Basic Linear Algebra Subprograms

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. Examples of BLAS libraries include: OpenBLAS, BLIS (BLAS-like Library Instantiation Software), Arm Performance Libraries, ATLAS, and Intel Math Kernel Library (MKL). BLIS is optimized for the AMD platform. ATLAS is a portable library that automatically optimizes itself for an arbitrary architecture. MKL 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 codice_1 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, and Julia.
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). 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.
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, , Level 2 operations quadratic time and Level 3 operations cubic time. 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), which defined only "vector operations" on strided arrays: dot products, vector norms, a generalized vector addition of the form
(called "codice_2", "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 (codice_3):
as well as a solver for in the linear equation
with being triangular. Design of the Level 2 BLAS started in 1984, with results published in 1988. The Level 2 subroutines are especially intended to improve performance of programs using BLAS on vector processors, where Level 1 BLAS are suboptimal "because they hide the matrix-vector nature of the operations from the compiler."
Level 3.
This level, formally published in 1990, contains "matrix-matrix operations", including a "general matrix multiplication" (codice_1), of the form
where and can optionally be transposed or hermitian-conjugated inside the routine, and all three matrices may be strided. The ordinary matrix multiplication can be performed by setting to one and to an all-zeros matrix of the appropriate size.
Also included in Level 3 are routines for computing
where 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, and because faster algorithms exist beyond the obvious repetition of matrix-vector multiplication, codice_1 is a prime target of optimization for BLAS implementers. E.g., by decomposing one or both of , into block matrices, codice_1 can be implemented recursively. This is one of the motivations for including the parameter, so the results of previous blocks can be accumulated. Note that this decomposition requires the special case which many implementations optimize for, thereby eliminating one multiplication for each value of . 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 cache on the system. 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 ATLAS. More recently, implementations by Kazushige Goto have shown that blocking only for the L2 cache, combined with careful amortizing of copying to contiguous memory to reduce TLB misses, is superior to ATLAS. A highly tuned implementation based on these ideas is part of the GotoBLAS, OpenBLAS and BLIS.
A common variation of is the , 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.
Sparse BLAS.
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.