As we know, polynomial filtering technique is efficient for accelerating convergence of standard eigenvalue problems, which, however, has not appeared for solving generalized eigenvalue problems. In this paper, by integrating the effectiveness and robustness of the Chebyshev polynomial filters, we propose the Chebyshev–Davidson method for computing some extreme eigenvalues and corresponding eigenvectors of generalized matrix pencils. In this method, both matrix factorizations and solving systems of linear equations are all avoided. Convergence analysis indicates that the Chebyshev–Davidson method achieves quadratic convergence locally in an ideal situation. Furthermore, numerical experiments are carried out to demonstrate the convergence properties and to show great superiority and robustness over some state-of-the art iteration methods.

众所周知，多项式滤波技术对于加速标准特征值问题的收敛是有效的，但是，对于解决广义特征值问题尚未出现。在本文中，通过集成Chebyshev多项式滤波器的有效性和鲁棒性，我们提出了Chebyshev-Davidson方法，用于计算广义矩阵铅笔的一些极限特征值和相应的特征向量。在这种方法中，都避免了矩阵分解和线性方程组的求解。收敛分析表明，切比雪夫-戴维森方法在理想情况下局部实现了二次收敛。此外，进行了数值实验，以证明收敛性，并显示出优于某些最新迭代方法的优越性和鲁棒性。