Efficient Eigenvalue and Singular Value Computations on Shared Memory Machines

We describe two techniques for speeding up eigenvalue and singular value computations on shared memory parallel computers. Depending on the information that is required, different steps in the overall process can be made more efficient. If only the eigenvalues or singluar values are sought then the reduction to condensed form may be done in two or more steps to make best use of optimized level-3 BLAS. If eigenvectors and/or singular vectors are required, too, then their accumulation can be sped up by another blocking technique. The efficiency of the blocked algorithms depends heavily on the values of certain control parameters. We also present a very simple performance model that allows selecting these parameters automatically. Keywords: Linear algebra; Eigenvalues and singular values; Reduction to condensed form; Hessenberg QR iteration; Blocked algorithms. 1 Introduction The problem of determining eigenvalues and associated eigenvectors (or singular values and vectors) of a matrix ..

MRRR-based Eigensolvers for Multi-core Processors and Supercomputers

The real symmetric tridiagonal eigenproblem is of outstanding importance in numerical computations; it arises frequently as part of eigensolvers for standard and generalized dense Hermitian eigenproblems that are based on a reduction to tridiagonal form. For its solution, the algorithm of Multiple Relatively Robust Representations (MRRR or MR3 in short) - introduced in the late 1990s - is among the fastest methods. To compute k eigenpairs of a real n-by-n tridiagonal T, MRRR only requires O(kn) arithmetic operations; in contrast, all the other practical methods require O(k^2 n) or O(n^3) operations in the worst case. This thesis centers around the performance and accuracy of MRRR.Comment: PhD thesi