Optimal Rank Matrix Algebras Preconditioners

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}
}

Keywords: Preconditioning Matrix algebras Toeplitz Hankel Clustering Fast discrete transforms