## MB01UX

### Computation of matrix expressions alpha T A or alpha A T, over A, T quasi-triangular

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

Purpose

```  To compute one of the matrix products

A : = alpha*op( T ) * A, or A : = alpha*A * op( T ),

where alpha is a scalar, A is an m-by-n matrix, T is a quasi-
triangular matrix, and op( T ) is one of

op( T ) = T   or   op( T ) = T',  the transpose of T.

```
Specification
```      SUBROUTINE MB01UX( SIDE, UPLO, TRANS, M, N, ALPHA, T, LDT, A, LDA,
\$                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         SIDE, TRANS, UPLO
INTEGER           INFO, LDA, LDT, LDWORK, M, N
DOUBLE PRECISION  ALPHA
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), DWORK(*), T(LDT,*)

```
Arguments

Mode Parameters

```  SIDE    CHARACTER*1
Specifies whether the upper quasi-triangular matrix H
appears on the left or right in the matrix product as
follows:
= 'L':  A := alpha*op( T ) * A;
= 'R':  A := alpha*A * op( T ).

UPLO    CHARACTER*1.
Specifies whether the matrix T is an upper or lower
quasi-triangular matrix as follows:
= 'U':  T is an upper quasi-triangular matrix;
= 'L':  T is a lower quasi-triangular matrix.

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

```
Input/Output Parameters
```  M       (input) INTEGER
The number of rows of the matrix A.  M >= 0.

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

ALPHA   (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then T is not
referenced and A need not be set before entry.

T       (input) DOUBLE PRECISION array, dimension (LDT,k)
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
On entry with UPLO = 'U', the leading k-by-k upper
Hessenberg part of this array must contain the upper
quasi-triangular matrix T. The elements below the
subdiagonal are not referenced.
On entry with UPLO = 'L', the leading k-by-k lower
Hessenberg part of this array must contain the lower
quasi-triangular matrix T. The elements above the
supdiagonal are not referenced.

LDT     INTEGER
The leading dimension of the array T.  LDT >= max(1,k),
where k is M when SIDE = 'L' and is N when SIDE = 'R'.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the matrix A.
On exit, the leading M-by-N part of this array contains
the computed product.

LDA     INTEGER
The leading dimension of the array A.  LDA >= max(1,M).

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 and ALPHA<>0,  DWORK(1)  returns the
optimal value of LDWORK.
On exit, if  INFO = -12,  DWORK(1)  returns the minimum
value of LDWORK.
This array is not referenced when alpha = 0.

LDWORK  The length of the array DWORK.
LDWORK >= 1,       if alpha =  0 or MIN(M,N) = 0;
LDWORK >= 2*(M-1), if SIDE  = 'L';
LDWORK >= 2*(N-1), if SIDE  = 'R'.
For maximal efficiency LDWORK should be at least
NOFF*N + M - 1,    if SIDE  = 'L';
NOFF*M + N - 1,    if SIDE  = 'R';
where NOFF is the number of nonzero elements on the
subdiagonal (if UPLO = 'U') or supdiagonal (if UPLO = 'L')
of T.

If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.

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

```
Method
```  The technique used in this routine is similiar to the technique
used in the SLICOT  subroutine MB01UW developed by Vasile Sima.
The required matrix product is computed in two steps. In the first
step, the triangle of T specified by UPLO is used; in the second
step, the contribution of the sub-/supdiagonal is added. If the
workspace can accommodate parts of A, a fast BLAS 3 DTRMM
operation is used in the first step.

```
References
```   Benner, P., Mehrmann, V., Sima, V., Van Huffel, S., and
Varga, A.
SLICOT - A subroutine library in systems and control theory.
In: Applied and computational control, signals, and circuits,
Vol. 1, pp. 499-539, Birkhauser, Boston, 1999.

```
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Example

Program Text

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Program Data
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Program Results
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