NDLI: Level-3 Cholesky Factorization Routines Improve Performance of Many Cholesky Algorithms

Content Provider	ACM Digital Library
Author	Herrero, José R. Dongarra, Jack J. Langou, Julien Waśniewski, Jerzy Gustavson, Fred G.
Copyright Year	2013
Abstract	Four routines called DPOTF3i, i = a,b,c,d, are presented. DPOTF3i are a novel type of level-3 BLAS for use by BPF $(\textit{B}locked$ $\textit{P}acked$ $\textit{F}ormat)$ 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 $\textit{nb},$ 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 $\textit{n}$ ≈ $\textit{nb}$ as well as results for large $\textit{n}$ comparing DBPTRF versus DPOTRF.
Starting Page	1
Ending Page	10
Page Count	10
File Format	PDF
ISSN	00983500
e-ISSN	15577295
DOI	10.1145/2427023.2427026
Volume Number	39
Issue Number	2
Journal	ACM Transactions on Mathematical Software (TOMS)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2013-02-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	BLAS Cache Blocking Cholesky factorization and solution LAPACK Complex Hermitian matrices Inplace transposition Novel blocked packed matrix data structures Positive definite matrices Real symmetric matrices
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Software

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f – Linear Algebra f01bnc nag complex cholesky (f01bnc) 1. Purpose nag complex cholesky (f01bnc) computes a Cholesky factorization of a complex positive-definite Hermitian matrix. 2. Specification

F – linear algebra f01bnc nag complex cholesky (f01bnc) 1. Purpose nag complex cholesky (f01bnc) computes a cholesky factorization of a complex positive-definite hermitian matrix. 2. Specification.

F07 – linear equations (lapack) f07gdc.

Level-3 Cholesky Factorization Routines Improve Performance of Many Cholesky Algorithms

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