The basic linear algebra subprograms, normally referred to as the BLAS, are routines for low-level operations such as dot products, matrix times vector, and matrix times matrix. Lawson et al. (1979) published the original set of 38 BLAS. The IMSL BLAS collection includes these 38 subprograms plus additional ones that extend their functionality. Since Dongarra et al. (1988 and 1990) published extensions to this set, it is customary to refer to the original 38 as Level 1 BLAS. The Level 1 operations are performed on one or two vectors of data. An extended set of subprograms perform operations involving a matrix and one or two vectors. These are called the Level 2 BLAS (see Specification of the Level 2 BLAS). An additional extended set of operations on matrices is called the Level 3 BLAS (see Specification of the Level 3 BLAS).

Users of the BLAS will often benefit from using versions of the BLAS supplied by hardware vendors, if available. This can provide for more efficient execution of many application programs. The BLAS provided by IMSL are written in FORTRAN. Those supplied by vendors may be written in other languages, such as assembler. The documentation given below for the BLAS is compatible with a vendor’s version of the BLAS that conforms to the published specifications.

The Level 1 BLAS do not follow the usual IMSL naming conventions. Instead, the names consist of a prefix of one or more of the letters “I”, “S”, “D”, “C”, and “Z”; a root name; and sometimes a suffix. For subprograms involving a mixture of data types, the output type is indicated by the first prefix letter. The suffix denotes a variant algorithm. The prefix denotes the type of the operation according to the following table:

Vector arguments have an increment parameter that specifies the storage space or stride between elements. The correspondence between the vectors x and y and the arguments SX and SY, and INCX and INCY is

Function subprograms SXYZ and DXYZ refer to a third vector argument z. The storage increment INCZ for z is defined like INCX and INCY. In the Level 1 BLAS, only positive values of INCX are allowed for operations that have a single vector argument. The loops in all of the Level 1 BLAS process the vector arguments in order of increasing i. For INCX, INCY, INCZ < 0, this implies processing in reverse storage order.

The function subprograms in the Level 1 BLAS are all illustrated by means of an assignment statement. For example, see SDOT. Any value of a function subprogram can be used in an expression or as a parameter passed to a subprogram as long as the data types agree.

The set of Level 1 BLAS are summarized in Table 6. This table also lists the page numbers where the subprograms are described in more detail.

With the definitions,

the subprogram descriptions assume the following FORTRAN declarations:

Since FORTRAN 77 does not include the type DOUBLE COMPLEX, subprograms with DOUBLE COMPLEX arguments are not available for all systems. Some systems use the declaration COMPLEX * 16 instead of DOUBLE COMPLEX.

In the following descriptions, the original BLAS are marked with an * in the left column.

†Higher precision accumulation used

These subprograms set xi ← a for i = 1, 2, …, N. If N ≤ 0, then the subprograms return immediately.

These subprograms set yi ← xi for i = 1, 2, …, N. If N ≤ 0, then the subprograms return immediately.

These subprograms set xi ← axi for i = 1, 2, …, N. If N ≤ 0, then the subprograms return immediately. CAUTION: For CSSCAL and ZDSCAL, the scalar quantity a is real and the vector x is complex.

These subprograms set yi ← axi for i = 1, 2, …, N. If N ≤ 0, then the subprograms return immediately. CAUTION: For CSVCAL and ZDVCAL, the scalar quantity a is real and the vector x is complex.

These subprograms set xi ← xi + a for i = 1, 2, …, N. If N ≤ 0, then the subprograms return immediately.

These subprograms set xi ← a ‑ xi for i = 1, 2, …, N. If N ≤ 0, then the subprograms return immediately.

These subprograms set yi ← axi + yi for i = 1, 2, …, N. If N ≤ 0, then the subprograms return immediately.

These subprograms perform the exchange yi ↔ xi for i = 1, 2, …, N. If N ≤ 0, then the subprograms return immediately.

The function subprograms SDOT, DDOT, CDOTU, and ZDOTU compute

The function subprograms CDOTC and ZDOTC compute

The suffix C indicates that the complex conjugates of xi are used. The suffix U indicates that the unconjugated values of xi are used. If N ≤ 0, then the subprograms return zero.

The function subprogram DSDOT computes

using double precision accumulation. The function subprograms CZDOTU and ZQDOTU compute

using double and quadruple complex accumulation, respectively. The function subprograms CZDOTC and ZQDOTC compute

using double and quadruple complex accumulation, respectively. If N ≤ 0, then the subprograms return zero.

The function subprograms SDSDOT, DQDDOT, CZUDOT, and ZQUDOT compute

using higher precision accumulation where SDSDOT uses double precision accumulation, DQDDOT uses quadruple precision accumulation, CZUDOT uses double complex accumulation, and ZQUDOT uses quadruple complex accumulation. The function subprograms CZCDOT and ZQCDOT compute

using double complex and quadruple complex accumulation, respectively. If N ≤ 0, then the subprograms return zero.

The variable DACC, a double precision array of length two, is used as a quadruple precision accumulator. DZACC, a double precision array of length four, is its complex analog. The function subprograms with a name ending in I initialize DACC to zero. All of the function subprograms then compute

and store the result in DACC. The result, converted to the precision of the function, is also returned as the function value. If N ≤ 0, then the function subprograms return zero.

These subprograms set zi ← xiyi for i = 1, 2, …, N. If N ≤ 0, then the subprograms return immediately.

These function subprograms compute

If N ≤ 0 then the subprograms return zero.

These function subprograms compute

If N ≤ 0, then the subprograms return zero.

The function subprograms SASUM and DASUM compute

The function subprograms SCASUM and DZASUM compute

If N ≤ 0, then the subprograms return zero. CAUTION: For SCASUM and DZASUM, the function subprogram returns a real value.

These function subprograms compute

If N ≤ 0, then the subprograms return zero. CAUTION: For SCNRM2 and DZNRM2, the function subprogram returns a real value.

These function subprograms compute

If N ≤ 0, then the subprograms return zero.

These function subprograms compute the smallest index i such that xi = min1 ≤ j ≤ Nxj. If N ≤ 0, then the subprograms return zero.

These function subprograms compute the smallest index i such that xi = max1 ≤ j ≤ Nxj. If N ≤ 0, then the subprograms return zero.

The function subprograms ISAMIN and IDAMIN compute the smallest index i such that ∣xi∣ = min1 ≤ j ≤ N∣xj∣. The function subprograms ICAMIN and IZAMIN compute the smallest index i such that

If N ≤ 0, then the subprograms return zero.

The function subprograms ISAMAX and IDAMAX compute the smallest index i such that
∣xi∣ = max1 ≤ j≤N∣xj∣. The function subprograms ICAMAX and IZAMAX compute the smallest index i such that

If N ≤ 0, then the subprograms return zero.

Given the values a and b, these subprograms compute

and

where r = σ(a2 + b2)1/2 and

Then, the values c, s and r satisfy the matrix equation

The introduction of σ is not essential to the computation of the Givens rotation matrix; but its use permits later stable reconstruction of c and s from just one stored number, an idea due to Stewart (1976). For this purpose, the subprogram also computes

In addition to returning c and s, the subprograms return r overwriting a, and z overwriting b.

Reconstruction of c and s from z can be done as follows:

If z = 1, then set c = 0 and s = 1

If ∣z∣ < 1, then set

If ∣z∣ > 1, then set

These subprograms compute

If N ≤ 0, then the subprograms return immediately. CAUTION: For CSROT and ZDROT, the scalar quantities c and s are real, and x and y are complex.

The input quantities d1, d2, x1 and y1 define a 2-vector [w1, z1]T by the following:

The subprograms determine the modified Givens rotation matrix H that transforms y1, and thus, z1 to zero. They also replace d1, d2 and x1 with

respectively. That is,

A representation of this matrix is stored in the array SPARAM or DPARAM. The form of the matrix H is flagged by PARAM(1).

PARAM(1) = 1. In this case,

and

The elements PARAM(3) and PARAM(4) are not changed.

PARAM(1) = 0. In this case,

and

The elements PARAM(2) and PARAM(5) are not changed.

PARAM(1) = ‑1. In this case, rescaling was done and

PARAM(2) = ‑2. In this case, H = I where I is the identity matrix. The elements PARAM(2), PARAM(3), PARAM(4) and PARAM(5) are not changed.

The values of d1, d2 and x1are changed to represent the effect of the transformation. The quantity y1, which would be zeroed by the transformation, is left unchanged.

The input value of d1 should be nonnegative, but d2 can be negative for the purpose of removing data from a least-squares problem.

See Lawson et al. (1979) for further details.

If PARAM(1) = 1.0, then these subprograms compute

If PARAM(1) = 0.0, then the subprograms compute

If PARAM(1) = 1.0, then the subprograms compute

If N ≤ 0 or if PARAM(1) = ‑2.0, then the subprograms return immediately. CAUTION: For CSROTM and ZDROTM, the scalar quantities PARAM(*) are real and x and y are complex.

For definitions of the matrix data structures used in the discussion below, see Reference Material. The Level 2 and Level 3 BLAS, like the Level 1 BLAS, do not follow the IMSL naming conventions. Instead, the names consist of a prefix of one of the letters “S”, “D” , “C” , or “Z”. Next is a root name denoting the kind of matrix. This is followed by a suffix indicating the type of the operation.1 The prefix denotes the type of operation according to the following table:

The root names for the kind of matrix:

The suffixes for the type of operation:

1IMSL does not support any extended precision versions of the Level 2 BLAS.

The specifications of the operations are provided by subprogram arguments of CHARACTER*1 data type. Both lower and upper case of the letter have the same meaning:

Note: See the “Triangular Mode” section in the Reference Material for definitions of these terms.

The subprograms for Level 2 and Level 3 BLAS that perform operations involving the expression βy or βC do not require that the contents of y or C be defined when β = 0. In that case, the expression βy or βC is defined to be zero. Note that for the _GEMV and _GBMV subprograms, the dimensions of the vectors x and y are implied by the specification of the operation. If TRANS = ’N’, the dimension of y is m; if TRANS = ’T’ or = ’C’, the dimension of y is n. The Level 2 and Level 3 BLAS are summarized in Table 9.2. This table also lists the page numbers where the subprograms are described in more detail.

Type and dimension for variables occurring in the subprogram specifications are

There is a lower bound on the leading dimension LDA. It must be ≥ the number of rows in the matrix that is contained in this array. Vector arguments have an increment parameter that specifies the storage space or stride between elements. The correspondence between the vector x, y and the arguments SX, SY and INCX, INCY is

In the Level 2 BLAS, only nonzero values of INCX, INCY are allowed for operations that have vector arguments. The Level 3 BLAS do not refer to INCX, INCY.

Type and dimension for variables occurring in the subprogram specifications are

Each of the integers K, M, N must be ≥ 0. It is an error if any of them are < 0. If any of them are = 0, the subprograms return immediately. There are lower bounds on the leading dimensions LDA, LDB, LDC. Each must be ≥ the number of rows in the matrix that is contained in this array.

Table 7 describes the Level 2 and 3 BLAS subprograms.

For all data types, A is an M × N matrix. These subprograms set y to one of the expressions: y ← αAx + βy, y ← αATx + βy, or for complex data,

The character flag TRANS determines the operation.

For all data types, A is an M × N matrix with NLCA lower codiagonals and NUCA upper codiagonals. The matrix is stored in band storage mode. These subprograms set y to one of the expressions: y ← αAx + βy, y ← αATx + βy, or for complex data,

The character flag TRANS determines the operation.

For complex types, A is an N × N matrix. These subprograms set y ← αAx + βy, where A is an Hermitian matrix. The matrix A is either referenced using the upper or lower triangular part. The character flag UPLO determines the part used.

For complex types, A is an N × N matrix. These subprograms set y ← αAx + βy, where A is an Hermitian matrix. The matrix A is either referenced using the packed upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N matrix with NCODA codiagonals. The matrix is stored in band Hermitian storage mode. These subprograms set y ← αAx + βy. The matrix A is either referenced using its upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N matrix. These subprograms set y ← αAx + βy, where A is a symmetric matrix. The matrix A is either referenced using the upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N matrix. These subprograms set y ← αAx + βy, where A is a packed triangular matrix. The matrix A is either referenced using the packed upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N matrix with NCODA codiagonals. The matrix is stored in band symmetric storage mode. These subprograms set y ← αAx + βy. The matrix A is either referenced using its upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N triangular matrix. These subprograms set x to one of the expressions: x ← Ax, x ← ATx, or for complex data,

The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAGNL determine the part of the matrix used and the operation performed.

For all data types, A is an N × N packed triangular matrix. These subprograms set x to one of the expressions: x ← Ax, x ← ATx, or for complex data,

The matrix A is either referenced using the packed upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAGNL determine the part of the matrix used and the operation performed.

For all data types, A is an N × N matrix with NCODA codiagonals. The matrix is stored in band triangular storage mode. These subprograms set x to one of the expressions: x ← Ax, x ← ATx, or for complex data,

The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAGNL determine the part of the matrix used and the operation performed.

For all data types, A is an N × N triangular matrix. These subprograms solve x for one of the expressions: x ← A-1x, x ← (A-1)Tx, or for complex data,

The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAGNL determine the part of the matrix used and the operation performed.

For all data types, A is an N × N triangular matrix with NCODA codiagonals. The matrix is stored in band triangular storage mode. These subprograms solve x for one of the expressions: x ← A-1x, x ← (A-T) -1x, or for complex data,

The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAGNL determine the part of the matrix used and the operation performed.

For all data types, A is an N × N packed triangular matrix. These subprograms solve x for one of the expressions: x ← A-1x, x ← (A-1)Tx, or for complex data,

The matrix A is either referenced using its packed upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAGNL determine the part of the matrix used and the operation performed.

For all data types, A is an M × N matrix. These subprograms set A ← A + αxyT.

For all data types, A is an M × N matrix. These subprograms set A ← A + αxyT.

For all data types, A is an M × N matrix. These subprograms set

For all data types, A is an N × N matrix. These subprograms set

where A is Hermitian. The matrix A is either referenced by its upper or lower triangular part. The character flag UPLO determines the part used. CAUTION: Notice the scalar parameter α is real, and the data in the matrix and vector are complex.

For all data types, A is an N × N matrix. These subprograms set

where A is packed Hermitian. The matrix A is either referenced by its upper or lower triangular part. The character flag UPLO determines the part used. CAUTION: Notice the scalar parameter α is real, and the data in the matrix and vector are complex.

For all data types, A is an N × N matrix. These subprograms set

where A is an Hermitian matrix. The matrix A is either referenced by its upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N matrix. These subprograms set

where A is a packed Hermitian matrix. The matrix A is either referenced by its upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N matrix. These subprograms set A ← A + αxxT where A is a symmetric matrix. The matrix A is either referenced by its upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N matrix. These subprograms set A ← A + αxxT where A is a packed symmetric matrix. The matrix A is either referenced using the packed upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N matrix. These subprograms set A ← A + αxxT where A is a packed Hermitian matrix. The matrix A is either referenced using the packed upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N matrix. These subprograms set A ← A + αxyT + αyxT, where A is a symmetric matrix. The matrix A is referenced by its upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, A is an N × N matrix. These subprograms set A ← A + αxyT + αyxT, where A is a packed symmetric matrix. The matrix A is referenced by its upper or lower triangular part. The character flag UPLO determines the part used.

For all data types, these subprograms set CM×N to one of the expressions:

The character flags TRANSA and TRANSB determine the operation to be performed. Each matrix product has dimensions that follow from the fact that C has dimension M × N.

For all data types, these subprograms set CM×N to one of the expressions: C ← αAB + βC or C ← αBA + βC, where A is a symmetric matrix. The matrix A is referenced either by its upper or lower triangular part. The character flags SIDE and UPLO determine the part of the matrix used and the operation performed.

For all data types, these subprograms set CM×N to one of the expressions: C ← αAB + βC or C ← αBA + βC, where A is an Hermitian matrix. The matrix A is referenced either by its upper or lower triangular part. The character flags SIDE and UPLO determine the part of the matrix used and the operation performed.

For all data types, these subprograms set CM×N to one of the expressions: C ← αAAT + βC or C ← αATA + βC. The matrix C is referenced either by its upper or lower triangular part. The character flags UPLO and TRANS determine the part of the matrix used and the operation performed. In subprogram CSYRK and ZSYRK, only values ’N’ or ’T’ are allowed for TRANS; ’C’ is not acceptable.

For all data types, these subprograms set CN× N to one of the expressions:

The matrix C is referenced either by its upper or lower triangular part. The character flags UPLO and TRANS determine the part of the matrix used and the operation performed. CAUTION: Notice the scalar parameters α and β are real, and the data in the matrices are complex. Only values ’N’or ’C’are allowed for TRANS; ’T’is not acceptable.

For all data types, these subprograms set CN× N to one of the expressions:

The matrix C is referenced either by its upper or lower triangular part. The character flags UPLO and TRANS determine the part of the matrix used and the operation performed. In subprogram CSYR2K and ZSYR2K, only values ’N’or ’T’ are allowed for TRANS; ’C’is not acceptable.

For all data types, these subprograms set CN× N to one of the expressions:

The matrix C is referenced either by its upper or lower triangular part. The character flags UPLO and TRANS determine the part of the matrix used and the operation performed. CAUTION: Notice the scalar parameter β is real, and the data in the matrices are complex. In subprogram CHER2K and ZHER2K, only values ’N’ or ’C’are allowed for TRANS; ’T’is not acceptable.

For all data types, these subprograms set BM×N to one of the_expressions:

or for complex data,

where A is a triangular matrix. The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags SIDE, UPLO, TRANSA, and DIAGNL determine the part of the matrix used and the operation performed.

For all data types, these subprograms set BM×N to one of the expressions:

or for complex data,

where A is a triangular matrix. The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags SIDE, UPLO, TRANSA, and DIAGNL determine the part of the matrix used and the operation performed.