## table of contents

single_blas_level3(3) | LAPACK | single_blas_level3(3) |

# NAME¶

single_blas_level3

# SYNOPSIS¶

## Functions¶

subroutine **sgemm** (TRANSA, TRANSB, M, N, K, ALPHA, A, LDA,
B, LDB, BETA, C, LDC)

**SGEMM** subroutine **ssymm** (SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB,
BETA, C, LDC)

**SSYMM** subroutine **ssyr2k** (UPLO, TRANS, N, K, ALPHA, A, LDA, B,
LDB, BETA, C, LDC)

**SSYR2K** subroutine **ssyrk** (UPLO, TRANS, N, K, ALPHA, A, LDA, BETA,
C, LDC)

**SSYRK** subroutine **strmm** (SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA,
A, LDA, B, LDB)

**STRMM** subroutine **strsm** (SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA,
A, LDA, B, LDB)

**STRSM**

# Detailed Description¶

This is the group of real LEVEL 3 BLAS routines.

# Function Documentation¶

## subroutine sgemm (character TRANSA, character TRANSB, integer M, integer N, integer K, real ALPHA, real, dimension(lda,*) A, integer LDA, real, dimension(ldb,*) B, integer LDB, real BETA, real, dimension(ldc,*) C, integer LDC)¶

**SGEMM**

**Purpose:**

SGEMM performs one of the matrix-matrix operations

C := alpha*op( A )*op( B ) + beta*C,

where op( X ) is one of

op( X ) = X or op( X ) = X**T,

alpha and beta are scalars, and A, B and C are matrices, with op( A )

an m by k matrix, op( B ) a k by n matrix and C an m by n matrix.

**Parameters**

*TRANSA*

TRANSA is CHARACTER*1

On entry, TRANSA specifies the form of op( A ) to be used in

the matrix multiplication as follows:

TRANSA = 'N' or 'n', op( A ) = A.

TRANSA = 'T' or 't', op( A ) = A**T.

TRANSA = 'C' or 'c', op( A ) = A**T.

*TRANSB*

TRANSB is CHARACTER*1

On entry, TRANSB specifies the form of op( B ) to be used in

the matrix multiplication as follows:

TRANSB = 'N' or 'n', op( B ) = B.

TRANSB = 'T' or 't', op( B ) = B**T.

TRANSB = 'C' or 'c', op( B ) = B**T.

*M*

M is INTEGER

On entry, M specifies the number of rows of the matrix

op( A ) and of the matrix C. M must be at least zero.

*N*

N is INTEGER

On entry, N specifies the number of columns of the matrix

op( B ) and the number of columns of the matrix C. N must be

at least zero.

*K*

K is INTEGER

On entry, K specifies the number of columns of the matrix

op( A ) and the number of rows of the matrix op( B ). K must

be at least zero.

*ALPHA*

ALPHA is REAL

On entry, ALPHA specifies the scalar alpha.

*A*

A is REAL array, dimension ( LDA, ka ), where ka is

k when TRANSA = 'N' or 'n', and is m otherwise.

Before entry with TRANSA = 'N' or 'n', the leading m by k

part of the array A must contain the matrix A, otherwise

the leading k by m part of the array A must contain the

matrix A.

*LDA*

LDA is INTEGER

On entry, LDA specifies the first dimension of A as declared

in the calling (sub) program. When TRANSA = 'N' or 'n' then

LDA must be at least max( 1, m ), otherwise LDA must be at

least max( 1, k ).

*B*

B is REAL array, dimension ( LDB, kb ), where kb is

n when TRANSB = 'N' or 'n', and is k otherwise.

Before entry with TRANSB = 'N' or 'n', the leading k by n

part of the array B must contain the matrix B, otherwise

the leading n by k part of the array B must contain the

matrix B.

*LDB*

LDB is INTEGER

On entry, LDB specifies the first dimension of B as declared

in the calling (sub) program. When TRANSB = 'N' or 'n' then

LDB must be at least max( 1, k ), otherwise LDB must be at

least max( 1, n ).

*BETA*

BETA is REAL

On entry, BETA specifies the scalar beta. When BETA is

supplied as zero then C need not be set on input.

*C*

C is REAL array, dimension ( LDC, N )

Before entry, the leading m by n part of the array C must

contain the matrix C, except when beta is zero, in which

case C need not be set on entry.

On exit, the array C is overwritten by the m by n matrix

( alpha*op( A )*op( B ) + beta*C ).

*LDC*

LDC is INTEGER

On entry, LDC specifies the first dimension of C as declared

in the calling (sub) program. LDC must be at least

max( 1, m ).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 3 Blas routine.

-- Written on 8-February-1989.

Jack Dongarra, Argonne National Laboratory.

Iain Duff, AERE Harwell.

Jeremy Du Croz, Numerical Algorithms Group Ltd.

Sven Hammarling, Numerical Algorithms Group Ltd.

## subroutine ssymm (character SIDE, character UPLO, integer M, integer N, real ALPHA, real, dimension(lda,*) A, integer LDA, real, dimension(ldb,*) B, integer LDB, real BETA, real, dimension(ldc,*) C, integer LDC)¶

**SSYMM**

**Purpose:**

SSYMM performs one of the matrix-matrix operations

C := alpha*A*B + beta*C,

or

C := alpha*B*A + beta*C,

where alpha and beta are scalars, A is a symmetric matrix and B and

C are m by n matrices.

**Parameters**

*SIDE*

SIDE is CHARACTER*1

On entry, SIDE specifies whether the symmetric matrix A

appears on the left or right in the operation as follows:

SIDE = 'L' or 'l' C := alpha*A*B + beta*C,

SIDE = 'R' or 'r' C := alpha*B*A + beta*C,

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the symmetric matrix A is to be

referenced as follows:

UPLO = 'U' or 'u' Only the upper triangular part of the

symmetric matrix is to be referenced.

UPLO = 'L' or 'l' Only the lower triangular part of the

symmetric matrix is to be referenced.

*M*

M is INTEGER

On entry, M specifies the number of rows of the matrix C.

M must be at least zero.

*N*

N is INTEGER

On entry, N specifies the number of columns of the matrix C.

N must be at least zero.

*ALPHA*

ALPHA is REAL

On entry, ALPHA specifies the scalar alpha.

*A*

A is REAL array, dimension ( LDA, ka ), where ka is

m when SIDE = 'L' or 'l' and is n otherwise.

Before entry with SIDE = 'L' or 'l', the m by m part of

the array A must contain the symmetric matrix, such that

when UPLO = 'U' or 'u', the leading m by m upper triangular

part of the array A must contain the upper triangular part

of the symmetric matrix and the strictly lower triangular

part of A is not referenced, and when UPLO = 'L' or 'l',

the leading m by m lower triangular part of the array A

must contain the lower triangular part of the symmetric

matrix and the strictly upper triangular part of A is not

referenced.

Before entry with SIDE = 'R' or 'r', the n by n part of

the array A must contain the symmetric matrix, such that

when UPLO = 'U' or 'u', the leading n by n upper triangular

part of the array A must contain the upper triangular part

of the symmetric matrix and the strictly lower triangular

part of A is not referenced, and when UPLO = 'L' or 'l',

the leading n by n lower triangular part of the array A

must contain the lower triangular part of the symmetric

matrix and the strictly upper triangular part of A is not

referenced.

*LDA*

LDA is INTEGER

On entry, LDA specifies the first dimension of A as declared

in the calling (sub) program. When SIDE = 'L' or 'l' then

LDA must be at least max( 1, m ), otherwise LDA must be at

least max( 1, n ).

*B*

B is REAL array, dimension ( LDB, N )

Before entry, the leading m by n part of the array B must

contain the matrix B.

*LDB*

LDB is INTEGER

On entry, LDB specifies the first dimension of B as declared

in the calling (sub) program. LDB must be at least

max( 1, m ).

*BETA*

BETA is REAL

On entry, BETA specifies the scalar beta. When BETA is

supplied as zero then C need not be set on input.

*C*

C is REAL array, dimension ( LDC, N )

Before entry, the leading m by n part of the array C must

contain the matrix C, except when beta is zero, in which

case C need not be set on entry.

On exit, the array C is overwritten by the m by n updated

matrix.

*LDC*

LDC is INTEGER

On entry, LDC specifies the first dimension of C as declared

in the calling (sub) program. LDC must be at least

max( 1, m ).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 3 Blas routine.

-- Written on 8-February-1989.

Jack Dongarra, Argonne National Laboratory.

Iain Duff, AERE Harwell.

Jeremy Du Croz, Numerical Algorithms Group Ltd.

Sven Hammarling, Numerical Algorithms Group Ltd.

## subroutine ssyr2k (character UPLO, character TRANS, integer N, integer K, real ALPHA, real, dimension(lda,*) A, integer LDA, real, dimension(ldb,*) B, integer LDB, real BETA, real, dimension(ldc,*) C, integer LDC)¶

**SSYR2K**

**Purpose:**

SSYR2K performs one of the symmetric rank 2k operations

C := alpha*A*B**T + alpha*B*A**T + beta*C,

or

C := alpha*A**T*B + alpha*B**T*A + beta*C,

where alpha and beta are scalars, C is an n by n symmetric matrix

and A and B are n by k matrices in the first case and k by n

matrices in the second case.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the array C is to be referenced as

follows:

UPLO = 'U' or 'u' Only the upper triangular part of C

is to be referenced.

UPLO = 'L' or 'l' Only the lower triangular part of C

is to be referenced.

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the operation to be performed as

follows:

TRANS = 'N' or 'n' C := alpha*A*B**T + alpha*B*A**T +

beta*C.

TRANS = 'T' or 't' C := alpha*A**T*B + alpha*B**T*A +

beta*C.

TRANS = 'C' or 'c' C := alpha*A**T*B + alpha*B**T*A +

beta*C.

*N*

N is INTEGER

On entry, N specifies the order of the matrix C. N must be

at least zero.

*K*

K is INTEGER

On entry with TRANS = 'N' or 'n', K specifies the number

of columns of the matrices A and B, and on entry with

TRANS = 'T' or 't' or 'C' or 'c', K specifies the number

of rows of the matrices A and B. K must be at least zero.

*ALPHA*

ALPHA is REAL

On entry, ALPHA specifies the scalar alpha.

*A*

A is REAL array, dimension ( LDA, ka ), where ka is

k when TRANS = 'N' or 'n', and is n otherwise.

Before entry with TRANS = 'N' or 'n', the leading n by k

part of the array A must contain the matrix A, otherwise

the leading k by n part of the array A must contain the

matrix A.

*LDA*

LDA is INTEGER

On entry, LDA specifies the first dimension of A as declared

in the calling (sub) program. When TRANS = 'N' or 'n'

then LDA must be at least max( 1, n ), otherwise LDA must

be at least max( 1, k ).

*B*

B is REAL array, dimension ( LDB, kb ), where kb is

k when TRANS = 'N' or 'n', and is n otherwise.

Before entry with TRANS = 'N' or 'n', the leading n by k

part of the array B must contain the matrix B, otherwise

the leading k by n part of the array B must contain the

matrix B.

*LDB*

LDB is INTEGER

On entry, LDB specifies the first dimension of B as declared

in the calling (sub) program. When TRANS = 'N' or 'n'

then LDB must be at least max( 1, n ), otherwise LDB must

be at least max( 1, k ).

*BETA*

BETA is REAL

On entry, BETA specifies the scalar beta.

*C*

C is REAL array, dimension ( LDC, N )

Before entry with UPLO = 'U' or 'u', the leading n by n

upper triangular part of the array C must contain the upper

triangular part of the symmetric matrix and the strictly

lower triangular part of C is not referenced. On exit, the

upper triangular part of the array C is overwritten by the

upper triangular part of the updated matrix.

Before entry with UPLO = 'L' or 'l', the leading n by n

lower triangular part of the array C must contain the lower

triangular part of the symmetric matrix and the strictly

upper triangular part of C is not referenced. On exit, the

lower triangular part of the array C is overwritten by the

lower triangular part of the updated matrix.

*LDC*

LDC is INTEGER

On entry, LDC specifies the first dimension of C as declared

in the calling (sub) program. LDC must be at least

max( 1, n ).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 3 Blas routine.

-- Written on 8-February-1989.

Jack Dongarra, Argonne National Laboratory.

Iain Duff, AERE Harwell.

Jeremy Du Croz, Numerical Algorithms Group Ltd.

Sven Hammarling, Numerical Algorithms Group Ltd.

## subroutine ssyrk (character UPLO, character TRANS, integer N, integer K, real ALPHA, real, dimension(lda,*) A, integer LDA, real BETA, real, dimension(ldc,*) C, integer LDC)¶

**SSYRK**

**Purpose:**

SSYRK performs one of the symmetric rank k operations

C := alpha*A*A**T + beta*C,

or

C := alpha*A**T*A + beta*C,

where alpha and beta are scalars, C is an n by n symmetric matrix

and A is an n by k matrix in the first case and a k by n matrix

in the second case.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the array C is to be referenced as

follows:

UPLO = 'U' or 'u' Only the upper triangular part of C

is to be referenced.

UPLO = 'L' or 'l' Only the lower triangular part of C

is to be referenced.

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the operation to be performed as

follows:

TRANS = 'N' or 'n' C := alpha*A*A**T + beta*C.

TRANS = 'T' or 't' C := alpha*A**T*A + beta*C.

TRANS = 'C' or 'c' C := alpha*A**T*A + beta*C.

*N*

N is INTEGER

On entry, N specifies the order of the matrix C. N must be

at least zero.

*K*

K is INTEGER

On entry with TRANS = 'N' or 'n', K specifies the number

of columns of the matrix A, and on entry with

TRANS = 'T' or 't' or 'C' or 'c', K specifies the number

of rows of the matrix A. K must be at least zero.

*ALPHA*

ALPHA is REAL

On entry, ALPHA specifies the scalar alpha.

*A*

A is REAL array, dimension ( LDA, ka ), where ka is

k when TRANS = 'N' or 'n', and is n otherwise.

Before entry with TRANS = 'N' or 'n', the leading n by k

part of the array A must contain the matrix A, otherwise

the leading k by n part of the array A must contain the

matrix A.

*LDA*

LDA is INTEGER

On entry, LDA specifies the first dimension of A as declared

in the calling (sub) program. When TRANS = 'N' or 'n'

then LDA must be at least max( 1, n ), otherwise LDA must

be at least max( 1, k ).

*BETA*

BETA is REAL

On entry, BETA specifies the scalar beta.

*C*

C is REAL array, dimension ( LDC, N )

Before entry with UPLO = 'U' or 'u', the leading n by n

upper triangular part of the array C must contain the upper

triangular part of the symmetric matrix and the strictly

lower triangular part of C is not referenced. On exit, the

upper triangular part of the array C is overwritten by the

upper triangular part of the updated matrix.

Before entry with UPLO = 'L' or 'l', the leading n by n

lower triangular part of the array C must contain the lower

triangular part of the symmetric matrix and the strictly

upper triangular part of C is not referenced. On exit, the

lower triangular part of the array C is overwritten by the

lower triangular part of the updated matrix.

*LDC*

LDC is INTEGER

On entry, LDC specifies the first dimension of C as declared

in the calling (sub) program. LDC must be at least

max( 1, n ).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 3 Blas routine.

-- Written on 8-February-1989.

Jack Dongarra, Argonne National Laboratory.

Iain Duff, AERE Harwell.

Jeremy Du Croz, Numerical Algorithms Group Ltd.

Sven Hammarling, Numerical Algorithms Group Ltd.

## subroutine strmm (character SIDE, character UPLO, character TRANSA, character DIAG, integer M, integer N, real ALPHA, real, dimension(lda,*) A, integer LDA, real, dimension(ldb,*) B, integer LDB)¶

**STRMM**

**Purpose:**

STRMM performs one of the matrix-matrix operations

B := alpha*op( A )*B, or B := alpha*B*op( A ),

where alpha is a scalar, B is an m by n matrix, A is a unit, or

non-unit, upper or lower triangular matrix and op( A ) is one of

op( A ) = A or op( A ) = A**T.

**Parameters**

*SIDE*

SIDE is CHARACTER*1

On entry, SIDE specifies whether op( A ) multiplies B from

the left or right as follows:

SIDE = 'L' or 'l' B := alpha*op( A )*B.

SIDE = 'R' or 'r' B := alpha*B*op( A ).

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the matrix A is an upper or

lower triangular matrix as follows:

UPLO = 'U' or 'u' A is an upper triangular matrix.

UPLO = 'L' or 'l' A is a lower triangular matrix.

*TRANSA*

TRANSA is CHARACTER*1

On entry, TRANSA specifies the form of op( A ) to be used in

the matrix multiplication as follows:

TRANSA = 'N' or 'n' op( A ) = A.

TRANSA = 'T' or 't' op( A ) = A**T.

TRANSA = 'C' or 'c' op( A ) = A**T.

*DIAG*

DIAG is CHARACTER*1

On entry, DIAG specifies whether or not A is unit triangular

as follows:

DIAG = 'U' or 'u' A is assumed to be unit triangular.

DIAG = 'N' or 'n' A is not assumed to be unit

triangular.

*M*

M is INTEGER

On entry, M specifies the number of rows of B. M must be at

least zero.

*N*

N is INTEGER

On entry, N specifies the number of columns of B. N must be

at least zero.

*ALPHA*

ALPHA is REAL

On entry, ALPHA specifies the scalar alpha. When alpha is

zero then A is not referenced and B need not be set before

entry.

*A*

A is REAL array, dimension ( LDA, k ), where k is m

when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'.

Before entry with UPLO = 'U' or 'u', the leading k by k

upper triangular part of the array A must contain the upper

triangular matrix and the strictly lower triangular part of

A is not referenced.

Before entry with UPLO = 'L' or 'l', the leading k by k

lower triangular part of the array A must contain the lower

triangular matrix and the strictly upper triangular part of

A is not referenced.

Note that when DIAG = 'U' or 'u', the diagonal elements of

A are not referenced either, but are assumed to be unity.

*LDA*

LDA is INTEGER

On entry, LDA specifies the first dimension of A as declared

in the calling (sub) program. When SIDE = 'L' or 'l' then

LDA must be at least max( 1, m ), when SIDE = 'R' or 'r'

then LDA must be at least max( 1, n ).

*B*

B is REAL array, dimension ( LDB, N )

Before entry, the leading m by n part of the array B must

contain the matrix B, and on exit is overwritten by the

transformed matrix.

*LDB*

LDB is INTEGER

On entry, LDB specifies the first dimension of B as declared

in the calling (sub) program. LDB must be at least

max( 1, m ).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 3 Blas routine.

-- Written on 8-February-1989.

Jack Dongarra, Argonne National Laboratory.

Iain Duff, AERE Harwell.

Jeremy Du Croz, Numerical Algorithms Group Ltd.

Sven Hammarling, Numerical Algorithms Group Ltd.

## subroutine strsm (character SIDE, character UPLO, character TRANSA, character DIAG, integer M, integer N, real ALPHA, real, dimension(lda,*) A, integer LDA, real, dimension(ldb,*) B, integer LDB)¶

**STRSM**

**Purpose:**

STRSM solves one of the matrix equations

op( A )*X = alpha*B, or X*op( A ) = alpha*B,

where alpha is a scalar, X and B are m by n matrices, A is a unit, or

non-unit, upper or lower triangular matrix and op( A ) is one of

op( A ) = A or op( A ) = A**T.

The matrix X is overwritten on B.

**Parameters**

*SIDE*

SIDE is CHARACTER*1

On entry, SIDE specifies whether op( A ) appears on the left

or right of X as follows:

SIDE = 'L' or 'l' op( A )*X = alpha*B.

SIDE = 'R' or 'r' X*op( A ) = alpha*B.

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the matrix A is an upper or

lower triangular matrix as follows:

UPLO = 'U' or 'u' A is an upper triangular matrix.

UPLO = 'L' or 'l' A is a lower triangular matrix.

*TRANSA*

TRANSA is CHARACTER*1

On entry, TRANSA specifies the form of op( A ) to be used in

the matrix multiplication as follows:

TRANSA = 'N' or 'n' op( A ) = A.

TRANSA = 'T' or 't' op( A ) = A**T.

TRANSA = 'C' or 'c' op( A ) = A**T.

*DIAG*

DIAG is CHARACTER*1

On entry, DIAG specifies whether or not A is unit triangular

as follows:

DIAG = 'U' or 'u' A is assumed to be unit triangular.

DIAG = 'N' or 'n' A is not assumed to be unit

triangular.

*M*

M is INTEGER

On entry, M specifies the number of rows of B. M must be at

least zero.

*N*

N is INTEGER

On entry, N specifies the number of columns of B. N must be

at least zero.

*ALPHA*

ALPHA is REAL

On entry, ALPHA specifies the scalar alpha. When alpha is

zero then A is not referenced and B need not be set before

entry.

*A*

A is REAL array, dimension ( LDA, k ),

where k is m when SIDE = 'L' or 'l'

and k is n when SIDE = 'R' or 'r'.

Before entry with UPLO = 'U' or 'u', the leading k by k

upper triangular part of the array A must contain the upper

triangular matrix and the strictly lower triangular part of

A is not referenced.

Before entry with UPLO = 'L' or 'l', the leading k by k

lower triangular part of the array A must contain the lower

triangular matrix and the strictly upper triangular part of

A is not referenced.

Note that when DIAG = 'U' or 'u', the diagonal elements of

A are not referenced either, but are assumed to be unity.

*LDA*

LDA is INTEGER

On entry, LDA specifies the first dimension of A as declared

in the calling (sub) program. When SIDE = 'L' or 'l' then

LDA must be at least max( 1, m ), when SIDE = 'R' or 'r'

then LDA must be at least max( 1, n ).

*B*

B is REAL array, dimension ( LDB, N )

Before entry, the leading m by n part of the array B must

contain the right-hand side matrix B, and on exit is

overwritten by the solution matrix X.

*LDB*

LDB is INTEGER

On entry, LDB specifies the first dimension of B as declared

in the calling (sub) program. LDB must be at least

max( 1, m ).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Level 3 Blas routine.

-- Written on 8-February-1989.

Jack Dongarra, Argonne National Laboratory.

Iain Duff, AERE Harwell.

Jeremy Du Croz, Numerical Algorithms Group Ltd.

Sven Hammarling, Numerical Algorithms Group Ltd.

# Author¶

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