On convergence to eigenvalues and eigenvectors in the block-Jacobi EVD algorithm with dynamic ordering
On convergence to eigenvalues and eigenvectors in the block-Jacobi EVD algorithm with dynamic ordering
复制标题
动态排序块-Jacobi EVD算法中特征值和特征向量的收敛
DOI：
10.1016/j.laa.2021.03.027
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发表时间：
2021
影响因子：
1.1
通讯作者：
Vajtersic Marian
中科院分区：
文献类型：
--
作者：
Yamamoto Yusaku;Oksa Gabriel;Vajtersic Marian
在求解Hermitian特征值问题的经典双边Jacobi方法的块形式中，迭代矩阵A（k）的非对角元素收敛于零。然而，这个事实本身并不一定保证A（k）收敛到固定的对角矩阵。对于累积酉变换矩阵Q（k）也是如此。我们证明了在一定的假设下，A（k）确实收敛于一个固定的对角矩阵，其对角元素是输入矩阵A的特征值。其次，它表明，对于一个简单的特征值，Q（k）的相应列收敛到相应的特征向量。对于重特征值或特征值簇，我们证明了由Q（k）的相应列构造的正交投影收敛于这些特征值所对应的特征空间上的正交投影.此外，适当的收敛界得到所有讨论的情况下。收敛性结果也适用于动态排序的并行块Jacobi方法。最后通过数值算例说明了所发展的理论。
In the block version of the classical two-sided Jacobi method for the Hermitian eigenvalue problem, the off-diagonal elements of iterated matrix A (k) converge to zero. However, this fact alone does not necessarily guarantee that A (k) converges to a fixed diagonal matrix. The same is true for the matrix of accumulated unitary transformations Q (k). We prove that under certain assumptions A (k) indeed converges to a fixed diagonal matrix, whose diagonal elements are the eigenvalues of the input matrix A. Next it is shown that for a simple eigenvalue the corresponding column of Q (k) converges to the corresponding eigenvector. For a multiple eigenvalue or a cluster of eigenvalues, we prove that the orthogonal projectors constructed from the corresponding columns of Q (k) converge to the orthogonal projector onto the eigenspace corresponding to those eigenvalues. Moreover, the appropriate convergence bounds are obtained for all discussed cases. Convergence results are also valid for the parallel block-Jacobi method with dynamic ordering. The developed theory is illustrated by numerical example.