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Level-3 Cholesky Factorization Routines Improve Performance of Many Cholesky Algorithms
Umeå University, Faculty of Science and Technology, Department of Computing Science. (IBM TJ Watson Res Ctr, Yorktown Hts, NY USA)
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2013 (English)In: ACM Transactions on Mathematical Software, ISSN 0098-3500, Vol. 39, no 2, 9- p.Article in journal (Refereed) Published
Abstract [en]

Four routines called DPOTF3i, i = a, b, c, d, are presented. DPOTF3i are a novel type of level-3 BLAS for use by BPF (Blocked Packed Format) Cholesky factorization and LAPACK routine DPOTRF. Performance of routines DPOTF3i are still increasing when the performance of Level-2 routine DPOTF2 of LAPACK starts decreasing. This is our main result and it implies, due to the use of larger block size nb, that DGEMM, DSYRK, and DTRSM performance also increases! The four DPOTF3i routines use simple register blocking. Different platforms have different numbers of registers. Thus, our four routines have different register blocking sizes. BPF is introduced. LAPACK routines for POTRF and PPTRF using BPF instead of full and packed format are shown to be trivial modifications of LAPACK POTRF source codes. We call these codes BPTRF. There are two variants of BPF: lower and upper. Upper BPF is "identical" to Square Block Packed Format (SBPF). "LAPACK" implementations on multicore processors use SBPF. Lower BPF is less efficient than upper BPF. Vector inplace transposition converts lower BPF to upper BPF very efficiently. Corroborating performance results for DPOTF3i versus DPOTF2 on a variety of common platforms are given for n approximate to nb as well as results for large n comparing DBPTRF versus DPOTRF.

Place, publisher, year, edition, pages
Association for Computing Machinery (ACM), 2013. Vol. 39, no 2, 9- p.
Keyword [en]
Algorithms, Performance, LAPACK, real symmetric matrices, complex Hermitian matrices, positive definite matrices, Cholesky factorization and solution, novel blocked packed matrix data structures, inplace transposition, Cache Blocking, BLAS
National Category
Computer Science Mathematics
URN: urn:nbn:se:umu:diva-67808DOI: 10.1145/2427023.2427026ISI: 000315458000003OAI: diva2:614234
Available from: 2013-04-03 Created: 2013-04-03 Last updated: 2013-04-03Bibliographically approved

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Gustavson, Fred G.
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