### Abstract

This paper proposes and studies the performance of a preconditioner used in the preconditioned conjugate gradient method for solving a class of symmetric positive definite systems, A_{p}x = b, which we call Lower Rank Extracted Systems (LRES). 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 circular matrix constructed from the elements of A_{N}. The preconditioner is shown to yield clustering in the spectrum of preconditioned matrix similar to the clustering results in iterative algorithms used to solve Toeplitz systems. The analysis further 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) | 4039-4044 |

Number of pages | 6 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 5 |

State | Published - Dec 1 2001 |

Externally published | Yes |

Event | 40th IEEE Conference on Decision and Control (CDC) - Orlando, FL, United States Duration: Dec 4 2001 → Dec 7 2001 |

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

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## Cite this

*Proceedings of the IEEE Conference on Decision and Control*,

*5*, 4039-4044.