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Krylov Subspace MethodsPrinciples and Analysis$
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Jörg Liesen and Zdenek Strakos

Print publication date: 2012

Print ISBN-13: 9780199655410

Published to Oxford Scholarship Online: January 2013

DOI: 10.1093/acprof:oso/9780199655410.001.0001

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Matching Moments and Model Reduction View

Matching Moments and Model Reduction View

(p.71) 3 Matching Moments and Model Reduction View
Krylov Subspace Methods

Jörg Liesen

Zdenek Strakos

Oxford University Press

The projected system matrix in Krylov subspace methods consists of moments of the original system matrix with respect to the initial residual(s). This hints that Krylov subspace methods can be viewed as matching moments model reduction. Through the simplified Stieltjes moment problem, orthogonal polynomials, continued fractions, and Jacobi matrices, we thus obtain the Gauss–Christoffel quadrature representation of the conjugate gradient method (CG). It is described how generalisations to the non-Hermitian case can easily be achieved using the Vorobyev method of moments. Finally, the described results and their historical roots are linked with the model reduction of large-scale dynamical systems. The chapter demonstrates the strong connection between Krylov subspace methods used in state-of-the-art numerical calculations and classical topics of analysis and approximation theory. Since moments represent very general objects, this suggests that Krylov subspace methods might have much wider applications beyond their immediate context of solving algebraic problems.

Keywords:   moment problem, model reduction, Hauss-Christoffel quadrature, continued fractions, orthogonal polynomials, Jacobi matrices, Vorobyev method of moments, estimates in quadratic forms, minimal partial realisation, P. l. Chebyshev, A. A. Markoff, T. J. Stieltjes

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