### Abstract

Original language | English |
---|---|

Pages (from-to) | 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 |

### Fingerprint

### Keywords

- preconditioning
- matrix algebras
- Toeplitz
- Hankel
- clustering
- fast discrete transforms

### Cite this

*Linear Algebra and its Applications*,

*438*(1), 405-427. https://doi.org/10.1016/j.laa.2012.07.042

}

*Linear Algebra and its Applications*, vol. 438, no. 1, pp. 405-427. https://doi.org/10.1016/j.laa.2012.07.042

**Optimal rank matrix algebras preconditioners.** / Tudisco, F.; Di Fiore, C.; Tyrtyshnikov, E. E.

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

JF - Linear Algebra and its Applications

SN - 0024-3795

IS - 1

ER -