Abstract
| Language | English |
|---|---|
| Pages | 405-427 |
| Number of pages | 23 |
| Journal | Linear Algebra and its Applications |
| Volume | 438 |
| Issue number | 1 |
| Early online date | 19 Sep 2012 |
| DOIs | |
| Publication status | Published - 1 Jan 2013 |
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Keywords
- preconditioning
- matrix algebras
- Toeplitz
- Hankel
- clustering
- fast discrete transforms
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Optimal rank matrix algebras preconditioners. / Tudisco, F.; Di Fiore, C.; Tyrtyshnikov, E. E.
In: Linear Algebra and its Applications, Vol. 438, No. 1, 01.01.2013, p. 405-427.Research output: Contribution to journal › Article
TY - JOUR
T1 - Optimal rank matrix algebras preconditioners
AU - Tudisco, F.
AU - Di Fiore, C.
AU - Tyrtyshnikov, E. E.
PY - 2013/1/1
Y1 - 2013/1/1
N2 - When a linear system Ax=y is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner P and move to the preconditioned system P-1Ax=P-1y. The use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate. The construction of optimal rank preconditioners is strongly related to the possibility of splitting A as A=P+R+E, where E is a small perturbation and R is of low rank (Tyrtyshnikov, 1996) [1]. In the present work we extend the black-dot algorithm for the computation of such splitting for P circulant (see Oseledets and Tyrtyshnikov, 2006 [2]), to the case where P is in A, for several known low-complexity matrix algebras A. The algorithm so obtained is particularly efficient when A is Toeplitz plus Hankel like. We finally discuss in detail the existence and the properties of the decomposition A=P+R+E when A is Toeplitz, also extending to the φ-circulant and Hartley-type cases some results previously known for P circulant.
AB - When a linear system Ax=y is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner P and move to the preconditioned system P-1Ax=P-1y. The use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate. The construction of optimal rank preconditioners is strongly related to the possibility of splitting A as A=P+R+E, where E is a small perturbation and R is of low rank (Tyrtyshnikov, 1996) [1]. In the present work we extend the black-dot algorithm for the computation of such splitting for P circulant (see Oseledets and Tyrtyshnikov, 2006 [2]), to the case where P is in A, for several known low-complexity matrix algebras A. The algorithm so obtained is particularly efficient when A is Toeplitz plus Hankel like. We finally discuss in detail the existence and the properties of the decomposition A=P+R+E when A is Toeplitz, also extending to the φ-circulant and Hartley-type cases some results previously known for P circulant.
KW - preconditioning
KW - matrix algebras
KW - Toeplitz
KW - Hankel
KW - clustering
KW - fast discrete transforms
UR - http://www.sciencedirect.com/science/journal/00243795
U2 - 10.1016/j.laa.2012.07.042
DO - 10.1016/j.laa.2012.07.042
M3 - Article
VL - 438
SP - 405
EP - 427
JO - Linear Algebra and its Applications
T2 - Linear Algebra and its Applications
JF - Linear Algebra and its Applications
SN - 0024-3795
IS - 1
ER -