BLAS Support - GNU Scientific Library -- Reference Manual
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## 13 BLAS Support

The Basic Linear Algebra Subprograms (blas) define a set of fundamental operations on vectors and matrices which can be used to create optimized higher-level linear algebra functionality.

The library provides a low-level layer which corresponds directly to the C-language blas standard, referred to here as “cblas”, and a higher-level interface for operations on GSL vectors and matrices. Users who are interested in simple operations on GSL vector and matrix objects should use the high-level layer described in this chapter. The functions are declared in the file `gsl_blas.h` and should satisfy the needs of most users.

Note that GSL matrices are implemented using dense-storage so the interface only includes the corresponding dense-storage blas functions. The full blas functionality for band-format and packed-format matrices is available through the low-level cblas interface. Similarly, GSL vectors are restricted to positive strides, whereas the the low-level cblas interface supports negative strides as specified in the blas standard.1

The interface for the `gsl_cblas` layer is specified in the file `gsl_cblas.h`. This interface corresponds to the blas Technical Forum's standard for the C interface to legacy blas implementations. Users who have access to other conforming cblas implementations can use these in place of the version provided by the library. Note that users who have only a Fortran blas library can use a cblas conformant wrapper to convert it into a cblas library. A reference cblas wrapper for legacy Fortran implementations exists as part of the cblas standard and can be obtained from Netlib. The complete set of cblas functions is listed in an appendix (see GSL CBLAS Library).

There are three levels of blas operations,

Level 1
  Vector operations, e.g. y = \alpha x + y
Level 2
  Matrix-vector operations, e.g. y = \alpha A x + \beta y
Level 3
  Matrix-matrix operations, e.g. C = \alpha A B + C

Each routine has a name which specifies the operation, the type of matrices involved and their precisions. Some of the most common operations and their names are given below,

DOT
  scalar product, x^T y
AXPY
  vector sum, \alpha x + y
MV
  matrix-vector product, A x
SV
  matrix-vector solve, inv(A) x
MM
  matrix-matrix product, A B
SM
  matrix-matrix solve, inv(A) B

The types of matrices are,

GE
  general
GB
  general band
SY
  symmetric
SB
  symmetric band
SP
  symmetric packed
HE
  hermitian
HB
  hermitian band
HP
  hermitian packed
TR
  triangular
TB
  triangular band
TP
  triangular packed

Each operation is defined for four precisions,

S
  single real
D
  double real
C
  single complex
Z
  double complex

Thus, for example, the name sgemm stands for “single-precision general matrix-matrix multiply” and zgemm stands for “double-precision complex matrix-matrix multiply”.

Note that the vector and matrix arguments to BLAS functions must not be aliased, as the results are undefined when the underlying arrays overlap (see Aliasing of arrays).

#### Footnotes

[1] In the low-level cblas interface, a negative stride accesses the vector elements in reverse order, i.e. the i-th element is given by (N-i)*|incx| for incx < 0.

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