广义块Toeplitz特征值问题的基于sine变换的预处理子(英文)

Block Sine Transform Preconditioner for Generalized Block Toeplitz Eigenvalue Problem

在求块Toeplitz矩阵束(A_(mn),B_(mn))特征值的Lanczos过程中,通过对移位块Toepltz矩阵A_(mn)-ρB_(mn)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.

We employ the block sine transform-based preconditioner to precondition the shifted block Toeplitz matrix Amn - ρBmn involved in the Lanczes method to compute the minimum eigenvalue of the generalized block Toeplitz eigenvalue problem Anmx=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.