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Titolo:
Preconditioners for ill-conditioned Toeplitz systems constructed from positive kernels
Autore:
Potts, D; Steidl, G;
Indirizzi:
Med Univ Lubeck, Inst Math, D-23560 Lubeck, Germany Med Univ Lubeck Lubeck Germany D-23560 nst Math, D-23560 Lubeck, Germany Univ Mannheim, Inst Math, D-68131 Mannheim, Germany Univ Mannheim Mannheim Germany D-68131 t Math, D-68131 Mannheim, Germany
Titolo Testata:
SIAM JOURNAL ON SCIENTIFIC COMPUTING
fascicolo: 5, volume: 22, anno: 2001,
pagine: 1741 - 1761
SICI:
1064-8275(20010208)22:5<1741:PFITSC>2.0.ZU;2-X
Fonte:
ISI
Lingua:
ENG
Soggetto:
CIRCULANT PRECONDITIONERS; SINE TRANSFORMS; FAST ALGORITHMS; MATRICES; COSINE;
Keywords:
ill-conditioned Toeplitz matrices; CG method; preconditioners; reproducing kernels;
Tipo documento:
Article
Natura:
Periodico
Settore Disciplinare:
Physical, Chemical & Earth Sciences
Citazioni:
46
Recensione:
Indirizzi per estratti:
Indirizzo: Potts, D Med Univ Lubeck, Inst Math, Wallstr 40, D-23560 Lubeck, Germany Med Univ Lubeck Wallstr 40 Lubeck Germany D-23560 ubeck, Germany
Citazione:
D. Potts e G. Steidl, "Preconditioners for ill-conditioned Toeplitz systems constructed from positive kernels", SIAM J SC C, 22(5), 2001, pp. 1741-1761

Abstract

In this paper, we are interested in the iterative solution of ill-conditioned Toeplitz systems generated by continuous nonnegative real-valued functions f with a finite number of zeros. We construct new w-circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f * K-N with a suitable positive reproducing kernel K-N. By the restriction to positive kernels we obtain positive definite preconditioners. Moreover, if f has only zeros of even order less than or equal to 2s, then we can prove that the property integral (pi)(-pi) t(2k) K-N(t) dt less than or equal to CN-2k (k = 0,..., s) of the kernel is necessary and sufficient to ensure the convergence of the PCG method in a number of iteration steps independent of the dimension N of the system. Our theoretical results were confirmed by numerical tests.

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Documento generato il 26/01/20 alle ore 01:09:24