Rectangular full packed format for cholesky's algorithm: Factorization, solution, and inversion

Fred G. Gustavson, Jerzy Waśniewski, Jack J. Dongarra, Julien Langou

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

We describe a new data format for storing triangular, symmetric, and Hermitian matrices called Rectangular Full Packed Format (RFPF). The standard two-dimensional arrays of Fortran and C (also known as full format) that are used to represent triangular and symmetric matrices waste nearly half of the storage space but provide high performance via the use of Level 3 BLAS. Standard packed format arrays fully utilize storage (array space) but provide low performance as there is no Level 3 packed BLAS. We combine the good features of packed and full storage using RFPF to obtain high performance via using Level 3 BLAS as RFPF is a standard full-format representation. Also, RFPF requires exactly the same minimal storage as packed the format. Each LAPACK full and/or packed triangular, symmetric, and Hermitian routine becomes a single new RFPF routine based on eight possible data layouts of RFPF. This new RFPF routine usually consists of two calls to the corresponding LAPACK full-format routine and two calls to Level 3 BLAS routines. This means no new software is required. As examples, we present LAPACK routines for Cholesky factorization, Cholesky solution, and Cholesky inverse computation in RFPF to illustrate this new work and to describe its performance on several commonly used computer platforms. Performance of LAPACK full routines using RFPF versus LAPACK full routines using the standard format for both serial and SMP parallel processing is about the same while using half the storage. Performance gains are roughly one to a factor of 43 for serial and one to a factor of 97 for SMP parallel times faster using vendor LAPACK full routines with RFPF than with using vendor and/or reference packed routines.

Original languageEnglish
Article number18
JournalACM Transactions on Mathematical Software
Volume37
Issue number2
DOIs
StatePublished - Apr 1 2010
Externally publishedYes

Keywords

  • BLAS
  • Cholesky factorization and solution
  • Complex Hermitian matrices
  • LAPACK
  • Linear algebra libraries
  • Novel packed matrix data structures
  • Positive definite matrices
  • Real symmetric matrices
  • Rectangular Full Packed Format
  • Recursive algorithms

Fingerprint

Dive into the research topics of 'Rectangular full packed format for cholesky's algorithm: Factorization, solution, and inversion'. Together they form a unique fingerprint.

Cite this