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 Mat. Sb., 2008, Volume 199, Number 2, Pages 27–48 (Mi msb3777)

Gauss–Arnoldi quadrature for $\langle(zI-A)^{-1}\varphi,\varphi\rangle$ and rational Padé-type approximation for Markov-type functions

L. A. Knizhnerman

Central Geophysical Expedition

Abstract: The efficiency of Gauss–Arnoldi quadrature for the calculation of the quantity $\langle(zI-A)^{-1}\varphi,\varphi\rangle$ is studied, where $A$ is a bounded operator in a Hilbert space and $\varphi$ is a non-trivial vector in this space. A necessary and a sufficient conditions are found for the efficiency of the quadrature in the case of a normal operator. An example of a non-normal operator for which this quadrature is inefficient is presented.
It is shown that Gauss–Arnoldi quadrature is related in certain cases to rational Padé-type approximation (with the poles at Ritz numbers) for functions of Markov type and, in particular, can be used for the localization of the poles of a rational perturbation. Error estimates are found, which can also be used when classical Padé approximation does not work or it may not be efficient.
Theoretical results and conjectures are illustrated by numerical experiments.
Bibliography: 44 titles.

DOI: https://doi.org/10.4213/sm3777

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English version:
Sbornik: Mathematics, 2008, 199:2, 185–206

Bibliographic databases:

UDC: 519.644+519.651+517.538.52
MSC: Primary 65J99, 41A21; Secondary 65F15

Citation: L. A. Knizhnerman, “Gauss–Arnoldi quadrature for $\langle(zI-A)^{-1}\varphi,\varphi\rangle$ and rational Padé-type approximation for Markov-type functions”, Mat. Sb., 199:2 (2008), 27–48; Sb. Math., 199:2 (2008), 185–206

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb3777
• https://doi.org/10.4213/sm3777
• http://mi.mathnet.ru/eng/msb/v199/i2/p27

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This publication is cited in the following articles:
1. S. P. Suetin, “Strong asymptotics of polynomials orthogonal with respect to a complex weight”, Sb. Math., 200:1 (2009), 77–93
2. S. P. Suetin, “On the Existence of Nonlinear Padé–Chebyshev Approximations for Analytic Functions”, Math. Notes, 86:2 (2009), 264–275
3. Bosuwan N., “on Montessus de Ballore'S Theorem For Nonlinear Pade-Orthogonal Approximants”, Jaen J. Approx., 8:2 (2016), 151–173
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