## Abstract

This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, Âx = b, which we call p-level lower rank extracted systems (p-level LRES), by the preconditioned conjugate gradient method. The study of these systems is motivated by the numerical approximation of integral equations with convolution kernels defined on arbitrary p-dimensional domains. This is in contrast to p-level Toeplitz systems which only apply to rectangular domains. The coefficient matrix, Â, is a principal submatrix of a p-level Toeplitz matrix, A, and the preconditioner for the preconditioned conjugate gradient algorithm is provided in terms of the inverse of a p-level circulant matrix constructed from the elements of A. The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix which leads to a substantial reduction in the computational cost of solving LRE systems.

Original language | English (US) |
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Pages (from-to) | 437-472 |

Number of pages | 36 |

Journal | Numerical Linear Algebra with Applications |

Volume | 13 |

Issue number | 6 |

DOIs | |

State | Published - Aug 2006 |

## Keywords

- Conjugate gradient method
- Convolution
- Integral equations
- Non rectangular domains
- Preconditions

## ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics