Базовые подпрограммы линейной алгебры (BLAS) и их реализации

Routines for performing common linear algebra operations

Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, 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 ("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 and its interface was standardized by the BLAS Technical (BLAST) Forum, whose latest BLAS report can be found on the netlib website. 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. 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 cuBLAS (NVIDIA GPU, GPGPU), rocBLAS (AMD GPU), and OpenBLAS. Examples of CPU based BLAS library branches include: OpenBLAS, BLIS (BLAS like Library Instantiation Software), Arm Performance Libraries, ATLAS, and Intel Math Kernel Library (iMKL). AMD maintains a fork of BLIS that is optimized for the AMD platform. ATLAS is a portable library that automatically optimizes itself for an arbitrary architecture. iMKL is a freeware and proprietary vendor library optimized for x86 and x86 64 with a performance emphasis on Intel processors. 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 gemm for its performance measurements. Many numerical software applications use BLAS compatible libraries to do linear algebra computations, including LAPACK, LINPACK, Armadillo, GNU Octave, Mathematica, MATLAB, NumPy, R, Julia and Lisp Stat.

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). 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 operating systems). Between 1973 and 1977, several of these kernel operations were identified. 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 scalars and vectors, the level 1 Basic Linear Algebra Subroutines (BLAS), was published in 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 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. The original BLAS concerned only densely stored vectors and matrices. Further extensions to BLAS, such as for sparse matrices, have been addressed.

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, O(n), Level 2 operations quadratic time and Level 3 operations cubic time. Modern BLAS implementations typically provide all three levels.

Armadillo 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. 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. It provides generic linear algebra subprograms (GLAS). It can be built on a CBLAS implementation.

Elemental Elemental is an open source software for distributed memory dense and sparse direct linear algebra and optimization. HASEM is a C++ template library, being able to solve linear equations and to compute eigenvalues. It is licensed under BSD License. LAMA The Library for Accelerated Math Applications (LAMA) is a C++ template library for writing numerical solvers targeting various kinds of hardware (e. g. GPUs 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.

Several extensions to BLAS for handling 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.