Abstract
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.
Original language | English |
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Pages (from-to) | 405-427 |
Number of pages | 23 |
Journal | Linear Algebra and its Applications |
Volume | 438 |
Issue number | 1 |
Early online date | 19 Sept 2012 |
DOIs | |
Publication status | Published - 1 Jan 2013 |
Keywords
- preconditioning
- matrix algebras
- Toeplitz
- Hankel
- clustering
- fast discrete transforms