LAPACK

LAPACK

Initial release	1992 (1992)
Stable release	3.8.0 / 12 November 2017 (2017-11-12)
Written in	Fortran 90
Type	Software library
License	BSD-new
Website	www.netlib.org/lapack/

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). 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.

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	Band matrix
GE	Matrix (i.e., unsymmetric, in some cases rectangular)
GG	general matrices, generalized problem (i.e., a pair of general matrices)
GT	Tridiagonal Matrix General Matrix
HB	(complex) Hermitian matrix Band matrix
HE	(complex) Hermitian matrix
HG	upper Hessenberg matrix, generalized problem (i.e. a Hessenberg and a Triangular matrix)
HP	(complex) Hermitian matrix, 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 matrix Band matrix
SP	Symmetric matrix, Packed storage matrix
ST	(real) Symmetric matrix Tridiagonal matrix
SY	Symmetric matrix
TB	Triangular matrix Band matrix
TG	triangular matrices, generalized problem (i.e., a pair of triangular matrices)
TP	Triangular matrix, Packed storage matrix
TR	Triangular matrix (or in some cases quasi-triangular)
TZ	Trapezoidal matrix
UN	(complex) Unitary matrix
UP	(complex) Unitary matrix, 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

See also

• 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

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