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Main Title: The Faber–Manteuffel theorem for linear operators
Author(s): Faber, Vance
Liesen, Jörg
Tichý, Petr
Type: Article
Language Code: en
Abstract: A short recurrence for orthogonalizing Krylov subspace bases for a matrix A exists if and only if the adjoint of A is a low-degree polynomial in A (i.e., A is normal of low degree). In the area of iterative methods, this result is known as the Faber–Manteuffel theorem [V. Faber and T. Manteuffel, SIAM J. Numer. Anal., 21 (1984), pp. 352–362]. Motivated by the description by J. Liesen and Z. Strakoš, we formulate here this theorem in terms of linear operators on finite dimensional Hilbert spaces and give two new proofs of the necessity part. We have chosen the linear operator rather than the matrix formulation because we found that a matrix-free proof is less technical. Of course, the linear operator result contains the Faber–Manteuffel theorem for matrices.
Issue Date: 2008
Date Available: 19-Dec-2017
DDC Class: 518 Numerische Analysis
Subject(s): cyclic subspaces
Krylov subspaces
orthogonal bases
short recurrences
normal matrices
Journal Title: SIAM Journal on Numerical Analysis
Publisher: Society for Industrial and Applied Mathematics
Publisher Place: Philadelphia, Pa
Volume: 46
Issue: 3
Publisher DOI: 10.1137/060678087
Page Start: 1323
Page End: 1337
EISSN: 1095-7170
ISSN: 0036-1429
Appears in Collections:FG Numerische Lineare Algebra » Publications

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