A Krylov--Schur Algorithm for Large Eigenproblems

Stewart, G. W.

doi:10.1137/s0895479800371529

Cited by 437 publications

(407 citation statements)

References 14 publications

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“…Since convergence may be slow, it is necessary to restart the method, that is, discard part of the information contained in the subspace and extend the subspace again. We use the Krylov-Schur restart [13]. We will not describe the algorithm in detail here, just enumerate the main computational kernels:…”

Section: Krylov Methods For Block-tridiagonal Matricesmentioning

confidence: 99%

GPU Implementation of Krylov Solvers for Block-Tridiagonal Eigenvalue Problems

Daviña

Román

2016

Parallel Processing and Applied Mathematics

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Abstract. In an eigenvalue problem defined by one or two matrices with block-tridiagonal structure, if only a few eigenpairs are required it is interesting to consider iterative methods based on Krylov subspaces, even if matrix blocks are dense. In this context, using the GPU for the associated dense linear algebra may provide high performance. We analyze this in an implementation done in the context of SLEPc, the Scalable Library for Eigenvalue Problem Computations. In the case of a generalized eigenproblem or when interior eigenvalues are computed with shift-andinvert, the main computational kernel is the solution of linear systems with a block-tridiagonal matrix. We explore possible implementations of this operation on the GPU, including a block cyclic reduction algorithm.

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Section: Krylov Methods For Block-tridiagonal Matricesmentioning

confidence: 99%

GPU Implementation of Krylov Solvers for Block-Tridiagonal Eigenvalue Problems

Daviña

Román

2016

Parallel Processing and Applied Mathematics

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“…For simplicity, we neglect this possibility in the following description and refer to [19] for details. Similar to the previous section, we will now discuss the efficient implementation of steps 4-6 of the Arnoldi method (Algorithm 1) applied to A −1 , and the expansion of the basis representation (26).…”

Section: Two-level Orthogonalization Arnoldi (Toar)mentioning

confidence: 99%

“…The Krylov-Schur method [26] is based on generalizing the Arnoldi decomposition (24) to a Krylov decomposition…”

Section: Krylov-schur Restartmentioning

confidence: 99%

“…has rank d + p. This is due to its relation to the truncated Krylov-Schur decomposition (34), which can be retransformed into a standard Arnoldi decomposition [26] of order p (with different starting vector) and hence Theorem 4 applies. By, e.g., a truncated SVD, we can find an orthonormal matrix W ∈ C (d+k)×(d+p) such that its columns span the row space of the matrix in (35).…”

Section: Krylov-schur Restartmentioning

confidence: 99%

“…This is exactly the form of the usual representation (26) in the TOAR method, allowing us to expand the truncated Arnoldi decomposition (34) using the techniques of Algorithm 6.…”

Section: Krylov-schur Restartmentioning

confidence: 99%

See 2 more Smart Citations

Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Kreßner

Román

2013

Numer. Linear Algebra Appl.

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Novel memory-efficient Arnoldi algorithms for solving matrix polynomial eigenvalue problems are presented. More specifically, we consider the case of matrix polynomials expressed in the Chebyshev basis, which is often numerically more appropriate than the standard monomial basis for a larger degree d. The standard way of solving polynomial eigenvalue problems proceeds by linearization, which increases the problem size by a factor d. Consequently, the memory requirements of Krylov subspace methods applied to the linearization grow by this factor. In this paper, we develop two variants of the Arnoldi method that build the Krylov subspace basis implicitly, in a way that only vectors of length equal to the size of the original problem need to be stored. The proposed variants are generalizations of the so called Q-Arnoldi and TOAR methods, which have been developed for the monomial case. We also show how the typical ingredients of a full implementation of the Arnoldi method, including shift-and-invert and restarting, can be incorporated. Numerical experiments are presented for matrix polynomials up to degree 30 arising from the interpolation of nonlinear eigenvalue problems which stem from boundary element discretizations of PDE eigenvalue problems.

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Limited‐memory polynomial methods for large‐scale matrix functions

Güttel,

Kressner,

Lund

2020

GAMM-Mitteilungen

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Matrix functions are a central topic of linear algebra, and problems requiring their numerical approximation appear increasingly often in scientific computing. We review various limited‐memory methods for the approximation of the action of a large‐scale matrix function on a vector. Emphasis is put on polynomial methods, whose memory requirements are known or prescribed a priori. Methods based on explicit polynomial approximation or interpolation, as well as restarted Arnoldi methods, are treated in detail. An overview of existing software is also given, as well as a discussion of challenging open problems.

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A Krylov--Schur Algorithm for Large Eigenproblems

Cited by 437 publications

References 14 publications

GPU Implementation of Krylov Solvers for Block-Tridiagonal Eigenvalue Problems

GPU Implementation of Krylov Solvers for Block-Tridiagonal Eigenvalue Problems

Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Limited‐memory polynomial methods for large‐scale matrix functions

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