Optimal rank matrix algebras preconditioners

• Published in: Linear algebra and its applications Vol. 438; no. 1; pp. 405 - 427
• Main Authors: Tudisco, F., Di Fiore, C., Tyrtyshnikov, E.E.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-01-2013
• Subjects: Clustering; Fast discrete transforms; Hankel; Matrix algebras; Preconditioning; Toeplitz; 65F50; 65T50; 65F10; Matrix algebras; Toeplitz; 65F15; Hankel; Fast discrete transforms; Clustering; Preconditioning; 65F05
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2012.07.042

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.