioner is always far superior to the spectral preconditioner (Numer. Linear Algebra Appl. 24(3):1–14, 2017), on matrices with highly clustered eigenvalues.
Eigenvalues SPD matrix Newton method Tuned preconditioner Incomplete Cholesky preconditioner
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This work has been supported by the Italian project CPDA155834/15: “Stable and efficient discretizations of the mechanics of faults” and by the Italian INdAM-GNCS Project Metodi numerici per problemi di ottimizzazione vincolata di grandi dimensioni e applicazioni (2017). We wish to thank the anonymous reviewers whose comments and suggestions helped improve the quality of the paper.
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