## table of contents

complex_blas_level3(3) | LAPACK | complex_blas_level3(3) |

# NAME¶

complex_blas_level3

# SYNOPSIS¶

## Functions¶

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

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

**CHEMM** subroutine **cher2k** (UPLO, TRANS, N, K, ALPHA, A, LDA, B,
LDB, BETA, C, LDC)

**CHER2K** subroutine **cherk** (UPLO, TRANS, N, K, ALPHA, A, LDA, BETA,
C, LDC)

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

**CSYMM** subroutine **csyr2k** (UPLO, TRANS, N, K, ALPHA, A, LDA, B,
LDB, BETA, C, LDC)

**CSYR2K** subroutine **csyrk** (UPLO, TRANS, N, K, ALPHA, A, LDA, BETA,
C, LDC)

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

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

**CTRSM**

# Detailed Description¶

This is the group of complex LEVEL 3 BLAS routines.

# Function Documentation¶

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

**CGEMM**

**Purpose:**

CGEMM 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 or op( X ) = X**H,

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**H.

*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**H.

*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 COMPLEX

On entry, ALPHA specifies the scalar alpha.

*A*

A is COMPLEX 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 COMPLEX 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 COMPLEX

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

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

*C*

C is COMPLEX 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 chemm (character SIDE, character UPLO, integer M, integer N, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB, complex BETA, complex, dimension(ldc,*) C, integer LDC)¶

**CHEMM**

**Purpose:**

CHEMM 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 an hermitian matrix and B and

C are m by n matrices.

**Parameters**

*SIDE*

SIDE is CHARACTER*1

On entry, SIDE specifies whether the hermitian 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 hermitian matrix A is to be

referenced as follows:

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

hermitian matrix is to be referenced.

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

hermitian 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 COMPLEX

On entry, ALPHA specifies the scalar alpha.

*A*

A is COMPLEX 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 hermitian 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 hermitian 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 hermitian

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 hermitian 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 hermitian 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 hermitian

matrix and the strictly upper triangular part of A is not

referenced.

Note that the imaginary parts of the diagonal elements need

not be set, they are assumed to be zero.

*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 COMPLEX 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 COMPLEX

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

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

*C*

C is COMPLEX 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 cher2k (character UPLO, character TRANS, integer N, integer K, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB, real BETA, complex, dimension(ldc,*) C, integer LDC)¶

**CHER2K**

**Purpose:**

CHER2K performs one of the hermitian rank 2k operations

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

or

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

where alpha and beta are scalars with beta real, C is an n by n

hermitian 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**H +

conjg( alpha )*B*A**H +

beta*C.

TRANS = 'C' or 'c' C := alpha*A**H*B +

conjg( alpha )*B**H*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 = 'C' or 'c', K specifies the number of rows of the

matrices A and B. K must be at least zero.

*ALPHA*

ALPHA is COMPLEX

On entry, ALPHA specifies the scalar alpha.

*A*

A is COMPLEX 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 COMPLEX 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 COMPLEX 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 hermitian 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 hermitian 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.

Note that the imaginary parts of the diagonal elements need

not be set, they are assumed to be zero, and on exit they

are set to zero.

*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.

-- Modified 8-Nov-93 to set C(J,J) to REAL( C(J,J) ) when BETA = 1.

Ed Anderson, Cray Research Inc.

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

**CHERK**

**Purpose:**

CHERK performs one of the hermitian rank k operations

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

or

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

where alpha and beta are real scalars, C is an n by n hermitian

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**H + beta*C.

TRANS = 'C' or 'c' C := alpha*A**H*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 = '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 COMPLEX 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 COMPLEX 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 hermitian 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 hermitian 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.

Note that the imaginary parts of the diagonal elements need

not be set, they are assumed to be zero, and on exit they

are set to zero.

*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.

-- Modified 8-Nov-93 to set C(J,J) to REAL( C(J,J) ) when BETA = 1.

Ed Anderson, Cray Research Inc.

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

**CSYMM**

**Purpose:**

CSYMM 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 COMPLEX

On entry, ALPHA specifies the scalar alpha.

*A*

A is COMPLEX 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 COMPLEX 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 COMPLEX

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

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

*C*

C is COMPLEX 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 csyr2k (character UPLO, character TRANS, integer N, integer K, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB, complex BETA, complex, dimension(ldc,*) C, integer LDC)¶

**CSYR2K**

**Purpose:**

CSYR2K 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.

*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', K specifies the number of rows of the

matrices A and B. K must be at least zero.

*ALPHA*

ALPHA is COMPLEX

On entry, ALPHA specifies the scalar alpha.

*A*

A is COMPLEX 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 COMPLEX 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 COMPLEX

On entry, BETA specifies the scalar beta.

*C*

C is COMPLEX 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 csyrk (character UPLO, character TRANS, integer N, integer K, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex BETA, complex, dimension(ldc,*) C, integer LDC)¶

**CSYRK**

**Purpose:**

CSYRK 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.

*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', K specifies the number of rows of the

matrix A. K must be at least zero.

*ALPHA*

ALPHA is COMPLEX

On entry, ALPHA specifies the scalar alpha.

*A*

A is COMPLEX 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 COMPLEX

On entry, BETA specifies the scalar beta.

*C*

C is COMPLEX 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 ctrmm (character SIDE, character UPLO, character TRANSA, character DIAG, integer M, integer N, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB)¶

**CTRMM**

**Purpose:**

CTRMM 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 or op( A ) = A**H.

**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**H.

*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 COMPLEX

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 COMPLEX 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 COMPLEX 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 ctrsm (character SIDE, character UPLO, character TRANSA, character DIAG, integer M, integer N, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB)¶

**CTRSM**

**Purpose:**

CTRSM 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 or op( A ) = A**H.

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**H.

*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 COMPLEX

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 COMPLEX 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 COMPLEX 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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