View source for Basic Linear Algebra Subprograms

{{Short description|Routines for performing common linear algebra operations}}
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'''Basic Linear Algebra Subprograms''' ('''BLAS''') is a [[specification (technical standard)|specification]] that prescribes a set of low-level routines for performing common [[linear algebra]] operations such as [[vector space|vector]] addition, [[scalar multiplication]], [[dot product]]s, linear combinations, and [[matrix multiplication]]. They are the ''[[de facto]]'' standard low-level routines for linear algebra libraries; the routines have bindings for both [[C (programming language)|C]] ("CBLAS interface") and [[Fortran]] ("BLAS interface"). Although the BLAS specification is general, BLAS implementations are often optimized for speed on a particular machine, so using them can bring substantial performance benefits. BLAS implementations will take advantage of special floating point hardware such as vector registers or [[SIMD]] instructions.

It originated as a Fortran library in 1979<ref name="lawson79">*{{cite journal |last1=Lawson |first1=C. L. |last2=Hanson |first2=R. J. |last3=Kincaid |first3=D. |last4=Krogh |first4=F. T. |title=Basic Linear Algebra Subprograms for FORTRAN usage |journal=ACM Trans. Math. Softw. |volume=5 |issue=3 |pages=308–323 |date=1979 |id=Algorithm 539 |doi=10.1145/355841.355847 |hdl=2060/19780018835|s2cid=6585321 |hdl-access=free }}</ref> and its interface was standardized by the BLAS Technical (BLAST) Forum, whose latest BLAS report can be found on the [[netlib]] website.<ref>{{Cite web |url=https://netlib.org/blas/blast-forum|title=BLAS Technical Forum |website=netlib.org |access-date=2017-07-07}}</ref> This Fortran library is known as the ''[[reference implementation]]'' (sometimes confusingly referred to as ''the'' BLAS library) and is not optimized for speed but is in the [[public domain]].<ref>[http://www.lahey.com/docs/blaseman_lin62.pdf blaseman] {{webarchive |url=https://web.archive.org/web/20161012014431/http://www.lahey.com/docs/blaseman_lin62.pdf |date=2016-10-12}} ''"The products are the implementations of the public domain BLAS (Basic Linear Algebra Subprograms) and LAPACK (Linear Algebra PACKage), which have been developed by groups of people such as Prof. Jack Dongarra, University of Tennessee, USA and all published on the WWW (URL: https://www.netlib.org/)."''{{dead link|date=October 2016 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>{{cite web |url=https://www.netlib.org/utk/people/JackDongarra/PAPERS/netlib-history6.pdf |title=Netlib and NA-Net: building a scientific computing community |author=Jack Dongarra |author2=Gene Golub |author3=Eric Grosse |author4=Cleve Moler |author5=Keith Moore |quote=The Netlib software repository was created in 1984 to facilitate quick distribution of public domain software routines for use in scientific computation. |publisher=netlib.org |access-date=2016-02-13}}</ref>

Most libraries that offer linear algebra routines conform to the BLAS interface, allowing library users to develop programs that are indifferent to the BLAS library being used.

Many BLAS libraries have been developed, targeting various different hardware platforms. Examples includes [[CUDA#Programming_abilities|cuBLAS]] (NVIDIA GPU, [[GPGPU]]), [[ROCm#rocBLAS_/_hipBLAS|rocBLAS]] (AMD GPU), and [[OpenBLAS]]. Examples of CPU-based BLAS library branches include: [[OpenBLAS]], [[BLIS (software)|BLIS (BLAS-like Library Instantiation Software)]], Arm Performance Libraries,<ref name="Arm Performance Libraries">{{cite web|date=2020 |title=Arm Performance Libraries |publisher=[[Arm]] |url=https://www.arm.com/products/development-tools/server-and-hpc/allinea-studio/performance-libraries |access-date=2020-12-16}}</ref> [[Automatically Tuned Linear Algebra Software|ATLAS]], and [[Intel Math Kernel Library]] (iMKL). AMD maintains a fork of BLIS that is optimized for the [[Advanced Micro Devices|AMD]] platform.<ref>{{Cite web|url=https://developer.amd.com/amd-aocl/blas-library/|title=BLAS Library}}</ref> ATLAS is a portable library that automatically optimizes itself for an arbitrary architecture. iMKL is a freeware<ref name="MKLfree">{{cite web |date=2015 |title=No Cost Options for Intel Math Kernel Library (MKL), Support yourself, Royalty-Free |publisher=[[Intel]] |url=https://software.intel.com/articles/free_mkl |access-date=31 August 2015}}</ref> and proprietary<ref name="MKLintel">{{cite web |date=2015 |title=Intel Math Kernel Library (Intel MKL) |publisher=[[Intel]] |url=https://software.intel.com/intel-mkl |access-date=25 August 2015}}</ref> vendor library optimized for x86 and x86-64 with a performance emphasis on [[Intel]] processors.<ref name="optnotice">{{cite web |year=2012 |title=Optimization Notice |publisher=[[Intel]] |url=https://software.intel.com/articles/optimization-notice |access-date=10 April 2013}}</ref> OpenBLAS is an open-source library that is hand-optimized for many of the popular architectures. The [[LINPACK benchmarks]] rely heavily on the BLAS routine <code>[[General Matrix Multiply|gemm]]</code> for its performance measurements.

Many numerical software applications use BLAS-compatible libraries to do linear algebra computations, including [[LAPACK]], [[LINPACK]], [[Armadillo (C++ library)|Armadillo]], [[GNU Octave]], [[Mathematica]],<ref>{{cite journal |author=Douglas Quinney |date=2003 |title=So what's new in Mathematica 5.0? |journal=MSOR Connections |volume=3 |number=4 |publisher=The Higher Education Academy |url=http://78.158.56.101/archive/msor/headocs/34mathematica5.pdf |url-status=dead |archive-url=https://web.archive.org/web/20131029204826/http://78.158.56.101/archive/msor/headocs/34mathematica5.pdf |archive-date=2013-10-29 }}</ref> [[MATLAB]],<ref>{{cite web |author=Cleve Moler |date=2000 |title=MATLAB Incorporates LAPACK |publisher=[[MathWorks]] |url=https://www.mathworks.com/company/newsletters/articles/matlab-incorporates-lapack.html |access-date=26 October 2013}}</ref> [[NumPy]],<ref name="cise">{{cite journal |title=The NumPy array: a structure for efficient numerical computation |author=Stéfan van der Walt |author2=S. Chris Colbert |author3=Gaël Varoquaux |name-list-style=amp |date=2011 |journal=Computing in Science and Engineering |volume=13 |issue=2 |pages=22–30 |arxiv=1102.1523|bibcode=2011CSE....13b..22V |doi=10.1109/MCSE.2011.37|s2cid=16907816 }}</ref> [[R (programming language)|R]], [[Julia (programming language)|Julia]] and Lisp-Stat.

==Background==
With the advent of numerical programming, sophisticated subroutine libraries became useful.  These libraries would contain subroutines for common high-level mathematical operations such as root finding, matrix inversion, and solving systems of equations. The language of choice was [[FORTRAN]]. The most prominent numerical programming library was [[IBM]]'s [[Scientific Subroutine Package]] (SSP).<ref>{{Cite journal | last1 = Boisvert | first1 = Ronald F. | year = 2000 | title = Mathematical software: past, present, and future | journal = Mathematics and Computers in Simulation | volume = 54 | issue = 4–5 | pages = 227–241 | doi = 10.1016/S0378-4754(00)00185-3 | arxiv = cs/0004004| bibcode = 2000cs........4004B | s2cid = 15157725 }}</ref>  These subroutine libraries allowed programmers to concentrate on their specific problems and avoid re-implementing well-known algorithms.  The library routines would also be better than average implementations; matrix algorithms, for example, might use full pivoting to get better numerical accuracy. The library routines would also have more efficient routines. For example, a library may include a program to solve a matrix that is upper triangular. The libraries would include single-precision and double-precision versions of some algorithms.

Initially, these subroutines used hard-coded loops for their low-level operations. For example, if a subroutine needed to perform a matrix multiplication, then the subroutine would have three nested loops. Linear algebra programs have many common low-level operations (the so-called "kernel" operations, not related to [[kernel (operating system)|operating system]]s).<ref>Even the SSP (which appeared around 1966) had some basic routines such as RADD (add rows), CADD (add columns), SRMA (scale row and add to another row), and RINT (row interchange). These routines apparently were not used as kernel operations to implement other routines such as matrix inversion.  See {{Citation |last=IBM |title=System/360 Scientific Subroutine Package, Version III, Programmer's Manual |edition=5th |publisher=International Business Machines |date=1970 |id=GH20-0205-4}}.</ref> Between 1973 and 1977, several of these kernel operations were identified.{{sfn|BLAST Forum|2001|p=1}}  These kernel operations became defined subroutines that math libraries could call.  The kernel calls had advantages over hard-coded loops: the library routine would be more readable, there were fewer chances for bugs, and the kernel implementation could be optimized for speed.  A specification for these kernel operations using [[scalar (mathematics)|scalars]] and [[vector space|vector]]s, the level-1 Basic Linear Algebra Subroutines (BLAS), was published in 1979.{{sfn|Lawson|Hanson|Kincaid|Krogh|1979}}  BLAS was used to implement the linear algebra subroutine library [[LINPACK]].

The BLAS abstraction allows customization for high performance. For example, LINPACK is a general purpose library that can be used on many different machines without modification. LINPACK could use a generic version of BLAS. To gain performance, different machines might use tailored versions of BLAS. As computer architectures became more sophisticated, [[vector processor|vector machines]] appeared. BLAS for a vector machine could use the machine's fast vector operations.  (While vector processors eventually fell out of favor, vector instructions in modern CPUs are essential for optimal performance in BLAS routines.)

Other machine features became available and could also be exploited.  Consequently, BLAS was augmented from 1984 to 1986 with level-2 kernel operations that concerned vector-matrix operations.  Memory hierarchy was also recognized as something to exploit. Many computers have [[cache memory]] that is much faster than main memory; keeping matrix manipulations localized allows better usage of the cache. In 1987 and 1988, the level 3 BLAS were identified to do matrix-matrix operations. The level 3 BLAS encouraged block-partitioned algorithms. The [[LAPACK]] library uses level 3 BLAS.{{sfn|BLAST Forum|2001|pp=1&ndash;2}}

The original BLAS concerned only densely stored vectors and matrices. Further extensions to BLAS, such as for sparse matrices, have been addressed.{{sfn|BLAST Forum|2001|p=2}}

==Functionality==
BLAS functionality is categorized into three sets of routines called "levels", which correspond to both the chronological order of definition and publication, as well as the degree of the polynomial in the complexities of algorithms; Level 1 BLAS operations typically take [[linear time]], {{math|''O''(''n'')}}, Level 2 operations quadratic time and Level 3 operations cubic time.{{r|level3}} Modern BLAS implementations typically provide all three levels.

===Level 1===
This level consists of all the routines described in the original presentation of BLAS (1979),<ref name="lawson79"/> which defined only ''vector operations'' on [[Stride of an array|strided arrays]]: [[dot product]]s, [[norm (mathematics)|vector norms]], a generalized vector addition of the form

:<math>\boldsymbol{y} \leftarrow \alpha \boldsymbol{x} + \boldsymbol{y}</math>

(called "<code>axpy</code>", "a x plus y") and several other operations.

===Level 2===
This level contains ''matrix-vector operations'' including, among other things, a ''ge''neralized ''m''atrix-''v''ector multiplication (<code>gemv</code>):

:<math>\boldsymbol{y} \leftarrow \alpha \boldsymbol{A} \boldsymbol{x} + \beta \boldsymbol{y}</math>

as well as a solver for {{math|'''''x'''''}} in the linear equation

:<math>\boldsymbol{T} \boldsymbol{x} = \boldsymbol{y}</math>

with {{math|'''''T'''''}} being triangular. Design of the Level 2 BLAS started in 1984, with results published in 1988.<ref name="dongarra88"/> The Level 2 subroutines are especially intended to improve performance of programs using BLAS on [[vector processor]]s, where Level 1 BLAS are suboptimal "because they hide the matrix-vector nature of the operations from the compiler."<ref name="dongarra88">{{cite journal |first1=Jack J. |last1=Dongarra |first2=Jeremy |last2=Du Croz |first3=Sven |last3=Hammarling |first4=Richard J. |last4=Hanson |title=An extended set of FORTRAN Basic Linear Algebra Subprograms |journal=ACM Trans. Math. Softw. |volume=14 |date=1988 |pages=1–17 |doi=10.1145/42288.42291 |citeseerx=10.1.1.17.5421 |s2cid=3579623 }}</ref>

===Level 3===
This level, formally published in 1990,<ref name="level3">{{Cite journal |last1=Dongarra |first1=Jack J. |last2=Du Croz |first2=Jeremy |last3=Hammarling |first3=Sven |last4=Duff |first4=Iain S. |title=A set of level 3 basic linear algebra subprograms |doi=10.1145/77626.79170 |date=1990 |journal=[[ACM Transactions on Mathematical Software]] |issn=0098-3500 |volume=16 |issue=1 |pages=1–17|s2cid=52873593 |doi-access=free }}</ref> contains ''matrix-matrix operations'', including a "general [[matrix multiplication]]" (<code>gemm</code>), of the form

:<math>\boldsymbol{C} \leftarrow \alpha \boldsymbol{A} \boldsymbol{B} + \beta \boldsymbol{C},</math>

where {{math|'''''A'''''}} and {{math|'''''B'''''}} can optionally be [[transpose]]d or [[hermitian conjugate|hermitian-conjugated]] inside the routine, and all three matrices may be strided. The ordinary matrix multiplication {{math|'''''A B'''''}} can be performed by setting {{math|''α''}} to one and {{math|'''''C'''''}} to an all-zeros matrix of the appropriate size.

Also included in Level 3 are routines for computing

:<math>\boldsymbol{B} \leftarrow \alpha \boldsymbol{T}^{-1} \boldsymbol{B},</math>

where {{math|'''''T'''''}} is a [[triangular matrix]], among other functionality.

Due to the ubiquity of matrix multiplications in many scientific applications, including for the implementation of the rest of Level 3 BLAS,<ref name="Geijn_2008"/> and because faster algorithms exist beyond the obvious repetition of matrix-vector multiplication, <code>gemm</code> is a prime target of optimization for BLAS implementers. E.g., by decomposing one or both of {{math|'''''A'''''}}, {{math|'''''B'''''}} into [[Block matrix|block matrices]], <code>gemm</code> can be [[Matrix multiplication algorithm#Divide-and-conquer algorithm|implemented recursively]]. This is one of the motivations for including the {{math|''β''}} parameter,{{dubious|Reason for beta parameter|date=January 2015}} so the results of previous blocks can be accumulated. Note that this decomposition requires the special case {{math|''β'' {{=}} 1}} which many implementations optimize for, thereby eliminating one multiplication for each value of {{math|'''''C'''''}}. This decomposition allows for better [[locality of reference]] both in space and time of the data used in the product. This, in turn, takes advantage of the [[CPU cache|cache]] on the system.<ref>{{Citation | last1=Golub | first1=Gene H. | author1-link=Gene H. Golub | last2=Van Loan | first2=Charles F. | author2-link=Charles F. Van Loan | title=Matrix Computations | publisher=Johns Hopkins | edition=3rd | isbn=978-0-8018-5414-9 |date=1996}}</ref> For systems with more than one level of cache, the blocking can be applied a second time to the order in which the blocks are used in the computation. Both of these levels of optimization are used in implementations such as [[Automatically Tuned Linear Algebra Software|ATLAS]]. More recently, implementations by [[Kazushige Goto]] have shown that blocking only for the [[L2 cache]], combined with careful [[amortized analysis|amortizing]] of copying to contiguous memory to reduce [[translation lookaside buffer|TLB]] misses, is superior to [[Automatically Tuned Linear Algebra Software|ATLAS]].<ref name="Kazushige_2008"/> A highly tuned implementation based on these ideas is part of the [[GotoBLAS]], [[OpenBLAS]] and [[BLIS (software)|BLIS]].

A common variation of {{code|gemm}} is the {{code|gemm3m}}, which calculates a complex product using "three real matrix multiplications and five real matrix additions instead of the conventional four real matrix multiplications and two real matrix additions", an algorithm similar to [[Strassen algorithm]] first described by Peter Ungar.<ref>{{cite journal |last1=Van Zee |first1=Field G. |last2=Smith |first2=Tyler M. |title=Implementing High-performance Complex Matrix Multiplication via the 3m and 4m Methods |journal=ACM Transactions on Mathematical Software |date=24 July 2017 |volume=44 |issue=1 |pages=1–36 |doi=10.1145/3086466|s2cid=25580883 }}</ref>

==Implementations==
; 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>
; Arm Performance Libraries: [[Arm Performance Libraries]], supporting Arm 64-bit [[AArch64]]-based processors, available from [[Arm Ltd.|Arm]].<ref name="Arm Performance Libraries"/>
; ATLAS: [[Automatically Tuned Linear Algebra Software]], an [[Open-source software|open source]] implementation of BLAS [[application programming interface|API]]s for [[C (programming language)|C]] and [[Fortran|Fortran 77]].<ref>{{Cite web|url=https://math-atlas.sourceforge.net/|title=Automatically Tuned Linear Algebra Software (ATLAS)|website=math-atlas.sourceforge.net|access-date=2017-07-07}}</ref>
; [[BLIS (software)|BLIS]]: BLAS-like Library Instantiation Software framework for rapid instantiation. Optimized for most modern CPUs. BLIS is a complete refactoring of the GotoBLAS that reduces the amount of code that must be written for a given platform.<ref>{{Citation|title=blis: BLAS-like Library Instantiation Software Framework|date=2017-06-30|url=https://github.com/flame/blis|publisher=flame|access-date=2017-07-07}}</ref><ref>{{Citation|title=BLIS GitHub Repository|date=15 October 2021|url=https://github.com/flame/blis}}</ref>
; C++ AMP BLAS: The [[C++ AMP]] BLAS Library is an [[Open-source software|open source]] implementation of BLAS for Microsoft's AMP language extension for Visual C++.<ref>{{Cite web|url=http://ampblas.codeplex.com/|title=C++ AMP BLAS Library|website=CodePlex|language=en|access-date=2017-07-07|archive-date=2017-07-08 |archive-url=https://web.archive.org/web/20170708151515/http://ampblas.codeplex.com/|url-status=dead}}</ref>
; cuBLAS: Optimized BLAS for NVIDIA based GPU cards, requiring few additional library calls.<ref>{{Cite news|url=https://developer.nvidia.com/cublas|title=cuBLAS|date=2013-07-29|work=NVIDIA Developer|access-date=2017-07-07|language=en}}</ref>
; NVBLAS: Optimized BLAS for NVIDIA based GPU cards, providing only Level 3 functions, but as direct drop-in replacement for other BLAS libraries.<ref>{{Cite news|url=https://docs.nvidia.com/cuda/nvblas/index.htmls|title=NVBLAS|date=2018-05-15|work=NVIDIA Developer|access-date=2018-05-15|language=en}}{{Dead link|date=September 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
; clBLAS: An [[OpenCL]] implementation of BLAS by AMD. Part of the AMD Compute Libraries.<ref name="github.com">{{Citation|title=clBLAS: a software library containing BLAS functions written in OpenCL|date=2017-07-03|url=https://github.com/clMathLibraries/clBLAS|publisher=clMathLibraries|access-date=2017-07-07}}</ref>
; clBLAST: A tuned [[OpenCL]] implementation of most of the BLAS api.<ref name="https://github.com/CNugteren/CLBlast">{{Citation|last=Nugteren|first=Cedric|title=CLBlast: Tuned OpenCL BLAS|date=2017-07-05|url=https://github.com/CNugteren/CLBlast|access-date=2017-07-07}}</ref>
; Eigen BLAS: A [[Fortran|Fortran 77]] and [[C (programming language)|C]] BLAS library implemented on top of the [[Mozilla license|MPL]]-licensed [[Eigen (C++ library)|Eigen library]], supporting [[x86]], [[x86-64]], [[ARM architecture|ARM (NEON)]], and [[PowerPC]] architectures.
; ESSL: [[IBM]]'s Engineering and Scientific Subroutine Library, supporting the [[PowerPC]] architecture under [[AIX operating system|AIX]] and [[Linux]].<ref name="https://www.ibm.com/support/knowledgecenter/en/SSFHY8/essl_welcome.html">{{Citation|title=IBM Knowledge Centre: Engineering and Scientific Subroutine Library|url=https://www.ibm.com/support/knowledgecenter/en/SSFHY8/essl_welcome.html}}</ref>
; [[GotoBLAS]]: [[Kazushige Goto]]'s BSD-licensed implementation of BLAS, tuned in particular for [[Intel]] [[Nehalem (microarchitecture)|Nehalem]]/[[Intel Atom|Atom]], [[VIA Technologies|VIA]] [[VIA Nano|Nanoprocessor]], [[AMD]] [[Opteron]].<ref name="GotoBLAS2"/>
;[[GNU Scientific Library]]: Multi-platform implementation of many numerical routines. Contains a CBLAS interface.
; HP MLIB: [[Hewlett-Packard|HP]]'s Math library supporting [[IA-64]], [[PA-RISC]], [[x86]] and [[Opteron]] architecture under [[HP-UX]] and [[Linux]].
; Intel MKL: The [[Intel]] [[Math Kernel Library]], supporting x86 32-bits and 64-bits, available free from [[Intel]].<ref name="MKLfree" /> Includes optimizations for Intel [[Pentium (brand)|Pentium]], [[Intel Core|Core]] and Intel [[Xeon]] CPUs and Intel [[Xeon Phi]]; support for [[Linux]], [[Microsoft Windows|Windows]] and [[macOS]].<ref>{{Cite web|url=https://software.intel.com/en-us/intel-mkl/|title=Intel Math Kernel Library (Intel MKL) {{!}} Intel Software|website=software.intel.com|language=en|access-date=2017-07-07}}</ref>
; MathKeisan: [[NEC Corporation|NEC]]'s math library, supporting [[NEC SX architecture]] under [[SUPER-UX]], and [[Itanium]] under [[Linux]]<ref>{{Cite web|url=https://www.mathkeisan.com/|title=MathKeisan|last=Mathkeisan|first=NEC|website=www.mathkeisan.com|language=en|access-date=2017-07-07}}</ref>
; Netlib BLAS: The official reference implementation on [[Netlib]], written in [[Fortran|Fortran 77]].<ref>{{Cite web|url=https://www.netlib.org/blas/|title=BLAS (Basic Linear Algebra Subprograms)|website=www.netlib.org|access-date=2017-07-07}}</ref>
; Netlib CBLAS: Reference [[C (programming language)|C]] interface to the BLAS. It is also possible (and popular) to call the Fortran BLAS from C.<ref>{{Cite web|url=https://www.netlib.org/blas|title=BLAS (Basic Linear Algebra Subprograms)|website=www.netlib.org|access-date=2017-07-07}}</ref>
; [[OpenBLAS]]: Optimized BLAS based on GotoBLAS, supporting [[x86]], [[x86-64]], [[MIPS architecture|MIPS]] and [[ARM architecture|ARM]] processors.<ref>{{Cite web|url=https://www.openblas.net/|title=OpenBLAS : An optimized BLAS library|website=www.openblas.net|access-date=2017-07-07}}</ref>
; PDLIB/SX: [[NEC Corporation|NEC]]'s Public Domain Mathematical Library for the NEC [[NEC SX architecture|SX-4]] system.<ref name=":0">{{cite web |url=http://www.nec.co.jp/hpc/mediator/sxm_e/software/61.html |title=PDLIB/SX: Business Solution &#124; NEC |access-date=2007-05-20 |url-status=dead |archive-url=https://web.archive.org/web/20070222154031/http://www.nec.co.jp/hpc/mediator/sxm_e/software/61.html |archive-date=2007-02-22 }}</ref>
; rocBLAS: Implementation that runs on [[AMD]] GPUs via [[ROCm]].<ref>{{Cite web|url=https://rocmdocs.amd.com/en/latest/ROCm_Tools/rocblas.html|title=rocBLAS|website=rocmdocs.amd.com|access-date=2021-05-21}}</ref>
;SCSL
: [[Silicon Graphics|SGI]]'s Scientific Computing Software Library contains BLAS and LAPACK implementations for SGI's [[Irix]] workstations.<ref>{{cite web |url=http://www.sgi.com/products/software/scsl.html |title=SGI - SCSL Scientific Library: Home Page |access-date=2007-05-20 |url-status=dead |archive-url=https://web.archive.org/web/20070513173030/http://www.sgi.com/products/software/scsl.html |archive-date=2007-05-13 }}</ref>
; Sun Performance Library: Optimized BLAS and LAPACK for [[SPARC]], [[Intel Core|Core]] and [[AMD64]] architectures under [[Oracle Solaris|Solaris]] 8, 9, and 10 as well as Linux.<ref>{{Cite web|url=https://www.oracle.com/technetwork/server-storage/solarisstudio/overview/index.html|title=Oracle Developer Studio|website=www.oracle.com|access-date=2017-07-07}}</ref>
; uBLAS: A generic [[C++]] template class library providing BLAS functionality. Part of the [[Boost library]]. It provides bindings to many hardware-accelerated libraries in a unifying notation. Moreover, uBLAS focuses on correctness of the algorithms using advanced C++ features.<ref>{{Cite web|url=https://www.boost.org/doc/libs/1_60_0/libs/numeric/ublas/doc/index.html|title=Boost Basic Linear Algebra - 1.60.0|website=www.boost.org|access-date=2017-07-07}}</ref>

=== Libraries using BLAS ===

; Armadillo: [[Armadillo (C++ library)|Armadillo]] is a C++ linear algebra library aiming towards a good balance between speed and ease of use. It employs template classes, and has optional links to BLAS/ATLAS and LAPACK. It is sponsored by [[NICTA]] (in Australia) and is licensed under a free license.<ref>{{Cite web|url=https://arma.sourceforge.net/|title=Armadillo: C++ linear algebra library|website=arma.sourceforge.net|access-date=2017-07-07}}</ref>
; [[LAPACK]]: LAPACK is a higher level Linear Algebra library built upon BLAS. Like BLAS, a reference implementation exists, but many alternatives like libFlame and MKL exist.
; Mir: An [[LLVM]]-accelerated generic numerical library for science and machine learning written in [[D (programming language)|D]]. It provides generic linear algebra subprograms (GLAS). It can be built on a CBLAS implementation.<ref>{{Cite web|url=https://github.com/libmir|title= Dlang Numerical and System Libraries|website= [[GitHub]]}}</ref>

==Similar libraries (not compatible with BLAS)==
{{see also|LAPACK#Similar projects}}
; Elemental: Elemental is an open source software for [[distributed memory|distributed-memory]] dense and sparse-direct linear algebra and optimization.<ref>{{Cite web|url=https://libelemental.org/|title=Elemental: distributed-memory dense and sparse-direct linear algebra and optimization — Elemental|website=libelemental.org|access-date=2017-07-07}}</ref>
; HASEM: is a C++ template library, being able to solve linear equations and to compute eigenvalues.  It is licensed under BSD License.<ref>{{Cite web|url=https://sourceforge.net/projects/hasem/|title=HASEM|website=SourceForge|date=17 August 2015 |language=en|access-date=2017-07-07}}</ref>
; LAMA: The Library for Accelerated Math Applications ([[Library for Accelerated Math Applications|LAMA]]) is a C++ template library for writing numerical solvers targeting various kinds of hardware (e.g. [[GPU]]s through [[CUDA]] or [[OpenCL]]) on [[distributed memory]] systems, hiding the hardware specific programming from the program developer
; MTL4: The [[Matrix Template Library]] version 4 is a generic [[C++]] template library providing sparse and dense BLAS functionality. MTL4 establishes an intuitive interface (similar to [[MATLAB]]) and broad applicability thanks to [[generic programming]].

==Sparse BLAS==
Several extensions to BLAS for handling [[Sparse matrix|sparse matrices]] have been suggested over the course of the library's history; a small set of sparse matrix kernel routines was finally standardized in 2002.<ref>{{cite journal |first1=Iain S. |last1=Duff |first2=Michael A. |last2=Heroux |first3=Roldan |last3=Pozo |title=An Overview of the Sparse Basic Linear Algebra Subprograms: The New Standard from the BLAS Technical Forum |journal= ACM Transactions on Mathematical Software|year=2002 |volume=28 |issue=2 |pages=239–267 |doi=10.1145/567806.567810|s2cid=9411006 }}</ref>

==Batched BLAS==
The traditional BLAS functions have been also ported to architectures that support large amounts of parallelism such as [[GPUs]]. Here, the traditional BLAS functions provide typically good performance for large matrices. However, when computing e.g., matrix-matrix-products of many small matrices by using the GEMM routine, those architectures show significant performance losses. To address this issue, in 2017 a batched version of the BLAS function has been specified.<ref name="dongarra17">{{cite journal |last1=Dongarra |first1=Jack |last2=Hammarling |first2=Sven |last3=Higham |first3=Nicholas J. |last4=Relton |first4=Samuel D. |last5=Valero-Lara |first5=Pedro |last6=Zounon |first6=Mawussi |title=The Design and Performance of Batched BLAS on Modern High-Performance Computing Systems |journal=Procedia Computer Science |volume=108 |pages=495–504 |date=2017 |doi=10.1016/j.procs.2017.05.138|doi-access=free |hdl=2117/106913 |hdl-access=free }}</ref>

Taking the GEMM routine from above as an example, the batched version performs the following computation simultaneously for many matrices:

<math>\boldsymbol{C}[k] \leftarrow \alpha \boldsymbol{A}[k] \boldsymbol{B}[k] + \beta \boldsymbol{C}[k] \quad  \forall k </math>

The index <math>k</math> in square brackets indicates that the operation is performed for all matrices <math>k</math> in a stack. Often, this operation is implemented for a strided batched memory layout where all matrices follow concatenated in the arrays <math>A</math>, <math>B</math> and <math>C</math>.

Batched BLAS functions can be a versatile tool and allow e.g. a fast implementation of [[exponential integrators]] and [[Magnus integrators]] that handle long integration periods with many time steps.<ref name="herb21">{{cite journal |last1=Herb |first1=Konstantin |last2=Welter |first2=Pol |title=Parallel time integration using Batched BLAS (Basic Linear Algebra Subprograms) routines |journal=Computer Physics Communications |volume=270 |pages=108181 |date=2022 |doi=10.1016/j.cpc.2021.108181 |arxiv=2108.07126|bibcode=2022CoPhC.27008181H |s2cid=237091802 }}</ref> Here, the [[matrix exponentiation]], the computationally expensive part of the integration, can be implemented in parallel for all time-steps by using Batched BLAS functions.

==See also==
* [[List of numerical libraries]]
* [[Math Kernel Library]], math library optimized for the [[Intel]] architecture; includes BLAS, LAPACK
* [[Numerical linear algebra]], the type of problem BLAS solves

==References==
{{reflist|refs=
<ref name="Kazushige_2008">{{cite journal |author-last1=Goto |author-first1=Kazushige |author-link1=Kazushige Goto |author-last2=van de Geijn |author-first2=Robert A. |author-link2=Robert van de Geijn |title=Anatomy of High-Performance Matrix Multiplication |date=2008 |journal=[[ACM Transactions on Mathematical Software]] |issn=0098-3500 |volume=34 |issue=3 |pages=12:1–12:25 |doi=10.1145/1356052.1356053 |citeseerx=10.1.1.111.3873|s2cid=9359223 }} (25 pages) [https://www.cs.utexas.edu/~flame/web/FLAMEPublications.html#Goto]</ref>
<ref name="GotoBLAS2">{{cite web |title=GotoBLAS2 |author-first=Kent |author-last=Milfeld |publisher=[[Texas Advanced Computing Center]] |url=http://www.tacc.utexas.edu/tacc-software/gotoblas2 |access-date=2024-03-17 |url-status=dead |archive-url=https://web.archive.org/web/20200323172521/https://www.tacc.utexas.edu/research-development/tacc-software/gotoblas2 |archive-date=2020-03-23}}</ref>
<ref name="Geijn_2008">{{cite journal |author-last1=Goto |author-first1=Kazushige |author-link1=Kazushige Goto |author-last2=van de Geijn |author-first2=Robert A. |author-link2=Robert van de Geijn |title=High-performance implementation of the level-3 BLAS |journal=[[ACM Transactions on Mathematical Software]] |volume=35 |issue=1 |pages=1–14 |date=2008 |doi=10.1145/1377603.1377607 |s2cid=14722514 |archive-url=https://web.archive.org/web/20170706142000/ftp://ftp.cs.utexas.edu/pub/techreports/tr06-23.pdf |archive-date=2017-07-06 |url-status=dead |url=ftp://ftp.cs.utexas.edu/pub/techreports/tr06-23.pdf}}</ref>
}}

==Further reading==
* {{Citation |author=BLAST Forum |title=Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard |date=21 August 2001 |publisher=University of Tennessee |location=Knoxville, TN }}
* {{Citation |last1=Dodson |first1= D. S. |last2=Grimes |first2=R. G. |title=Remark on algorithm 539: Basic Linear Algebra Subprograms for Fortran usage |journal=ACM Trans. Math. Softw. |volume=8 |issue= 4 |pages=403&ndash;404 |year=1982 |doi= 10.1145/356012.356020|s2cid= 43081631 }}
* {{Citation |last=Dodson |first=D. S. |title=Corrigendum: Remark on "Algorithm 539: Basic Linear Algebra Subroutines for FORTRAN usage" |journal=ACM Trans. Math. Softw. |volume=9 |page=140 |year=1983 |doi= 10.1145/356022.356032|s2cid=22163977 |doi-access=free }}
* J. J. Dongarra, J. Du Croz, S. Hammarling, and R. J. Hanson, Algorithm 656: An extended set of FORTRAN Basic Linear Algebra Subprograms, ACM Trans. Math. Softw., 14 (1988), pp.&nbsp;18&ndash;32.
* J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, A set of Level 3 Basic Linear Algebra Subprograms, ACM Trans. Math. Softw., 16 (1990), pp.&nbsp;1&ndash;17.
* J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, Algorithm 679: A set of Level 3 Basic Linear Algebra Subprograms, ACM Trans. Math. Softw., 16 (1990), pp.&nbsp;18&ndash;28.
; New BLAS
* L. S. Blackford, J. Demmel, J. Dongarra, I. Duff, S. Hammarling, G. Henry, M. Heroux, L. Kaufman, A. Lumsdaine, A. Petitet, R. Pozo, K. Remington, R. C. Whaley, An Updated Set of Basic Linear Algebra Subprograms (BLAS), ACM Trans. Math. Softw., 28-2 (2002), pp.&nbsp;135&ndash;151.
* J. Dongarra, Basic Linear Algebra Subprograms Technical Forum Standard, International Journal of High Performance Applications and Supercomputing, 16(1) (2002), pp.&nbsp;1&ndash;111, and International Journal of High Performance Applications and Supercomputing, 16(2) (2002), pp.&nbsp;115&ndash;199.

==External links==
* [https://www.netlib.org/blas/ BLAS homepage] on Netlib.org
* [https://www.netlib.org/blas/faq.html BLAS FAQ]
* [https://www.netlib.org/lapack/lug/node145.html BLAS Quick Reference Guide] from LAPACK Users' Guide
* [https://web.archive.org/web/20061009230911/http://history.siam.org/oralhistories/lawson.htm Lawson Oral History] One of the original authors of the BLAS discusses its creation in an oral history interview. Charles L. Lawson Oral history interview by Thomas Haigh, 6 and 7 November 2004, San Clemente, California. Society for Industrial and Applied Mathematics, Philadelphia, PA.
* [https://web.archive.org/web/20061009230904/http://history.siam.org/oralhistories/dongarra.htm Dongarra Oral History] In an oral history interview, Jack Dongarra explores the early relationship of BLAS to LINPACK, the creation of higher level BLAS versions for new architectures, and his later work on the ATLAS system to automatically optimize BLAS for particular machines. Jack Dongarra, Oral history interview by Thomas Haigh, 26 April 2005, University of Tennessee, Knoxville TN. Society for Industrial and Applied Mathematics, Philadelphia, PA
*[https://stackoverflow.com/questions/1303182/how-does-blas-get-such-extreme-performance How does BLAS get such extreme performance?] Ten naive 1000&times;1000 matrix multiplications (10<sup>10</sup> floating point multiply-adds) takes 15.77&nbsp;seconds on 2.6&nbsp;GHz processor; BLAS implementation takes 1.32&nbsp;seconds.
* An Overview of the Sparse Basic Linear Algebra Subprograms: The New Standard from the BLAS Technical Forum {{doi-inline|10.1145/567806.567810}}

{{Numerical linear algebra}}
{{Linear algebra}}

[[Category:Numerical linear algebra]]
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