Performance of the parallel block Jacobi method with dynamic ordering for the symmetric eigenvalue problem

Abstract

We investigate the performance of the parallel block Jacobi method for the symmetric eigenvalue problem with dynamic ordering both theoretically and experimentally. First, we present an improved global convergence theorem of the method that takes into account the effect of annihilating multiple blocks at once. Next, we compare the dynamic ordering with two representative parallel cyclic orderings experimentally and show that the former can speedup the convergence for ill-conditioned matrices considerably with little extra cost.

Combinatorial preconditioning for accelerating the convergence of the parallel block Jacobi method for the symmetric eigenvalue problem

“…Let us denote the matrix at the kth iteration step by A (k) and its (I, J) block by A (k) I,J . Now, we focus on the so-called parallel block Jacobi method with dynamic ordering, which has proved effective in terms of convergence [5,6]. At the kth step of this method, we choose an off-diagonal block A…”

“…It is based on an algorithm to find the maximal weight matching (MWM) of a perfect graph. See [6] for details. In line 10, we apply P…”

“…Also, in line 14, these orthogonal transformations are accumulated into an n × n orthogonal matrix V (k) , which converges to an orthogonal eigenvector matrix of A as A (k) converges to a diagonal matrix. The readers are referred to [6] for more detailed explanation of the algorithm. Compute w I,J = ∥A (k) I,J ∥ F for 1 ≤ I < J ≤ q.…”

“…It is expected that the parallel block Jacobi method with dynamic ordering can reduce the off-diagonal norm off(A (k) ) = i̸ =j a 2 ij efficiently, because it chooses an off-diagonal block with the largest Frobenius norm for annihilation under the constraint that it can be annihilated in parallel with the already chosen blocks. In fact, numerical experiments show that it converges faster than the cyclic block Jacobi method, which is also a popular variant of the block Jacobi method [6].…”

“…In this paper, we propose to accelerate the convergence of the block Jacobi method by introducing combinatorial preconditioning. We focus on the parallel block Jacobi method with dynamic ordering [5,6]. In this variant, the matrix is partitioned into square blocks of appropriate size.…”