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Nonnegative Diagonals and High Performance on Low-Profile Matrices from Householder QR
The Householder reflections used in LAPACK's $QR$ factorization leave positive and negative real entries along $R$'s diagonal. This is sufficient for most applications of $QR$ factorizations, but a few require that $R$ have a nonnegative diagonal. This note describes a new Householder gene...
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Published in: | SIAM journal on scientific computing 2009-01, Vol.31 (4), p.2832-2841 |
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Main Authors: | , , , |
Format: | Article |
Language: | English |
Subjects: | |
Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
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Summary: | The Householder reflections used in LAPACK's $QR$ factorization leave positive and negative real entries along $R$'s diagonal. This is sufficient for most applications of $QR$ factorizations, but a few require that $R$ have a nonnegative diagonal. This note describes a new Householder generation routine to produce a nonnegative diagonal. Additionally, we find that scanning for trailing zeros in the generated reflections leads to large performance improvements when applying reflections with many trailing zeros. Factoring low-profile matrices, those with nonzero entries mostly near the diagonal (e.g., band matrices), now require far fewer operations. For example, $QR$ factorization of matrices with profile width $b$ that are stored densely in an $n\times n$ matrix improves from $O(n^3)$ to $O(n^2+nb^2)$. These routines are in LAPACK 3.2. |
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ISSN: | 1064-8275 1095-7197 |
DOI: | 10.1137/080725763 |