A Fully Pipelined Multishift QR Algorithm for Parallel Solution of Symmetric Tridiagonal Eigenproblems

Abstract

In this paper, we propose a fully pipelined multishift QR algorithm to compute all the eigenvalues of a symmetric tridiagonal matrix on parallel machines. Existing approaches for parallelizing the tridiagonal QR algorithm, such as the conventional multishift QR algorithm and the deferred shift QR algorithm, have suffered from either inefficiency of processor utilization or deterioration of convergence properties. In contrast, our algorithm realizes both efficient processor utilization and improved convergence properties at the same time by adopting a new shifting strategy. Numerical experiments on a shared memory parallel machine (Fujitsu PrimePower HPC2500) with 32 processors show that our algorithm is up to 1.9 times faster than the conventional multishift algorithm and up to 1.7 times faster than the deferred shift algorithm.