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The BLAS (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. There are three levels within the BLAS library:

The Level 1 BLAS perform scalar, vector and vectorvector operations

The Level 2 BLAS perform matrixvector operations

The Level 3 BLAS perform matrixmatrix operations
As the BLAS are efficient, portable, and widely available, they are commonly used in the development of high quality linear algebra software, LAPACK for example.
Popular BLAS (Basic Linear Algebra Subprograms) Implementations include:
 Intel's MKL (used in MKL DNN)
 OpenBLAS
 NetLib's BLAS
 BLAS++
 BLIS
Use
BLAS Libraries are used for the following:
 Speed up computations dealing with matrix and other operations
 Used within other optimized libraries such as MKL DNN
 Used within popular frameworks like TensorFlow
Level 1 BLAS routines
 SROTG  setup Givens rotation
 SROTMG  setup modified Givens rotation
 SROT  apply Givens rotation
 SROTM  apply modified Givens rotation
 SSWAP  swap x and y
 SSCAL  x = a*x
 SCOPY  copy x into y
 SAXPY  y = a*x + y
 SDOT  dot product
 SDSDOT  dot product with extended precision accumulation
 SNRM2  Euclidean norm
 SCNRM2 Euclidean norm
 SASUM  sum of absolute values
 ISAMAX  index of max abs value
 DOUBLE
 DROTG  setup Givens rotation
 DROTMG  setup modified Givens rotation
 DROT  apply Givens rotation
 DROTM  apply modified Givens rotation
 DSWAP  swap x and y
 DSCAL  x = a*x
 DCOPY  copy x into y
 DAXPY  y = a*x + y
 DDOT  dot product
 DSDOT  dot product with extended precision accumulation
 DNRM2  Euclidean norm
 DZNRM2  Euclidean norm
 DASUM  sum of absolute values
 IDAMAX  index of max abs value
 CROTG  setup Givens rotation
 CSROT  apply Givens rotation
 CSWAP  swap x and y
 CSCAL  x = a*x
 CSSCAL  x = a*x
 CCOPY  copy x into y
 CAXPY  y = a*x + y
 CDOTU  dot product
 CDOTC  dot product, conjugating the first vector
 SCASUM  sum of absolute values
 ICAMAX  index of max abs value
 DOUBLE COMLPEX
 ZROTG  setup Givens rotation
 ZDROTF  apply Givens rotation
 ZSWAP  swap x and y
 ZSCAL  x = a*x
 ZDSCAL  x = a*x
 ZCOPY  copy x into y
 ZAXPY  y = a*x + y
 ZDOTU  dot product
 ZDOTC  dot product, conjugating the first vector
 DZASUM  sum of absolute values
 IZAMAX  index of max abs value
Level 2 BLAS routines
 SGEMV  matrix vector multiply
 SGBMV  banded matrix vector multiply
 SSYMV  symmetric matrix vector multiply
 SSBMV  symmetric banded matrix vector multiply
 SSPMV  symmetric packed matrix vector multiply
 STRMV  triangular matrix vector multiply
 STBMV  triangular banded matrix vector multiply
 STPMV  triangular packed matrix vector multiply
 STRSV  solving triangular matrix problems
 STBSV  solving triangular banded matrix problems
 STPSV  solving triangular packed matrix problems
 SGER  performs the rank 1 operation A := alphaxy' + A
 SSYR  performs the symmetric rank 1 operation A := alphaxx' + A
 SSPR  symmetric packed rank 1 operation A := alphaxx' + A
 SSYR2  performs the symmetric rank 2 operation, A := alphaxy' + alphayx' + A
 SSPR2  performs the symmetric packed rank 2 operation, A := alphaxy' + alphayx' + A
 DGEMV  matrix vector multiply
 DGBMV  banded matrix vector multiply
 DSYMV  symmetric matrix vector multiply
 DSBMV  symmetric banded matrix vector multiply
 DSPMV  symmetric packed matrix vector multiply
 DTRMV  triangular matrix vector multiply
 DTBMV  triangular banded matrix vector multiply
 DTPMV  triangular packed matrix vector multiply
 DTRSV  solving triangular matrix problems
 DTBSV  solving triangular banded matrix problems
 DTPSV  solving triangular packed matrix problems
 DGER  performs the rank 1 operation A := alphaxy' + A
 DSYR  performs the symmetric rank 1 operation A := alphaxx' + A
 DSPR  symmetric packed rank 1 operation A := alphaxx' + A
 DSYR2  performs the symmetric rank 2 operation, A := alphaxy' + alphayx' + A
 DSPR2  performs the symmetric packed rank 2 operation, A := alphaxy' + alphayx' + A
 CGEMV  matrix vector multiply
 CGBMV  banded matrix vector multiply
 CHEMV  hermitian matrix vector multiply
 CHBMV  hermitian banded matrix vector multiply
 CHPMV  hermitian packed matrix vector multiply
 CTRMV  triangular matrix vector multiply
 CTBMV  triangular banded matrix vector multiply
 CTPMV  triangular packed matrix vector multiply
 CTRSV  solving triangular matrix problems
 CTBSV  solving triangular banded matrix problems
 CTPSV  solving triangular packed matrix problems
 CGERU  performs the rank 1 operation A := alphaxy' + A
 CGERC  performs the rank 1 operation A := alphaxconjg( y' ) + A
 CHER  hermitian rank 1 operation A := alphaxconjg(x') + A
 CHPR  hermitian packed rank 1 operation A := alphaxconjg( x' ) + A
 CHER2  hermitian rank 2 operation
 CHPR2  hermitian packed rank 2 operation
 ZGEMV  matrix vector multiply
 ZGBMV  banded matrix vector multiply
 ZHEMV  hermitian matrix vector multiply
 ZHBMV  hermitian banded matrix vector multiply
 ZHPMV  hermitian packed matrix vector multiply
 ZTRMV  triangular matrix vector multiply
 ZTBMV  triangular banded matrix vector multiply
 ZTPMV  triangular packed matrix vector multiply
 ZTRSV  solving triangular matrix problems
 ZTBSV  solving triangular banded matrix problems
 ZTPSV  solving triangular packed matrix problems
 ZGERU  performs the rank 1 operation A := alphaxy' + A
 ZGERC  performs the rank 1 operation A := alphaxconjg( y' ) + A
 ZHER  hermitian rank 1 operation A := alphaxconjg(x') + A
 ZHPR  hermitian packed rank 1 operation A := alphaxconjg( x' ) + A
 ZHER2  hermitian rank 2 operation
 ZHPR2  hermitian packed rank 2 operation
Level 3 BLAS routines
 SGEMM  matrix matrix multiply
 SSYMM  symmetric matrix matrix multiply
 SSYRK  symmetric rankk update to a matrix
 SSYR2K  symmetric rank2k update to a matrix
 STRMM  triangular matrix matrix multiply
 STRSM  solving triangular matrix with multiple right hand sides
 DGEMM  matrix matrix multiply
 DSYMM  symmetric matrix matrix multiply
 DSYRK  symmetric rankk update to a matrix
 DSYR2K  symmetric rank2k update to a matrix
 DTRMM  triangular matrix matrix multiply
 DTRSM  solving triangular matrix with multiple right hand sides
 CGEMM  matrix matrix multiply
 CSYMM  symmetric matrix matrix multiply
 CHEMM  hermitian matrix matrix multiply
 CSYRK  symmetric rankk update to a matrix
 CHERK  hermitian rankk update to a matrix
 CSYR2K  symmetric rank2k update to a matrix
 CHER2K  hermitian rank2k update to a matrix
 CTRMM  triangular matrix matrix multiply
 CTRSM  solving triangular matrix with multiple right hand sides
 ZGEMM  matrix matrix multiply
 ZSYMM  symmetric matrix matrix multiply
 ZHEMM  hermitian matrix matrix multiply
 ZSYRK  symmetric rankk update to a matrix
 ZHERK  hermitian rankk update to a matrix
 ZSYR2K  symmetric rank2k update to a matrix
 ZHER2K  hermitian rank2k update to a matrix
 ZTRMM  triangular matrix matrix multiply
 ZTRSM  solving triangular matrix with multiple right hand sides