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

• Published in: Numerical algorithms Vol. 80; no. 2; pp. 635 - 660
• Main Authors: Rodríguez-Sánchez, Rafael, Catalán, Sandra, Herrero, José R., Quintana-Ortí, Enrique S., Tomás, Andrés E.
• Format: Journal Article Publication
• Language: English
• Published: New York Springer US 01.02.2019; Springer Nature B.V
• Subjects: Algebra; Algebras, Linear; Algorithms; Bandwidths; Computer Science; Critical path; Decomposition; Eigenvalues; Informàtica; Informàtica teòrica; Linear algebra; Look-ahead; Matrices; Matrius (Matemàtica); Numeric Computing; Numerical Analysis; Original Paper; Reduction; Singular value decomposition; Symmetric eigenvalue problems; Theory of Computation; Two-sided reduction to compact band form; Àlgebra lineal; Àrees temàtiques de la UPC; Look-ahead; Symmetric eigenvalue problems; Two-sided reduction to compact band form; Singular value decomposition
• ISSN: 1017-1398; 1572-9265; 1572-9265
• DOI: 10.1007/s11075-018-0500-8

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 Basic Linear Algebra Subroutines (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.