广义块Toeplitz特征值问题的基于sine变换的预处理子

在求块Toeplitz矩阵束（Amn，Bmn）特征值的Lanczos过程中，通过对移位块Toepltz矩阵Amn-ρBmn进行基于sine变换的块预处理，从而改进了位移块Toeplitz矩阵的谱分布，加速了Lanczos过程的收敛速度．该块预处理方法能通过快速算法有效快速执行．本文证明了预处理后Lanczos过程收敛迅速，并通过实验证明该算法求解大规模矩阵问题尤其有效．

Block Sine Transform Preconditioner for Generalized Block Toeplitz Eigenvalue Problem

We employ the block sine transform-based preconditioner to precondition the shifted block Toeplitz matrix Amn - pBmn involved in the Lanczos method to compute the minimum eigenvalue of the generalized block Toeplitz eigenvalue problem Amn x=λBmnx, where Amn and Bmn are partitioned into m×m blocks with order n. The block sine transform-based preconditioner can improve the spectral distribution of the shifted block Toeplitz matrix and, hence, can speed up the convergence rate of the preconditioned Lanczos method.The block sine transform-based preconditioner can be implemented efficiently by the fast transform algorithm.A convergence analysis shows that the preconditioned Lanczos method converges sufficiently fast, and numerical results show that this method is highly effective for large matrix.