# herk - Man Page

{he,sy}rk: Hermitian/symmetric rank-k update

## Synopsis

### Functions

subroutine cherk (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)
CHERK
subroutine csyrk (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)
CSYRK
subroutine dsyrk (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)
DSYRK
subroutine ssyrk (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)
SSYRK
subroutine zherk (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)
ZHERK
subroutine zsyrk (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)
ZSYRK

## Function Documentation

### 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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.```

Definition at line 172 of file cherk.f.

### 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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.```

Definition at line 166 of file csyrk.f.

### subroutine dsyrk (character uplo, character trans, integer n, integer k, double precision alpha, double precision, dimension(lda,*) a, integer lda, double precision beta, double precision, dimension(ldc,*) c, integer ldc)

DSYRK

Purpose:

``` DSYRK  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 DOUBLE PRECISION.
On entry, ALPHA specifies the scalar alpha.```

A

```          A is DOUBLE PRECISION 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 DOUBLE PRECISION.
On entry, BETA specifies the scalar beta.```

C

```          C is DOUBLE PRECISION 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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.```

Definition at line 168 of file dsyrk.f.

### 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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.```

Definition at line 168 of file ssyrk.f.

### subroutine zherk (character uplo, character trans, integer n, integer k, double precision alpha, complex*16, dimension(lda,*) a, integer lda, double precision beta, complex*16, dimension(ldc,*) c, integer ldc)

ZHERK

Purpose:

``` ZHERK  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 DOUBLE PRECISION .
On entry, ALPHA specifies the scalar alpha.```

A

```          A is COMPLEX*16 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 DOUBLE PRECISION.
On entry, BETA specifies the scalar beta.```

C

```          C is COMPLEX*16 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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 DBLE( C(J,J) ) when BETA = 1.
Ed Anderson, Cray Research Inc.```

Definition at line 172 of file zherk.f.

### subroutine zsyrk (character uplo, character trans, integer n, integer k, complex*16 alpha, complex*16, dimension(lda,*) a, integer lda, complex*16 beta, complex*16, dimension(ldc,*) c, integer ldc)

ZSYRK

Purpose:

``` ZSYRK  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*16
On entry, ALPHA specifies the scalar alpha.```

A

```          A is COMPLEX*16 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*16
On entry, BETA specifies the scalar beta.```

C

```          C is COMPLEX*16 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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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.```

Definition at line 166 of file zsyrk.f.

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