Please use this identifier to cite or link to this item: http://dx.doi.org/10.14279/depositonce-6568
|Main Title:||Orthogonal Hessenberg reduction and orthogonal Krylov subspace bases|
Saylor, Paul E.
|Abstract:||We study necessary and sufficient conditions that a nonsingular matrix A can be B-orthogonally reduced to upper Hessenberg form with small bandwidth. By this we mean the existence of a decomposition AV=VH, where H is upper Hessenberg with few nonzero bands, and the columns of V are orthogonal in an inner product generated by a hermitian positive definite matrix B. The classical example for such a decomposition is the matrix tridiagonalization performed by the hermitian Lanczos algorithm, also called the orthogonal reduction to tridiagonal form. Does there exist such a decomposition when A is nonhermitian? In this paper we completely answer this question. The related (but not equivalent) question of necessary and sufficient conditions on A for the existence of short-term recurrences for computing B-orthogonal Krylov subspace bases was completely answered by the fundamental theorem of Faber and Manteuffel [SIAM J. Numer. Anal.}, 21 (1984), pp. 352--362]. We give a detailed analysis of B-normality, the central condition in both the Faber--Manteuffel theorem and our main theorem, and show how the two theorems are related. Our approach uses only elementary linear algebra tools. We thereby provide new insights into the principles behind Krylov subspace methods, that are not provided when more sophisticated tools are employed.|
|DDC Class:||DDC::500 Naturwissenschaften und Mathematik::510 Mathematik::518 Numerische Analysis|
DDC::500 Naturwissenschaften und Mathematik::510 Mathematik::512 Algebra
Krylov subspace methods
|Usage rights:||Terms of German Copyright Law|
|Journal Title:||SIAM Journal on Numerical Analysis|
|Publisher:||Society for Industrial and Applied Mathematics|
|Publisher Place:||Philadelphia, Pa|
|Appears in Collections:||Fachgebiet Numerische Lineare Algebra » Publications|
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