LAPACK ("Linear Algebra Package") is a standard software library for numerical linear algebra. It provides routines for solving systems of linear equations and linear least squares, eigenvalue problems, and singular value decomposition. It also includes routines to implement the associated matrix factorizations such as LU, QR, Cholesky and Schur decomposition. LAPACK was originally written in FORTRAN 77, but moved to Fortran 90 in version 3.2 (2008).[1] The routines handle both real and complex matrices in both single and double precision.

LAPACK was designed as the successor to the linear equations and linear least-squares routines of LINPACK and the eigenvalue routines of EISPACK. LINPACK, written in the 1970s and 1980s, was designed to run on the then-modern vector computers with shared memory. LAPACK, in contrast, was designed to effectively exploit the caches on modern cache-based architectures, and thus can run orders of magnitude faster than LINPACK on such machines, given a well-tuned BLAS implementation. LAPACK has also been extended to run on distributed memory systems in later packages such as ScaLAPACK and PLAPACK.[2]

LAPACK is licensed under a three-clause BSD style license, a permissive free software license with few restrictions.

Naming scheme

Subroutines in LAPACK have a naming convention which makes the identifiers very compact. This was necessary as the first Fortran standards only supported identifiers up to six characters long, so the names had to be shortened to fit into this limit.

A LAPACK subroutine name is in the form pmmaaa, where:

p is a one-letter code denoting the type of numerical constants used. S, D stand for real floating point arithmetic respectively in single and double precision, while C and Z stand for complex arithmetic with respectively single and double precision. The newer version, LAPACK95, uses generic subroutines in order to overcome the need to explicitly specify the data type.

mm is a two-letter code denoting the kind of matrix expected by the algorithm. The codes for the different kind of matrices are reported below; the actual data are stored in a different format depending on the specific kind; e.g., when the code DI is given, the subroutine expects a vector of length n containing the elements on the diagonal, while when the code GE is given, the subroutine expects an n×n array containing the entries of the matrix.

aaa is a one- to three-letter code describing the actual algorithm implemented in the subroutine, e.g. SV denotes a subroutine to solve linear system, while R denotes a rank-1 update.

For example, the subroutine to solve a linear system with a general (non-structured) matrix using real double-precision arithmetic is called DGESV.

Matrix types in the LAPACK naming scheme Name Description

BD bidiagonal matrix

DI diagonal matrix

GB general band matrix

GE general matrix (i.e., unsymmetric, in some cases rectangular)

GG general matrices, generalized problem (i.e., a pair of general matrices)

GT general tridiagonal matrix

HB (complex) Hermitian band matrix

HE (complex) Hermitian matrix

HG upper Hessenberg matrix, generalized problem (i.e. a Hessenberg and a triangular matrix)

HP (complex) Hermitian, packed storage matrix

HS upper Hessenberg matrix

OP (real) orthogonal matrix, packed storage matrix

OR (real) orthogonal matrix

PB symmetric matrix or Hermitian matrix positive definite band

PO symmetric matrix or Hermitian matrix positive definite

PP symmetric matrix or Hermitian matrix positive definite, packed storage matrix

PT symmetric matrix or Hermitian matrix positive definite tridiagonal matrix

SB (real) symmetric band matrix

SP symmetric, packed storage matrix

ST (real) symmetric matrix tridiagonal matrix

SY symmetric matrix

TB triangular band matrix

TG triangular matrices, generalized problem (i.e., a pair of triangular matrices)

TP triangular, packed storage matrix

TR triangular matrix (or in some cases quasi-triangular)

TZ trapezoidal matrix

UN (complex) unitary matrix

UP (complex) unitary, packed storage matrix

Details on this scheme can be found in the Naming scheme section in LAPACK Users' Guide.

Use with other programming languages

Many programming environments today support the use of libraries with C binding. The LAPACK routines can be used like C functions if a few restrictions are observed.

Several alternative language bindings are also available:

Armadillo for C++

IT++ for C++

LAPACK++ for C++

Lacaml for OCaml

CLapack for C

SciPy for Python

Implementations

As with BLAS, LAPACK is frequently forked or rewritten to provide better performance on specific systems. Some of the implementations are:

Accelerate

Apple's framework for macOS and iOS, which includes tuned versions of BLAS and LAPACK.[3][4]

Netlib LAPACK

The official LAPACK.

Netlib ScaLAPACK

Scalable (multicore) LAPACK, built on top of PBLAS.

Intel MKL

Intel's Math routines for their x86 CPUs.

OpenBLAS

Open-source reimplementation of BLAS and LAPACK.

Since LAPACK uses BLAS for the heavy-lifting, just linking to a better-tuned BLAS implementation usually improves the performance sufficiently. As a result, LAPACK is not reimplemented as often as BLAS is.

Similar projects

See also: Basic Linear Algebra Subprograms § Similar libraries (not compatible with BLAS)

These projects provide a similar functionality to LAPACK, but the main interface differs from that of LAPACK:

Libflame

A dense linear algebra library. Has a LAPACK-compatible wrapper. Can be used with any BLAS, although BLIS is the preferred implementation.[5]

Eigen

A header library for linear algebra. Has a BLAS and a partial LAPACK implementation for compatibility.

MAGMA

Matrix Algebra on GPU and Multicore Architectures (MAGMA) project develops a dense linear algebra library similar to LAPACK but for heterogeneous and hybrid architectures including multicore systems accelerated with GPGPUs.

PLASMA

The Parallel Linear Algebra for Scalable Multi-core Architectures (PLASMA) project is a modern replacement of LAPACK for multi-core architectures. PLASMA is a software framework for development of asynchronous operations and features out of order scheduling with a runtime scheduler called QUARK that may be used for any code that expresses its dependencies with a directed acyclic graph.[6]

See also

Free and open-source software portal

List of numerical libraries

Math Kernel Library (MKL)

NAG Numerical Library

SLATEC, a FORTRAN 77 library of mathematical and statistical routines

QUADPACK, a FORTRAN 77 library for numerical integration

References

"LAPACK 3.2 Release Notes". 16 November 2008.

"PLAPACK: Parallel Linear Algebra Package". www.cs.utexas.edu. University of Texas at Austin. 12 June 2007. Retrieved 20 April 2017.

"Guides and Sample Code". developer.apple.com. Retrieved 2017-07-07.

"Guides and Sample Code". developer.apple.com. Retrieved 2017-07-07.

"amd/libflame: High-performance object-based library for DLA computations". GitHub. AMD. 25 August 2020.

"ICL". icl.eecs.utk.edu. Retrieved 2017-07-07.

Further reading

Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.; Demmel, J.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKenney, A.; Sorensen, D. (1999). LAPACK Users' Guide (Third ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics. ISBN 0-89871-447-8.

External links

Official website on Netlib.org

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

Hellenica World - Scientific Library

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