Look-ahead in the two-sided reduction to compact band forms for symmetric eigenvalue problems and the SVD

Abstract

We address the reduction to compact band forms, via unitary similarity transformations, for the solution of symmetric eigenvalue problems and the computation of the singular value decomposition (SVD). Concretely, in the first case we revisit the reduction to symmetric band form while, for the second case, we propose a similar alternative, which transforms the original matrix to (unsymmetric) band form, replacing the conventional reduction method that produces a triangular-band output. In both cases, we describe algorithmic variants of the standard Level-3 BLAS-based procedures, enhanced with look-ahead, to overcome the performance bottleneck imposed by the panel factorization. Furthermore, our solutions employ an algorithmic block size that differs from the target bandwidth, illustrating the important performance benefits of this decision. Finally, we show that our alternative compact band form for the SVD is key to introduce an effective look-ahead strategy into the corresponding reduction procedure.

Fine‐grain task‐parallel algorithms for matrix factorizations and inversion on many‐threaded CPUs

“…In Reference 16, we proposed to specialize the initial reduction in order to obtain a band matrix with upper/lower bandwidth $$$ w $$$. The BSB algorithm is presented in Figure 7, assuming for simplicity that $$$ {b}_p=w $$$.…”