On Moment Methods in Krylov Subspaces

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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.

Original languageEnglish
Pages (from-to)478-482
Number of pages5
JournalDoklady Mathematics
Issue number3
Publication statusPublished - Nov 2020


  • conjugate direction algorithms
  • Krylov subspace
  • moment method
  • parametric Lanczos orthogonalization




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