A New Parallel Symmetric Tridiagonal Eigensolver Based on Bisection and Inverse Iteration Algorithms for Shared-Memory Multi-core Processors

Abstract

In order to accelerate the subset computation of eigenpairs for real symmetric tridiagonal matrices on sharedmemory multi-core processors, a parallel symmetric tridiagonal eigensolver is proposed, which computes eigenvalues of target matrices using the parallel bisection algorithm and computes the corresponding eigenvectors using the block inverse iteration algorithm with reorthogonalization (BIR algorithm). The BIR algorithm is based on the simultaneous inverse iteration (SI) algorithm, which is a variant of the inverse iteration algorithm, and is introduced to a block parameter. Since the BIR algorithm is mainly composed of the matrix multiplications, the proposed eigensolver is expected to accelerate the computation of eigenpairs even on massively parallel computers. Numerical experiments on shared-memory multi-core processors show that the BIR algorithm is faster than the SI algorithm and achieves the good parallel efficiency. In addition, many cases of the numerical experiments also show that the proposed eigensolver, including the parallel bisection and the BIR algorithm, is more accurate than the parallel implementation of other eigensolvers, such as the QR iteration algorithm, the divide-and-conquer algorithm, and the multiple relatively robust representations algorithm.