Numerical algorithms for undamped gyroscopic systems

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 (−λ2iM + λiG + K)xi = 0, for i = 1, …, 2n, where M, G, and K are real n × n matrices, and M is symmetric positive definite (denoted by M> 0), G is skew symmetric, and either K> 0 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.