Editing LAPACK

{{Short description|Software library for numerical linear algebra}}
{{Infobox software
| name                   = LAPACK (Netlib reference implementation)
| logo                   = LAPACK logo.svg
| logo size              = 120px
| screenshot             =
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| collapsible            =
| author                 =
| developer              =
| released               = {{Start date and age|1992}}
| latest release version = {{Wikidata|property|reference|edit|P348}}
| latest release date    = {{Start date and age|{{wikidata|qualifier|P348|P577}}|df=yes}}
| latest preview version =
| latest preview date    =
| programming language   = [[Fortran 90]]
| operating system       =
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| size                   =
| language               =
| genre                  = [[Software library]]
| license                = [[BSD-new]]
| website                = <!-- Wikidata -->
}}

'''LAPACK''' ("Linear Algebra Package") is a standard [[software library]] for [[numerical linear algebra]]. It provides [[subroutine|routines]] for solving [[systems of linear equations]] and [[numerical methods for linear least squares|linear least squares]], [[eigendecomposition of a matrix|eigenvalue problems]], and [[singular value decomposition]]. It also includes routines to implement the associated [[matrix factorization]]s such as [[LU decomposition|LU]], [[QR decomposition|QR]], [[Cholesky decomposition|Cholesky]] and [[Schur decomposition]].<ref name="LAPACK User Manual">{{cite book|last1=Anderson|first1=E.|last2=Bai|first2=Z.|last3=Bischof|first3=C.|last4=Blackford|first4=S.|last5=Demmel|first5=J.|author-link5=James Demmel|last6=Dongarra|first6=J.|author-link6=Jack Dongarra|last7=Du Croz|first7=J.|last8=Greenbaum|first8=A.|author-link8=Anne Greenbaum|last9=Hammarling|first9=S.|last10=McKenney|first10=A.|last11=Sorensen|first11=D.
| title =LAPACK Users' Guide
| edition = Third
| publisher = Society for Industrial and Applied Mathematics
| year = 1999
| location = Philadelphia, PA
| isbn = 0-89871-447-8
| url = https://www.netlib.org/lapack/lug/
| access-date=28 May 2022
}}</ref> LAPACK was originally written in [[FORTRAN 77]], but moved to [[Fortran 90]] in version 3.2 (2008).<ref>{{cite web|url=https://www.netlib.org/lapack/lapack-3.2.html|title=LAPACK 3.2 Release Notes|date=16 November 2008}}</ref> The routines handle both [[real number|real]] and [[complex number|complex]] matrices in both [[single precision|single]] and [[double precision]]. LAPACK relies on an underlying [[Basic Linear Algebra Subprograms|BLAS]] implementation to provide efficient and portable computational building blocks for its routines.<ref name="LAPACK User Manual" />{{rp|at=[https://www.netlib.org/lapack/lug/node65.html "The BLAS as the Key to Portability"]}}

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 processor|vector computer]]s with shared memory.  LAPACK, in contrast, was designed to effectively exploit the [[CPU cache|cache]]s on modern cache-based architectures and the [[instruction-level parallelism]] of modern [[superscalar processor]]s,<ref name="LAPACK User Manual" />{{rp|at=[https://www.netlib.org/lapack/lug/node61.html "Factors that Affect Performance"]}} and thus can run orders of magnitude faster than LINPACK on such machines, given a well-tuned [[Basic Linear Algebra Subprograms|BLAS]] implementation.<ref name="LAPACK User Manual" />{{rp|at=[https://www.netlib.org/lapack/lug/node65.html "The BLAS as the Key to Portability"]}} LAPACK has also been extended to run on [[distributed memory]] systems in later packages such as [[ScaLAPACK]] and PLAPACK.<ref>{{cite web|access-date=20 April 2017| date=12 June 2007| title=PLAPACK: Parallel Linear Algebra Package| url=https://www.cs.utexas.edu/users/plapack/| website=www.cs.utexas.edu| publisher=[[University of Texas at Austin]]}}</ref>

Netlib LAPACK is licensed under a three-clause [[BSD licenses|BSD style]] license, a [[permissive free software license]] with few restrictions.<ref>{{cite web |title=LICENSE.txt |url=https://www.netlib.org/lapack/LICENSE.txt |website=Netlib |access-date=28 May 2022}}</ref>

==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.<ref name="LAPACK User Manual" />{{rp|at=[https://www.netlib.org/lapack/lug/node24.html "Naming Scheme"]}}

A LAPACK subroutine name is in the form <code>pmmaaa</code>, where:

* <code>p</code> is a one-letter code denoting the type of numerical constants used. <code>S</code>, <code>D</code> stand for real [[floating-point arithmetic]] respectively in single and double precision, while <code>C</code> and <code>Z</code> stand for [[complex number|complex arithmetic]] with respectively single and double precision. The newer version, LAPACK95, uses [[generic function|generic]] subroutines in order to overcome the need to explicitly specify the data type.
* <code>mm</code> 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 <code>DI</code> is given, the subroutine expects a vector of length <code>n</code> containing the elements on the diagonal, while when the code <code>GE</code> is given, the subroutine expects an {{math|''n''×''n''}} array containing the entries of the matrix.
* <code>aaa</code> is a one- to three-letter code describing the actual algorithm implemented in the subroutine, e.g. <code>SV</code> denotes a subroutine to solve [[linear system]], while <code>R</code> 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 <code>DGESV</code>.<ref name="LAPACK User Manual" />{{Rp|at=[https://www.netlib.org/lapack/lug/node26.html "Linear Equations"]}}

{|class="wikitable"
|+ Matrix types in the LAPACK naming scheme
|-
! Name
! Description
|-
| BD
| [[bidiagonal matrix]]
|-
| DI
| [[diagonal matrix]]
|-
| GB
| general [[band matrix]]
|-
| GE
| general [[matrix (mathematics)|matrix]] (i.e., [[symmetric matrix|unsymmetric]], in some cases rectangular)
|-
| GG
| general matrices, generalized problem (i.e., a pair of general matrices)
|-
| GT
| general [[tridiagonal matrix]]
|-
| HB
| ([[Complex number|complex]]) [[Hermitian matrix|Hermitian]] [[band matrix]]
|-
| HE
| ([[Complex number|complex]]) [[Hermitian matrix]]
|-
| HG
| [[upper Hessenberg matrix]], generalized problem (i.e. a Hessenberg and a [[triangular matrix]])
|-
| HP
| ([[Complex number|complex]]) [[Hermitian matrix|Hermitian]], [[packed storage matrix]]
|-
| HS
| [[upper Hessenberg matrix]]
|-
| OP
| ([[Real number|real]]) [[orthogonal matrix]], [[packed storage matrix]]
|-
| OR
| ([[Real number|real]]) [[orthogonal matrix]]
|-
| PB
| [[symmetric matrix]] or [[Hermitian matrix]] [[positive definite matrix|positive definite]] band
|-
| PO
| [[symmetric matrix]] or [[Hermitian matrix]] [[positive definite matrix|positive definite]]
|-
| PP
| [[symmetric matrix]] or [[Hermitian matrix]] [[positive definite matrix|positive definite]], [[packed storage matrix]]
|-
| PT
| [[symmetric matrix]] or [[Hermitian matrix]] [[positive definite matrix|positive definite]] [[tridiagonal matrix]]
|-
| SB
| ([[Real number|real]]) [[symmetric matrix|symmetric]] [[band matrix]]
|-
| SP
| [[Symmetric matrix|symmetric]], [[packed storage matrix]]
|-
| ST
| ([[Real number|real]]) [[symmetric matrix]] [[tridiagonal matrix]]
|-
| SY
| [[symmetric matrix]]
|-
| TB
| [[Triangular matrix|triangular]] [[band matrix]]
|-
| TG
| [[Triangular matrix|triangular matrices]], generalized problem (i.e., a pair of [[triangular matrix|triangular matrices]])
|-
| TP
| [[Triangular matrix|triangular]], [[packed storage matrix]]
|-
| TR
| [[triangular matrix]] (or in some cases quasi-triangular)
|-
| TZ
| [[trapezoidal matrix]]
|-
| UN
| ([[Complex number|complex]]) [[unitary matrix]]
|-
| UP
| ([[Complex number|complex]]) [[unitary matrix|unitary]], [[packed storage matrix]]
|}

==Use with other programming languages and libraries==
Many programming environments today support the use of libraries with [[C (programming language)|C]] binding (LAPACKE, a standardised C interface,<ref>{{cite web |title=The LAPACKE C Interface to LAPACK |url=https://netlib.org/lapack/lapacke.html |website=LAPACK — Linear Algebra PACKage |access-date=2024-09-22 }}</ref> has been part of LAPACK since version 3.4.0<ref>{{cite web |title=LAPACK 3.4.0 |url=https://netlib.org/lapack/lapack-3.4.0.html |website=LAPACK — Linear Algebra PACKage |access-date=2024-09-22 }}</ref>), allowing LAPACK routines to be used directly so long as a few restrictions are observed. Additionally, many other software libraries and tools for scientific and numerical computing are built on top of LAPACK, such as [[R (programming language)|R]],<ref>{{Cite web |title=R: LAPACK Library |url=https://stat.ethz.ch/R-manual/R-patched/library/base/html/La_library.html |access-date=2022-03-19 |website=stat.ethz.ch }}</ref> [[MATLAB]],<ref>{{cite web |title=LAPACK in MATLAB |url=https://www.mathworks.com/help/matlab/math/lapack-in-matlab.html |website=Mathworks Help Center |access-date=28 May 2022 }}</ref> and [[SciPy]].<ref>{{cite web |title=Low-level LAPACK functions |url=https://docs.scipy.org/doc/scipy/reference/linalg.lapack.html |website=SciPy v1.8.1 Manual |access-date=28 May 2022 }}</ref>

Several alternative [[language binding]]s are also available:

* [[Armadillo (C++ library)|Armadillo]] for [[C++]]
* [[IT++]] for C++
* [[LAPACK++]] for C++
* Lacaml for [[OCaml]]
* [[SciPy]] for [[Python (programming language)|Python]]
* Gonum for [[Go (programming language)|Go]]
* [https://metacpan.org/pod/PDL::LinearAlgebra PDL::LinearAlgebra] for [[Perl Data Language]]
* [https://metacpan.org/pod/Math::Lapack Math::Lapack] for [[Perl]]
* [https://github.com/2xmax/NLapack NLapack] for [[.NET_Framework|.NET]]
* [https://github.com/DanielMartensson/CControl CControl] for [[C (programming language)|C]] in embedded systems
* [https://docs.rs/lapack/latest/lapack/ lapack] for [[rust (programming language)|rust]]

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

; Accelerate: [[Apple Inc.|Apple]]'s framework for [[macOS]] and [[IOS (Apple)|iOS]], which includes tuned versions of [[BLAS]] and LAPACK.<ref>{{Cite web|url=https://developer.apple.com/library/mac/#releasenotes/Performance/RN-vecLib/|title=Guides and Sample Code|website=developer.apple.com|access-date=2017-07-07}}</ref><ref>{{Cite web|url=https://developer.apple.com/library/ios/#documentation/Accelerate/Reference/AccelerateFWRef/|title=Guides and Sample Code|website=developer.apple.com|access-date=2017-07-07}}</ref>
; 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.
; Gonum LAPACK: A partial native [[Go (programming language)|Go]] implementation.

Since LAPACK typically calls underlying BLAS routines to perform the bulk of its computations, simply linking to a better-tuned BLAS implementation can be enough to significantly improve performance. As a result, LAPACK is not reimplemented as often as BLAS is.

=== Similar projects ===
These projects provide a similar functionality to LAPACK, but with a main interface differing from that of LAPACK: 

; Libflame: A dense linear algebra library. Has a LAPACK-compatible wrapper. Can be used with any BLAS, although [[BLIS (software)|BLIS]] is the preferred implementation.<ref>{{cite web |title=amd/libflame: High-performance object-based library for DLA computations |url=https://github.com/amd/libflame |website=GitHub |publisher=AMD |date=25 August 2020}}</ref>
; [[Eigen (C++ library)|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 [[GPGPU]]s.
; 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]].<ref>{{Cite web |url=http://icl.eecs.utk.edu/ |title=ICL |website=icl.eecs.utk.edu |access-date=2017-07-07 }}</ref>

{{See also|Basic Linear Algebra Subprograms#Similar libraries (not compatible with BLAS)}}

==See also==
{{Portal|Free and open-source software}}
* [[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==
{{Reflist}}

{{Numerical linear algebra}}

[[Category:Fortran libraries]]
[[Category:Free software programmed in Fortran]]
[[Category:Numerical linear algebra]]
[[Category:Numerical software]]
[[Category:Software using the BSD license]]