On convergence to eigenvalues and eigenvectors in the block-Jacobi EVD algorithm with dynamic ordering

Abstract

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.