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Statements

Subject Item
n2:RIV%2F67985807%3A_____%2F11%3A00358802%21RIV12-AV0-67985807
rdf:type
n4:Vysledek skos:Concept
dcterms:description
Let $A$ be a nonsingular complex matrix and $b$ and $c$ be complex vectors. We investigates approaches for efficient approximations of the bilinear form $c^*A^{-1}b$. Equivalently, we wish to approximate the scalar value $c^*x$, where $x$ solves the linear system $Ax = b$. Here the matrix $A$ can be very large or its elements can be too costly to compute so that $A$ is not explicitly available and it is used only in the form of the matrix-vector product. Therefore a direct method is not an option. For $A$ Hermitian positive definite, $b^*A^{-1}b$ can be efficiently approximated as a by-product of the conjugate-gradient iterations, which is mathematically equivalent to the matching moment approximations computed via the Gauss–Christoffel quadrature. We propose a new method using the biconjugate gradient iterations which is applicable to the general complex case. The proposed approach is compared with existing ones using analytic arguments and numerical experiments. Let $A$ be a nonsingular complex matrix and $b$ and $c$ be complex vectors. We investigates approaches for efficient approximations of the bilinear form $c^*A^{-1}b$. Equivalently, we wish to approximate the scalar value $c^*x$, where $x$ solves the linear system $Ax = b$. Here the matrix $A$ can be very large or its elements can be too costly to compute so that $A$ is not explicitly available and it is used only in the form of the matrix-vector product. Therefore a direct method is not an option. For $A$ Hermitian positive definite, $b^*A^{-1}b$ can be efficiently approximated as a by-product of the conjugate-gradient iterations, which is mathematically equivalent to the matching moment approximations computed via the Gauss–Christoffel quadrature. We propose a new method using the biconjugate gradient iterations which is applicable to the general complex case. The proposed approach is compared with existing ones using analytic arguments and numerical experiments.
dcterms:title
On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b
skos:prefLabel
On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b
skos:notation
RIV/67985807:_____/11:00358802!RIV12-AV0-67985807
n4:predkladatel
n5:ico%3A67985807
n3:aktivita
n16:Z n16:P
n3:aktivity
P(GA201/09/0917), P(IAA100300802), Z(AV0Z10300504), Z(MSM0021620839)
n3:cisloPeriodika
2
n3:dodaniDat
n12:2012
n3:domaciTvurceVysledku
n13:2085674
n3:druhVysledku
n19:J
n3:duvernostUdaju
n8:S
n3:entitaPredkladatele
n11:predkladatel
n3:idSjednocenehoVysledku
218006
n3:idVysledku
RIV/67985807:_____/11:00358802
n3:jazykVysledku
n18:eng
n3:klicovaSlova
bilinear forms; scattering amplitude; method of moments; Krylov subspace methods; conjugate gradient method; biconjugate gradient method; Lanczos algorithm; Arnoldi algorithm; Gauss-Christoffel quadrature; model reduction
n3:klicoveSlovo
n7:method%20of%20moments n7:biconjugate%20gradient%20method n7:Lanczos%20algorithm n7:Gauss-Christoffel%20quadrature n7:bilinear%20forms n7:Arnoldi%20algorithm n7:Krylov%20subspace%20methods n7:conjugate%20gradient%20method n7:model%20reduction n7:scattering%20amplitude
n3:kodStatuVydavatele
US - Spojené státy americké
n3:kontrolniKodProRIV
[B5714B90A55B]
n3:nazevZdroje
SIAM Journal on Scientific Computing
n3:obor
n14:BA
n3:pocetDomacichTvurcuVysledku
1
n3:pocetTvurcuVysledku
2
n3:projekt
n20:IAA100300802 n20:GA201%2F09%2F0917
n3:rokUplatneniVysledku
n12:2011
n3:svazekPeriodika
33
n3:tvurceVysledku
Strakoš, Z. Tichý, Petr
n3:wos
000289973500005
n3:zamer
n9:AV0Z10300504 n9:MSM0021620839
s:issn
1064-8275
s:numberOfPages
23
n15:doi
10.1137/090753723