## MB01RD

### Computation of matrix expression alpha R + beta A X trans(A), R, X symmetric

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

Purpose

```  To compute the matrix formula
_
R = alpha*R + beta*op( A )*X*op( A )',
_
where alpha and beta are scalars, R, X, and R are symmetric
matrices, A is a general matrix, and op( A ) is one of

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

The result is overwritten on R.

```
Specification
```      SUBROUTINE MB01RD( UPLO, TRANS, M, N, ALPHA, BETA, R, LDR, A, LDA,
\$                   X, LDX, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         TRANS, UPLO
INTEGER           INFO, LDA, LDR, LDWORK, LDX, M, N
DOUBLE PRECISION  ALPHA, BETA
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), DWORK(*), R(LDR,*), X(LDX,*)

```
Arguments

Mode Parameters

```  UPLO    CHARACTER*1                                         _
Specifies which triangles of the symmetric matrices R, R,
and X are given as follows:
= 'U':  the upper triangular part is given;
= 'L':  the lower triangular part is given.

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

```
Input/Output Parameters
```  M       (input) INTEGER           _
The order of the matrices R and R and the number of rows
of the matrix op( A ).  M >= 0.

N       (input) INTEGER
The order of the matrix X and the number of columns of the
the matrix op( A ).  N >= 0.

ALPHA   (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry, except when R is identified with X in
the call (which is possible only in this case).

BETA    (input) DOUBLE PRECISION
The scalar beta. When beta is zero then A and X are not
referenced.

R       (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry with UPLO = 'U', the leading M-by-M upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R; the strictly
lower triangular part of the array is used as workspace.
On entry with UPLO = 'L', the leading M-by-M lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R; the strictly
upper triangular part of the array is used as workspace.
On exit, the leading M-by-M upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
_
the computed matrix R. If beta <> 0, the remaining
strictly triangular part of this array contains the
corresponding part of the matrix expression
beta*op( A )*T*op( A )', where T is the triangular matrix
defined in the Method section.

LDR     INTEGER
The leading dimension of array R.  LDR >= MAX(1,M).

A       (input) DOUBLE PRECISION array, dimension (LDA,k)
where k is N when TRANS = 'N' and is M when TRANS = 'T' or
TRANS = 'C'.
On entry with TRANS = 'N', the leading M-by-N part of this
array must contain the matrix A.
On entry with TRANS = 'T' or TRANS = 'C', the leading
N-by-M part of this array must contain the matrix A.

LDA     INTEGER
The leading dimension of array A.  LDA >= MAX(1,l),
where l is M when TRANS = 'N' and is N when TRANS = 'T' or
TRANS = 'C'.

X       (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
On exit, each diagonal element of this array has half its
input value, but the other elements are not modified.

LDX     INTEGER
The leading dimension of array X.  LDX >= MAX(1,N).

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, the leading M-by-N part of this
array (with the leading dimension MAX(1,M)) returns the
matrix product beta*op( A )*T, where T is the triangular
matrix defined in the Method section.
This array is not referenced when beta = 0.

LDWORK  The length of the array DWORK.
LDWORK >= MAX(1,M*N), if  beta <> 0;
LDWORK >= 1,          if  beta =  0.

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

```
Method
```  The matrix expression is efficiently evaluated taking the symmetry
into account. Specifically, let X = T + T', with T an upper or
lower triangular matrix, defined by

T = triu( X ) - (1/2)*diag( X ),  if UPLO = 'U',
T = tril( X ) - (1/2)*diag( X ),  if UPLO = 'L',

where triu, tril, and diag denote the upper triangular part, lower
triangular part, and diagonal part of X, respectively. Then,

op( A )*X*op( A )' = B + B',

where B := op( A )*T*op( A )'. Matrix B is not symmetric, but it
can be written as tri( B ) + stri( B ), where tri denotes the
triangular part specified by UPLO, and stri denotes the remaining
strictly triangular part. Let R = V + V', with V defined as T
above. Then, the required triangular part of the result can be
written as

alpha*V + beta*tri( B )  + beta*(stri( B ))' +
alpha*diag( V ) + beta*diag( tri( B ) ).

```
References
```  None.

```
Numerical Aspects
```  The algorithm requires approximately

2             2
M x N + 1/2 x N x M

operations.

```
```  None
```
Example

Program Text

```  None
```
Program Data
```  None
```
Program Results
```  None
```