Rational Gauss Quadrature

Pranić, Miroslav S.; Reichel, Lothar

doi:10.1137/120902161

Cited by 10 publications

(8 citation statements)

References 18 publications

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“…As a consequence, it is not related to short recurrences. We also remark that similar properties have been derived in [52,Th. 3.1] for a different kind of rational Krylov subspaces.…”

Section: Applicationssupporting

confidence: 82%

“…However, while in the (standard) Arnoldi process this matrix is tridiagonal, the matrix J m is generally full. A detailed analysis of the structure and decay properties of the entries of J m can be found in [51]; see also [52]. The matrix J m appears in projection methods for solving problems such as linear and quadratic matrix equations and matrix functions evaluations, and it seems to require the whole matrix Q m for its computation.…”

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confidence: 99%

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The Short-term Rational Lanczos Method and Applications

Palitta¹,

Pozza²,

Simoncini³

2021

Preprint

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Rational Krylov subspaces have become a reference tool in dimension reduction procedures for several application problems. When data matrices are symmetric, a short-term recurrence can be used to generate an associated orthonormal basis. In the past this procedure was abandoned because it requires twice the number of linear system solves per iteration than with the classical long-term method. We propose an implementation that allows one to obtain key rational subspace matrices without explicitly storing the whole orthonormal basis, with a moderate computational overhead associated with sparse system solves. Several applications are discussed to illustrate the advantages of the proposed procedure.

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“…As a consequence, it is not related to short recurrences. We also remark that similar properties have been derived in [52,Th. 3.1] for a different kind of rational Krylov subspaces.…”

Section: Applicationssupporting

confidence: 82%

mentioning

confidence: 99%

The Short-term Rational Lanczos Method and Applications

Palitta¹,

Pozza²,

Simoncini³

2021

Preprint

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“…As already mentioned, a polynomial Krylov space K m+1 (A, v ) with orthonormal basis V m+1 is related to a decomposition of the form [29,36]. We prefer to work with the pencil H m , K m instead of the semiseparable plus diagonal representation since the former is widely used in practice.…”

Section: A Rational Implicit Q Theoremmentioning

confidence: 99%

Generalized Rational Krylov Decompositions with an Application to Rational Approximation

Berljafa¹,

Güttel

2015

SIAM J. Matrix Anal. & Appl.

102

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Abstract. Generalized rational Krylov decompositions are matrix relations which, under certain conditions, are associated with rational Krylov spaces. We study the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. Transformations on rational Krylov decompositions allow for changing the poles of a rational Krylov space without recomputation, and two algorithms are presented for this task. Using such transformations we develop a rational Krylov method for rational least squares fitting. Numerical experiments indicate that the proposed method converges fast and robustly. A MATLAB toolbox with implementations of the presented algorithms and experiments is provided.Key words. rational Krylov decomposition, inverse eigenvalue problem, rational approximation AMS subject classifications. 15A22, 65F15, 65F18, 30E10 DOI. 10.1137/140998081 1. Introduction. Numerical methods based on rational Krylov spaces have become an indispensable tool of scientific computing. Rational Krylov spaces were initially proposed by Ruhe in the 1980s for the purpose of solving large sparse eigenvalue problems [37,39,40]. Since then many more applications have been found in model order reduction [22,17], large-scale matrix functions and matrix equations [13,15,1,26,27], and nonlinear eigenvalue problems [41,30,47,28], to name a few.In this paper we study various algebraic properties of rational Krylov spaces, using as a starting point a generalized rational Krylov decomposition

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“…Therefore, the rational symmetric Lanczos method may require significantly fewer steps than the standard symmetric Lanczos method to deliver an approximation of (1.1) of desired quality. Because of this, the development and application of rational Lanczos methods has received considerable attention in the literature; see, e.g., [3,4,7,8,15,21,22,24,28]. The main drawback of the rational Lanczos method is the already mentioned need to solve linear systems of equations.…”

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confidence: 99%

“…}, then the matrix H p+q−1 ∈ R (p+q−1)×(p+q−1) is pentadiagonal. The structure of H p+q−1 obtained when two or more negative powers of A are used consecutively can be deduced from [28,Theorem 2.2].…”

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confidence: 99%

Convergence rates for inverse-free rational approximation of matrix functions

Jagels

Mach

Reichel

et al. 2016

Linear Algebra and its Applications

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This article deduces geometric convergence rates for approximating matrix functions via inversefree rational Krylov methods. In applications one frequently encounters matrix functions such as the matrix exponential or matrix logarithm; often the matrix under consideration is too large to compute the matrix function directly, and Krylov subspace methods are used to determine a reduced problem. If many evaluations of a matrix function of the form f(A)v with a large matrix A are required, then it may be advantageous to determine a reduced problem using rational Krylov subspaces. These methods may give more accurate approximations of f(A)v with subspaces of smaller dimension than standard Krylov subspace methods. Unfortunately, the system solves required to construct an orthogonal basis for a rational Krylov subspace may create numerical difficulties and/or require excessive computing time. This paper investigates a novel approach to determine an orthogonal basis of an approximation of a rational Krylov subspace of (small) dimension from a standard orthogonal Krylov subspace basis of larger dimension. The approximation error will depend on properties of the matrix A and on the dimension of the original standard Krylov subspace. We show that our inverse-free method for approximating the rational Krylov subspace converges geometrically (for increasing dimension of the standard Krylov subspace) to a rational Krylov subspace. The convergence rate may be used to predict the dimension of the standard Krylov subspace necessary to obtain a certain accuracy in the approximation. Computed examples illustrate the theory developed. Original language English Pages (from-to)291-310

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Rational Gauss Quadrature

Cited by 10 publications

References 18 publications

The Short-term Rational Lanczos Method and Applications

The Short-term Rational Lanczos Method and Applications

Generalized Rational Krylov Decompositions with an Application to Rational Approximation

Convergence rates for inverse-free rational approximation of matrix functions

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