陀螺特征值问题的广义SRR子空间迭代法及其加速法

THE GENERALIZED SRR-SUBSPACE ITERATION METHOD COMBINED WITH EXPEDITING ALGORITHM FOR GYROSCOPIC EIGENVALUE PROBLEM

本文基于非Hermitian矩阵的Schur—Rayleigh-Ritz加速的子空间迭代法,构造了状态空间表示的陀螺特征值问题广义SRR子空间迭代法,这是与由质量阵和刚度阵构成的广义特征值问题的子空间迭代法平行的一种算法。在迭代中充分利用陀螺特征值问题的反对称特性,使得投影及SRR加速过程得到极大的简化。在采用Schur型对幂加速进行分析的过程中发现,Meirovitch所构造的陀螺特征值问题求解方法的本质是反对称Schur型的平方乃是对称Schur型。这一发现揭示了对称与反对称特征值问题之间的内在联系,反映了保守系特征值问题的优良数学品质。本文构造了Meirovitch法中利用稀疏性方法,改变了认为它破坏稀疏性的看法。文中算例证实了方法的有效性。

Based on the Schur-Rayleigh-Ritz accelerated subspace iteration method for non-Hermitian matrix, a version of generalized SRR-subspace iteration algorithm is constructed for the large scale gyroscopic eigenvalue problem in state space, the method is paralell to the sub-space iteration for the generalized eigenvalue problem constituted by a mass matrix and a stiffness matrix. The projection and Schur-Rayleigh-Ritz acceleration step are dramatically simplified by utilizing the anti-symmetric property of the gyrosopic eigenvalue problem. In analizing the accelerated power method by Schur form. It turns out that the essence of Meirovitch's method is that the square of the antisymmetric Schur form is a symmetric one. From this character, the relationship betweethe symmetric and the antisymmetric eigenvalue problem can be found, which reflects the inherent property of the corresponding problem. The approach of how to make use of the sparsity of system matrices in Meirovitch's method is introduced, which is a correction to the point of view that Meirovitch's method deteriorates the sparsity of the system. Numerical examples are presented to verify the effectiveness of the method suggested.