Affiliation:
1. The University of Texas at Austin
2. Universitat Jaume I
Abstract
We show how both the tridiagonal and bidiagonal QR algorithms can be restructured so that they become rich in operations that can achieve near-peak performance on a modern processor. The key is a novel, cache-friendly algorithm for applying multiple sets of Givens rotations to the eigenvector/singular vector matrix. This algorithm is then implemented with optimizations that: (1) leverage vector instruction units to increase floating-point throughput, and (2) fuse multiple rotations to decrease the total number of memory operations. We demonstrate the merits of these new QR algorithms for computing the Hermitian eigenvalue decomposition (EVD) and singular value decomposition (SVD) of dense matrices when all eigenvectors/singular vectors are computed. The approach yields vastly improved performance relative to traditional QR algorithms for these problems and is competitive with two commonly used alternatives---Cuppen’s Divide-and-Conquer algorithm and the method of Multiple Relatively Robust Representations---while inheriting the more modest
O
(
n
) workspace requirements of the original QR algorithms. Since the computations performed by the restructured algorithms remain essentially identical to those performed by the original methods, robust numerical properties are preserved.
Funder
Microsoft
UTAustin-Portugal Colab program
Office of Cyberinfrastructure
Division of Computing and Communication Foundations
Publisher
Association for Computing Machinery (ACM)
Subject
Applied Mathematics,Software
Cited by
11 articles.
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