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Partitioned Triangular Tridiagonalization

Partitioned Triangular Tridiagonalization TOM00049 ACM (Typeset by SPi, Manila, Philippines) 1 of 16 February 23, 2011 38 Partitioned Triangular Tridiagonalization I MIROSLAV ROZLOZN K, Academy of Sciences of the Czech Republic GIL SHKLARSKI, Microsoft SIVAN TOLEDO, Tel-Aviv University We present a partitioned algorithm for reducing a symmetric matrix to a tridiagonal form, with partial pivoting. That is, the algorithm computes a factorization PA PT = LT L T , where, P is a permutation matrix, L is lower triangular with a unit diagonal and entries ™ magnitudes bounded by 1, and T is symmetric and tridiagonal. The algorithm is based on the basic (nonpartitioned) methods of Parlett and Reid and of Aasen. We show that our factorization algorithm is componentwise backward stable (provided that the growth factor is not too large), with a similar behavior to that of Aasen ™s basic algorithm. Our implementation also computes the Q R factorization of T and solves linear systems of equations using the computed factorization. The partitioning allows our algorithm to exploit modern computer architectures (in particular, cache memories and high-performance BLAS libraries). Experimental results demonstrate that our algorithms achieve approximately the same level of performance as the partitioned Bunch-Kaufman factor and solve routines http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Mathematical Software (TOMS) Association for Computing Machinery

Partitioned Triangular Tridiagonalization

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Publisher
Association for Computing Machinery
Copyright
Copyright © 2011 by ACM Inc.
ISSN
0098-3500
DOI
10.1145/1916461.1916462
Publisher site
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Abstract

TOM00049 ACM (Typeset by SPi, Manila, Philippines) 1 of 16 February 23, 2011 38 Partitioned Triangular Tridiagonalization I MIROSLAV ROZLOZN K, Academy of Sciences of the Czech Republic GIL SHKLARSKI, Microsoft SIVAN TOLEDO, Tel-Aviv University We present a partitioned algorithm for reducing a symmetric matrix to a tridiagonal form, with partial pivoting. That is, the algorithm computes a factorization PA PT = LT L T , where, P is a permutation matrix, L is lower triangular with a unit diagonal and entries ™ magnitudes bounded by 1, and T is symmetric and tridiagonal. The algorithm is based on the basic (nonpartitioned) methods of Parlett and Reid and of Aasen. We show that our factorization algorithm is componentwise backward stable (provided that the growth factor is not too large), with a similar behavior to that of Aasen ™s basic algorithm. Our implementation also computes the Q R factorization of T and solves linear systems of equations using the computed factorization. The partitioning allows our algorithm to exploit modern computer architectures (in particular, cache memories and high-performance BLAS libraries). Experimental results demonstrate that our algorithms achieve approximately the same level of performance as the partitioned Bunch-Kaufman factor and solve routines

Journal

ACM Transactions on Mathematical Software (TOMS) – Association for Computing Machinery

Published: Feb 1, 2011

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