# Optimal Rank Matrix Algebras Preconditioners

Francesco Tudisco,
Carmine Di Fiore,
Eugene E. Tyrtyshnikov,

**Linear Algebra and its Applications**,
438 :
405--427
(2013)

### 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 $P$. 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. In the present work we extend the black-dot algorithm for the computation of such splitting for $P$ circulant, to the case where $P$ is in $L$, for several known low-complexity matrix algebras $L$. 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 phi-circulant and Hartley-type cases some results previously known for $P$ circulant.

### Please cite this work as:

```
@article{tudisco2013optimal,
title={Optimal rank matrix algebras preconditioners},
author={Tudisco, Francesco and Di Fiore, Carmine and Tyrtyshnikov, E. Eugene},
journal={Linear Algebra and its Applications},
volume={438},
number={1},
pages={405--427},
year={2013},
publisher={Elsevier}
}
```

**Links:**
doi
arxiv

Keywords:
Preconditioning
Matrix algebras
Toeplitz
Hankel
Clustering
Fast discrete transforms