## MB01YD

### Symmetric rank k operation C := alpha op( A ) op( A )' + beta C, C symmetric

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To perform the symmetric rank k operations

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

where alpha and beta are scalars, C is an n-by-n symmetric matrix,
op( A ) is an n-by-k matrix, and op( A ) is one of

op( A ) = A   or   op( A ) = A'.

The matrix A has l nonzero codiagonals, either upper or lower.

```
Specification
```      SUBROUTINE MB01YD( UPLO, TRANS, N, K, L, ALPHA, BETA, A, LDA, C,
\$                   LDC, INFO )
C     .. Scalar Arguments ..
CHARACTER          TRANS, UPLO
INTEGER            INFO, LDA, LDC, K, L, N
DOUBLE PRECISION   ALPHA, BETA
C     .. Array Arguments ..
DOUBLE PRECISION   A( LDA, * ), C( LDC, * )

```
Arguments

Mode Parameters

```  UPLO    CHARACTER*1
Specifies which triangle of the symmetric matrix C
is given and computed, as follows:
= 'U':  the upper triangular part is given/computed;
= 'L':  the lower triangular part is given/computed.
UPLO also defines the pattern of the matrix A (see below).

TRANS   CHARACTER*1
Specifies the form of op( A ) to be used, as follows:
= 'N':  op( A ) = A;
= 'T':  op( A ) = A';
= 'C':  op( A ) = A'.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrix C.  N >= 0.

K       (input) INTEGER
The number of columns of the matrix op( A ).  K >= 0.

L       (input) INTEGER
If UPLO = 'U', matrix A has L nonzero subdiagonals.
If UPLO = 'L', matrix A has L nonzero superdiagonals.
MAX(0,NR-1) >= L >= 0, if UPLO = 'U',
MAX(0,NC-1) >= L >= 0, if UPLO = 'L',
where NR and NC are the numbers of rows and columns of the
matrix A, respectively.

ALPHA   (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then the array A is
not referenced.

BETA    (input) DOUBLE PRECISION
The scalar beta. When beta is zero then the array C need
not be set before entry.

A       (input) DOUBLE PRECISION array, dimension (LDA,NC), where
NC is K when TRANS = 'N', and is N otherwise.
If TRANS = 'N', the leading N-by-K part of this array must
contain the matrix A, otherwise the leading K-by-N part of
this array must contain the matrix A.
If UPLO = 'U', only the upper triangular part and the
first L subdiagonals are referenced, and the remaining
subdiagonals are assumed to be zero.
If UPLO = 'L', only the lower triangular part and the
first L superdiagonals are referenced, and the remaining
superdiagonals are assumed to be zero.

LDA     INTEGER
The leading dimension of array A.  LDA >= max(1,NR),
where NR = N, if TRANS = 'N', and NR = K, otherwise.

C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix C.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix C.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the updated matrix C.

LDC     INTEGER
The leading dimension of array C.  LDC >= MAX(1,N).

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
Method
```  The calculations are efficiently performed taking the symmetry
and structure into account.

```
```  The matrix A may have the following patterns, when n = 7, k = 5,
and l = 2 are used for illustration:

UPLO = 'U', TRANS = 'N'         UPLO = 'L', TRANS = 'N'

[ x x x x x ]                   [ x x x 0 0 ]
[ x x x x x ]                   [ x x x x 0 ]
[ x x x x x ]                   [ x x x x x ]
A = [ 0 x x x x ],              A = [ x x x x x ],
[ 0 0 x x x ]                   [ x x x x x ]
[ 0 0 0 x x ]                   [ x x x x x ]
[ 0 0 0 0 x ]                   [ x x x x x ]

UPLO = 'U', TRANS = 'T'         UPLO = 'L', TRANS = 'T'

[ x x x x x x x ]               [ x x x 0 0 0 0 ]
[ x x x x x x x ]               [ x x x x 0 0 0 ]
A = [ x x x x x x x ],          A = [ x x x x x 0 0 ].
[ 0 x x x x x x ]               [ x x x x x x 0 ]
[ 0 0 x x x x x ]               [ x x x x x x x ]

If N = K, the matrix A is upper or lower triangular, for L = 0,
and upper or lower Hessenberg, for L = 1.

This routine is a specialization of the BLAS 3 routine DSYRK.
BLAS 1 calls are used when appropriate, instead of in-line code,
in order to increase the efficiency. If the matrix A is full, or
its zero triangle has small order, an optimized DSYRK code could
be faster than MB01YD.

```
Example

Program Text

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Program Data
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Program Results
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