## Abstract

We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ^{2} - Aλ - B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.

Original language | English |
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Pages (from-to) | 52-81 |

Number of pages | 30 |

Journal | Linear Algebra and Its Applications |

Volume | 415 |

Issue number | 1 |

DOIs | |

State | Published - May 1 2006 |

### Bibliographical note

Funding Information:Keywords: Quadratic matrix polynomial; Krylov subspace; Quadratic eigenvalue problem; Model reduction ∗ Corresponding author. E-mail addresses: lhoff@ms.uky.edu (L. Hoffnung), rcli@ms.uky.edu (R.-C. Li), qye@ms.uky.edu (Q. Ye). 1 Supported in part by the NSF grant nos. CCR-9875201 and CCR-0098133. 2 Supported in part by NSF CAREER award grant no. CCR-9875201 and by NSF grant no. DMS-0510664. 3 Supported in part by NSF grant no. CCR-0098133 and NSF grant no. DMS-0411502.

## Keywords

- Krylov subspace
- Model reduction
- Quadratic eigenvalue problem
- Quadratic matrix polynomial

## ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics