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Eurasian Math. J., 2012, Volume 3, Number 4, Pages 53–80 (Mi emj105)  

This article is cited in 1 scientific paper (total in 1 paper)

Krylov subspace methods of approximate solving differential equations from the point of view of functional calculus

V. G. Kurbatova, I. V. Kurbatovab

a Finance University under the Government of the Russian Federation, Lipetsk, Russia
b Air Force Academy named after professor N. E. Zhukovsky and Y. A. Gagarin, Voronezh, Russia

Abstract: The paper deals with projection methods of approximate solving the problem
$$ Fx'=Gx+bu(t),\qquad y=\langle x,d\rangle $$
which consist in passage to the reduced-order problem
$$ \widehat F\hat x'=\widehat G\hat x+\hat bu(t),\qquad \hat y=\langle\hat x,\hat d\rangle, $$
where
$$ \widehat F=\Lambda FV,\qquad\widehat G=\Lambda GV,\qquad\hat b=\Lambda b,\qquad\hat d=V^*d. $$
It is shown that, if $V$ and $\Lambda$ are constructed on the basis of Krylov's subspaces, a projection method is equivalent to the replacement in the formula expressing the impulse response via the exponential function of the pencil $\lambda\mapsto\lambda F-G$, of the exponential function by its rational interpolation satisfying some interpolation conditions. Special attention is paid to the case when $F$ is not invertible.

Keywords and phrases: Krylov subspaces, Lanczos and Arnoldi methods, differential-algebraic equation, reduced-order system, functional calculus, rational interpolation, operator pencil, pseudoresolvent.

Full text: PDF file (586 kB)
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Bibliographic databases:
MSC: 65L80, 47A58, 41A20
Received: 20.11.2012
Language:

Citation: V. G. Kurbatov, I. V. Kurbatova, “Krylov subspace methods of approximate solving differential equations from the point of view of functional calculus”, Eurasian Math. J., 3:4 (2012), 53–80

Citation in format AMSBIB
\Bibitem{KurKur12}
\by V.~G.~Kurbatov, I.~V.~Kurbatova
\paper Krylov subspace methods of approximate solving differential equations from the point of view of functional calculus
\jour Eurasian Math. J.
\yr 2012
\vol 3
\issue 4
\pages 53--80
\mathnet{http://mi.mathnet.ru/emj105}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3040687}
\zmath{https://zbmath.org/?q=an:1267.65098}


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    This publication is cited in the following articles:
    1. M. N. Oreshina, “Spectral decomposition of normal operator in real Hilbert space”, Ufa Math. J., 9:4 (2017), 85–96  mathnet  crossref  isi  elib
  • Eurasian Mathematical Journal
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