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## NAME

```       DSYRK  -  perform  one  of  the  symmetric  rank  k  operations    C :=
alpha*A*A’ + beta*C,

```

## SYNOPSIS

```       SUBROUTINE DSYRK ( UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC )

CHARACTER*1  UPLO, TRANS

INTEGER      N, K, LDA, LDC

DOUBLE       PRECISION ALPHA, BETA

DOUBLE       PRECISION A( LDA, * ), C( LDC, * )

```

## PURPOSE

```       DSYRK  performs one of the symmetric rank k operations

or

C := alpha*A’*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   - 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.

Unchanged on exit.

TRANS  - CHARACTER*1.
On entry,  TRANS  specifies the operation  to  be  performed  as
follows:

TRANS = ’N’ or ’n’   C := alpha*A*A’ + beta*C.

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

TRANS = ’C’ or ’c’   C := alpha*A’*A + beta*C.

Unchanged on exit.

N      - INTEGER.
On  entry,  N specifies the order of the matrix C.  N must be at
least zero.  Unchanged on exit.

K      - 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.  Unchanged on exit.

ALPHA  - DOUBLE PRECISION.
On  entry, ALPHA specifies the scalar alpha.  Unchanged on exit.

A      - DOUBLE PRECISION array of 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.
Unchanged on exit.

LDA    - 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 ).  Unchanged on exit.

BETA   - DOUBLE PRECISION.
On entry, BETA specifies the scalar beta.  Unchanged on exit.

C      - DOUBLE PRECISION array of 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    - 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 ).  Unchanged on exit.

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.
```