TY - JOUR

T1 - On Moment Methods in Krylov Subspaces

AU - Il’in, V. P.

N1 - Funding Information: This work was supported by the Russian Foundation for Basic Research, project no. 18-01-00295. Publisher Copyright: © 2020, Pleiades Publishing, Ltd. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2020/11

Y1 - 2020/11

N2 - Moment methods in Krylov subspaces for solving symmetric systems of linear algebraic equations (SLAEs) are considered. A family of iterative algorithms is proposed based on generalized Lanczos orthogonalization with an initial vector $${{v}^{0}}$$ chosen regardless of the initial residual. By applying this approach, a series of SLAEs with the same matrix, but with different right-hand sides can be solved using a single set of basis vectors. Additionally, it is possible to implement generalized moment methods that reduce to block Krylov algorithms using a set of linearly independent guess vectors v10,..,. The performance of algorithm implementations is improved by reducing the number of matrix multiplications and applying efficient parallelization of vector operations. It is shown that the applicability of moment methods can be extended using preconditioning to various classes of algebraic systems: indefinite, incompatible, asymmetric, and complex, including non-Hermitian ones.

AB - Moment methods in Krylov subspaces for solving symmetric systems of linear algebraic equations (SLAEs) are considered. A family of iterative algorithms is proposed based on generalized Lanczos orthogonalization with an initial vector $${{v}^{0}}$$ chosen regardless of the initial residual. By applying this approach, a series of SLAEs with the same matrix, but with different right-hand sides can be solved using a single set of basis vectors. Additionally, it is possible to implement generalized moment methods that reduce to block Krylov algorithms using a set of linearly independent guess vectors v10,..,. The performance of algorithm implementations is improved by reducing the number of matrix multiplications and applying efficient parallelization of vector operations. It is shown that the applicability of moment methods can be extended using preconditioning to various classes of algebraic systems: indefinite, incompatible, asymmetric, and complex, including non-Hermitian ones.

KW - conjugate direction algorithms

KW - Krylov subspace

KW - moment method

KW - parametric Lanczos orthogonalization

UR - http://www.scopus.com/inward/record.url?scp=85102503403&partnerID=8YFLogxK

U2 - 10.1134/S1064562420060241

DO - 10.1134/S1064562420060241

M3 - Article

AN - SCOPUS:85102503403

VL - 102

SP - 478

EP - 482

JO - Doklady Mathematics

JF - Doklady Mathematics

SN - 1064-5624

IS - 3

ER -