### Abstract

The goal of solving an algebraic Riccati equation is to find the stable invariant subspace corresponding to all the eigenvalues lying in the open left-half plane. The purpose of this paper is to propose a structure-preserving Lanczos-type algorithm incorporated with shift and invert techniques, named shift-inverted J-Lanczos algorithm, for computing the stable invariant subspace for large sparse Hamiltonian matrices. The algorithm is based on the J-tridiagonalization procedure of a Hamiltonian matrix using symplectic similarity transformations. We give a detailed analysis on the convergence behavior of the J-Lanczos algorithm and present error bound analysis and Paige-type theorem. Numerical results for the proposed algorithm applied to a practical example arising from the position and velocity control for a string of high-speed vehicles are reported.

Original language | English |
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Pages (from-to) | 23-40 |

Number of pages | 18 |

Journal | Computers and Mathematics with Applications |

Volume | 33 |

Issue number | 10 |

DOIs | |

State | Published - 1 Jan 1997 |

### Keywords

- Hamiltonian matrix
- J-Lanczos algorithm
- J-tridiagonalization
- Riccati equation
- SR factorization
- Sympletic matrix

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## Cite this

*Computers and Mathematics with Applications*,

*33*(10), 23-40. https://doi.org/10.1016/S0898-1221(97)00074-6