广义特征值问题求解的改进Ritz向量法

Improved Ritz vector method for generalized eigenvalue problems

从提高算法的稳定性和计算效率入手,采取迭代及防止漏根、多根的措施,对传统的Ritz向量法进行改进,提出改进的Ritz向量法。此算法仅需生成r维的Krylov空间,大大降低投影矩阵阶数,减少投影矩阵特征值计算时间。引入重正交方案和模态比较法,并给出Ritz向量块宽q与生成步数r的建议取值。最后通过四参数的谱变换法,不但提高了该算法的稳定性和计算效率,也拓宽了Ritz向量法的适用范围。并用算例证明该算法的优越性。

To improve stability and efficiency, the improved Ritz vector method was presented, iteration and some effective measures avoiding root leakage or extra roots were adopted. The traditional Ritz vector method was improved. This new method only needed r-dimensional Krylov space to be constituted, greatly reduced the order number of projection matrix, decreased the eigenvalue computing time of projection matrix. Repeated orthogonality scheme and mode compare method were introduced. The values of block width q and iteration step r for Ritz vector were suggested. At last, with a four-parameter spectral transformation method, the computation efficiency and stability were greatly increased. At the same time, the applicable range of Ritz vector method was broadened. The numerical examples were given to demonstrate the validity and efficiency of the proposed method.