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On the sensitivity of Lanczos recursions to the spectrum

โœ Scribed by Vladimir Druskin; Liliana Borcea; Leonid Knizhnerman


Publisher
Elsevier Science
Year
2005
Tongue
English
Weight
320 KB
Volume
396
Category
Article
ISSN
0024-3795

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๐Ÿ“œ SIMILAR VOLUMES


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For a given real n ร— n matrix A and initial vectors v 1 and w 1 , we examine the sensitivity of the tridiagonal matrix T and the biorthogonal sets of vectors of the Lanczos reduction to small changes in A, v 1 and w 1 . We also consider the sensitivity of the developing Krylov subspaces.

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We prove strictly monotonic error decrease in the Euclidian norm of the Krylov subspace approximation of exp(A)ฯ•, where ฯ• and A are respectively a vector and a symmetric matrix. In addition, we show that the norm of the approximate solution grows strictly monotonically with the subspace dimension.

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Stability of passing from Gaussian quadrature data to the Lanczos recurrence coefficients is considered. Special attention is paid to estimates explicitly expressed in terms of quadrature data and not having weights in denominators. It has been shown that the recent approach, exploiting integral rep