A Parallel Bisection and Inverse Iteration Solver for a Subset of Eigenpairs of Symmetric Band Matrices

Abstract

The tridiagonalization and its back-transformation for computing eigenpairs of real symmetric dense matrices are known to be the bottleneck of the execution time in parallel processing owing to the communication cost and the number of floating-point operations. To overcome this problem, we focus on real symmetric band eigensolvers proposed by Gupta and Murata since their eigensolvers are composed of the bisection and inverse iteration algorithms and do not include neither the tridiagonalization of real symmetric band matrices nor its back-transformation. In this paper, the following parallel solver for computing a subset of eigenpairs of real symmetric band matrices is proposed on the basis of Murata’s eigensolver: the desired eigenvalues of the target band matrices are computed directly by using parallel Murata’s bisection algorithm. The corresponding eigenvectors are computed by using block inverse iteration algorithm with reorthogonalization, which can be parallelized with lower communication cost than the inverse iteration algorithm. Numerical experiments on shared-memory multi-core processors show that the proposed eigensolver is faster than the conventional solvers.