Error Bounds for Lanczos-Based Matrix Function Approximation

Chen, Tyler; Greenbaum, Anne; Musco, Cameron; Musco, Christopher

doi:10.1137/21m1427784

Cited by 7 publications

(4 citation statements)

References 35 publications

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“…In [CGMM23] an algorithm called the Lanczos method for optimal rational function approximation (Lanczos-OR) was developed. For rational functions of the form (7), when k > deg(n), Lanczos-OR produces iterates…”

Section: Comparison To Lanczos-ormentioning

confidence: 99%

“…This suggests that, for this class of functions, the A-norm is more natural. As in work on the Lanczos-OR method [CGMM23], it may be easier to prove strong near optimality bounds for different norms, perhaps depending on r.…”

Section: Construction Of Hard Instances / Refined Upper Boundsmentioning

confidence: 99%

“…We remark that by applying Lanczos-OR to term in the partial fraction decomposition of the approximating rational function, an approximation to the sign can be obtained[CGMM23]. This iterate seems to avoid the oscillations of Lanczos-FA.…”

mentioning

confidence: 95%

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Near-Optimality Guarantees for Approximating Rational Matrix Functions by the Lanczos Method

Amsel¹,

Chen²,

Greenbaum³

et al. 2023

Preprint

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We study the Lanczos method for approximating the action of a symmetric matrix function f (A) on a vector b (Lanczos-FA). For the function A −1 , it is known that the error of Lanczos-FA after k iterations matches the error of the best approximation from the Krylov subspace of degree k when A is positive definite. We prove that the same holds, up to a multiplicative approximation factor, when f is a rational function with no poles in the interval containing A's eigenvalues. The approximation factor depends the degree of f 's denominator and the condition number of A, but not on the number of iterations k. Experiments confirm that our bound accurately predicts the convergence of Lanczos-FA. Moreover, we believe that our result provides strong theoretical justification for the excellent practical performance that has long by observed of the Lanczos method, both for approximating rational functions and functions like A −1/2 b that are well approximated by rationals.

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Section: Comparison To Lanczos-ormentioning

confidence: 99%

Section: Construction Of Hard Instances / Refined Upper Boundsmentioning

confidence: 99%

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Near-Optimality Guarantees for Approximating Rational Matrix Functions by the Lanczos Method

Amsel¹,

Chen²,

Greenbaum³

et al. 2023

Preprint

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show abstract

“…This phenomenon is analogous to the fact that, when solving a positive definite linear system of equations, the conjugate gradient algorithm can converge significantly faster than Chebyshev iteration. Sharper error bounds for Lanczos-based Gaussian quadrature can be obtained through non-Gaussian quadrature rules [BFG96], interlacing type properties of Gaussian quadrature rules [CTU21], as well as reduction to the error of a certain linear system [Che+22].…”

Section: Main Boundsmentioning

confidence: 99%

Randomized matrix-free quadrature for spectrum and spectral sum approximation

Chen¹,

Trogdon²,

Ubaru³

2022

Preprint

Self Cite

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We study randomized matrix-free quadrature algorithms for spectrum and spectral sum approximation. The algorithms studied are characterized by the use of a Krylov subspace method to approximate independent and identically distributed samples of v H f [A]v, where v is an isotropic random vector, A is a Hermitian matrix, and f [A] is a matrix function. This class of algorithms includes the kernel polynomial method and stochastic Lanczos quadrature, two widely used methods for approximating spectra and spectral sums. Our analysis, discussion, and numerical examples provide a unified framework for understanding randomized matrix-free quadrature and shed light on the commonalities and tradeoffs between them. Moreover, this framework provides new insights into the practical implementation and use of these algorithms, particularly with regards to parameter selection in the kernel polynomial method.

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Error bounds for the approximation of matrix functions with rational Krylov methods

Simunec

2024

Numerical Linear Algebra App

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We obtain an expression for the error in the approximation of and with rational Krylov methods, where is a symmetric matrix, is a vector and the function admits an integral representation. The error expression is obtained by linking the matrix function error with the error in the approximate solution of shifted linear systems using the same rational Krylov subspace, and it can be exploited to derive both a priori and a posteriori error bounds. The error bounds are a generalization of the ones given in Chen et al. for the Lanczos method for matrix functions. A technique that we employ in the rational Krylov context can also be applied to refine the bounds for the Lanczos case.

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Error Bounds for Lanczos-Based Matrix Function Approximation

Cited by 7 publications

References 35 publications

Near-Optimality Guarantees for Approximating Rational Matrix Functions by the Lanczos Method

Near-Optimality Guarantees for Approximating Rational Matrix Functions by the Lanczos Method

Randomized matrix-free quadrature for spectrum and spectral sum approximation

Error bounds for the approximation of matrix functions with rational Krylov methods

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