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Statements

Subject Item
n2:RIV%2F67985807%3A_____%2F13%3A00390209%21RIV13-AV0-67985807
rdf:type
n16:Vysledek skos:Concept
dcterms: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.
dcterms: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
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
skos:notation
RIV/67985807:_____/13:00390209!RIV13-AV0-67985807
n16:predkladatel
n17:ico%3A67985807
n3:aktivita
n5:Z n5:P
n3:aktivity
P(IAA100300802), Z(AV0Z10300504)
n3:cisloPeriodika
2
n3:dodaniDat
n18:2013
n3:domaciTvurceVysledku
n10:2085674
n3:druhVysledku
n11:J
n3:duvernostUdaju
n9:S
n3:entitaPredkladatele
n12:predkladatel
n3:idSjednocenehoVysledku
93670
n3:idVysledku
RIV/67985807:_____/13:00390209
n3:jazykVysledku
n20:eng
n3:klicovaSlova
conjugate gradients; norm of the error; bounds for the error norm
n3:klicoveSlovo
n14:bounds%20for%20the%20error%20norm n14:conjugate%20gradients n14:norm%20of%20the%20error
n3:kodStatuVydavatele
NL - Nizozemsko
n3:kontrolniKodProRIV
[4DFD979005A3]
n3:nazevZdroje
Numerical Algorithms
n3:obor
n6:BA
n3:pocetDomacichTvurcuVysledku
1
n3:pocetTvurcuVysledku
2
n3:projekt
n4:IAA100300802
n3:rokUplatneniVysledku
n18:2013
n3:svazekPeriodika
62
n3:tvurceVysledku
Meurant, G. Tichý, Petr
n3:wos
000314058700001
n3:zamer
n13:AV0Z10300504
s:issn
1017-1398
s:numberOfPages
29
n19:doi
10.1007/s11075-012-9591-9