求解大型对称特征值问题的改进的块Davidson方法

Improved Block Davidson Method for Solving Large Symmetric Eigenproblems

块Davidson方法是求解大型对称矩阵特征值问题的一种有效方法.但对一些特征值问题,当Ritz值收敛以后,该方法并不能保证Ritz向量也同时收敛.因此,为加速块Davidson方法的收敛性,研究了块Davidson方法的重新开始技术,将精化策略和收缩技术应用于块Davidson方法,提出了收缩的精化块Davidson方法.数值试验结果及理论分析均表明,新方法比块Davidson和块Lanczos方法有更好的收敛效果,对计算大型对称矩阵的一些极端特征对是有效的.

Block Davidson method is effective for computing the eigenvalues of large symmetric matrices. However, the corresponding Ritz vectors obtained by block Davidson method always converge more slowly than the Ritz values. In order to solve this problem, a refined block Davidson method with deflation was proposed, which combines refined strategy and deflation technique with block Davidson method. Numerical experiments show that the new algorithm proposed is much more efficient than block Davidson and Lanczos algorithms in improving convergency and calculating extreme eigenpairs of large symmetric matrices.