### Abstract

The solutions of a gyroscopic vibrating system oscillating about an equilibrium position, with no external applied forces and no damping forces, are completely determined by the quadratic eigenvalue problem (-λ^{2} _{i}M + λ_{i}G + K)x_{i} = 0, for i = 1, . . . , 2n, where M, G, and K are real n x n matrices, and M is symmetric positive definite (denoted by M > 0), G is skew symmetric, and either K > O or -K > 0. Gyroscopic system in motion about a stable equilibrium position (with -K > 0) are well understood. Two Lanczos-type algorithms, the pseudo skew symmetric Lanczos algorithm and the J-Lanczos algorithm, are studied for computing some extreme eigenpairs for solving gyroscopic systems in motion about an unstable equilibrium position (with K > 0). Shift and invert strategies, error bounds, implementation issues, and numerical results for both algorithms are presented in details.

Original language | English |
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Pages (from-to) | 49-66 |

Number of pages | 18 |

Journal | Computers and Mathematics with Applications |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 1999 |

### Keywords

- Generalized eigenvalue problem
- Gyroscopic system
- Hamiltonian matrix
- Lanczos algorithm
- Quadratic eigenvalue problem

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## Cite this

*Computers and Mathematics with Applications*,

*37*(1), 49-66. https://doi.org/10.1016/S0898-1221(98)00241-7