About: On computing quadrature-based bounds for the A-norm of the error in conjugate gradients

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
Description	In their original paper, Golub and Meurant (BIT 37:687–705, 1997) suggest to compute bounds for the A-norm of the error in the conjugate gradient (CG) method using Gauss, Gauss-Radau and Gauss-Lobatto quadratures. The quadratures are computed using the (1,1)-entry of the inverse of the corresponding Jacobi matrix (or its rank-one or rank-two modifications). The resulting algorithm called CGQL computes explicitly the entries of the Jacobi matrix and its modifications from the CG coefficients. In this paper, we use the fact that CG computes the Cholesky decomposition of the Jacobi matrix which is given implicitly. For Gauss-Radau and Gauss-Lobatto quadratures, instead of computing the entries of the modified Jacobi matrices, we directly compute the entries of the Cholesky decompositions of the (modified) Jacobi matrices. This leads to simpler formulas in comparison to those used in CGQL. In their original paper, Golub and Meurant (BIT 37:687–705, 1997) suggest to compute bounds for the A-norm of the error in the conjugate gradient (CG) method using Gauss, Gauss-Radau and Gauss-Lobatto quadratures. The quadratures are computed using the (1,1)-entry of the inverse of the corresponding Jacobi matrix (or its rank-one or rank-two modifications). The resulting algorithm called CGQL computes explicitly the entries of the Jacobi matrix and its modifications from the CG coefficients. In this paper, we use the fact that CG computes the Cholesky decomposition of the Jacobi matrix which is given implicitly. For Gauss-Radau and Gauss-Lobatto quadratures, instead of computing the entries of the modified Jacobi matrices, we directly compute the entries of the Cholesky decompositions of the (modified) Jacobi matrices. This leads to simpler formulas in comparison to those used in CGQL. (en)
Title	On computing quadrature-based bounds for the A-norm of the error in conjugate gradients On computing quadrature-based bounds for the A-norm of the error in conjugate gradients (en)
skos:prefLabel	On computing quadrature-based bounds for the A-norm of the error in conjugate gradients On computing quadrature-based bounds for the A-norm of the error in conjugate gradients (en)
skos:notation	RIV/67985807:_____/13:00390209!RIV13-AV0-67985807
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(IAA100300802), Z(AV0Z10300504)
http://linked.open...iv/cisloPeriodika	2
http://linked.open...vai/riv/dodaniDat	2013
http://linked.open...aciTvurceVysledku	Tichý, Petr
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Ústav informatiky AV ČR, v. v. i.
http://linked.open...dnocenehoVysledku	93670
http://linked.open...ai/riv/idVysledku	RIV/67985807:_____/13:00390209
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	conjugate gradients; norm of the error; bounds for the error norm (en)
http://linked.open.../riv/klicoveSlovo	conjugate gradients bounds for the error norm norm of the error
http://linked.open...odStatuVydavatele	NL - Nizozemsko
http://linked.open...ontrolniKodProRIV	[4DFD979005A3]
http://linked.open...i/riv/nazevZdroje	Numerical Algorithms
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	2 (xsd:int)
http://linked.open...vavai/riv/projekt	Theory of Krylov subspace methods and its relationship to other mathematical disciplines
http://linked.open...UplatneniVysledku	2013
http://linked.open...v/svazekPeriodika	62
http://linked.open...iv/tvurceVysledku	Tichý, Petr Meurant, G.
http://linked.open...ain/vavai/riv/wos	000314058700001
http://linked.open...n/vavai/riv/zamer	Informatika pro informační společnost: modely, algoritmy, aplikace
issn	1017-1398
number of pages	29 (xsd:int)
http://bibframe.org/vocab/doi	10.1007/s11075-012-9591-9

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