## Abstract

This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, A _{p}x = b, which we call lower rank extracted systems (LRES), by the preconditioned conjugate gradient method. These systems correspond to integral equations with convolution kernels defined on a union of many line segments in contrast to only one line segment in the case of Toeplitz systems. The p × p matrix, A_{p}, is shown to be a principal submatrix of a larger N × N Toeplitz matrix, A_{N}. The preconditioner is provided in terms of the inverse of a 2N × 2N circulant matrix constructed from the elements of A_{N}- The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix similar to the clustering results for iterative algorithms used to solve Toeplitz systems. The analysis also demonstrates that the computational expense to solve LRE systems is reduced to O(N log N).

Original language | English (US) |
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Pages (from-to) | 9-32 |

Number of pages | 24 |

Journal | Numerical Linear Algebra with Applications |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2005 |

## Keywords

- Convolution integral equations
- Domain geometry
- Preconditioning

## ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics