## MB01RY

### Computation of a triangle of matrix expression alpha R + beta H B or alpha R + beta B H, H upper Hessenberg matrix

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

Purpose

```  To compute either the upper or lower triangular part of one of the
matrix formulas
_
R = alpha*R + beta*op( H )*B,                               (1)
_
R = alpha*R + beta*B*op( H ),                               (2)
_
where alpha and beta are scalars, H, B, R, and R are m-by-m
matrices, H is an upper Hessenberg matrix, and op( H ) is one of

op( H ) = H   or   op( H ) = H',  the transpose of H.

The result is overwritten on R.

```
Specification
```      SUBROUTINE MB01RY( SIDE, UPLO, TRANS, M, ALPHA, BETA, R, LDR, H,
\$                   LDH, B, LDB, DWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         SIDE, TRANS, UPLO
INTEGER           INFO, LDB, LDH, LDR, M
DOUBLE PRECISION  ALPHA, BETA
C     .. Array Arguments ..
DOUBLE PRECISION  B(LDB,*), DWORK(*), H(LDH,*), R(LDR,*)

```
Arguments

Mode Parameters

```  SIDE    CHARACTER*1
Specifies whether the Hessenberg matrix H appears on the
left or right in the matrix product as follows:
_
= 'L':  R = alpha*R + beta*op( H )*B;
_
= 'R':  R = alpha*R + beta*B*op( H ).

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

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

```
Input/Output Parameters
```  M       (input) INTEGER           _
The order of the matrices R, R, H and B.  M >= 0.

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

BETA    (input) DOUBLE PRECISION
The scalar beta. When beta is zero then H and B 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 matrix R; the strictly lower
triangular part of the array is not referenced.
On entry with UPLO = 'L', the leading M-by-M lower
triangular part of this array must contain the lower
triangular part of the matrix R; the strictly upper
triangular part of the array is not referenced.
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.

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

H       (input) DOUBLE PRECISION array, dimension (LDH,M)
On entry, the leading M-by-M upper Hessenberg part of
this array must contain the upper Hessenberg part of the
matrix H.
The elements below the subdiagonal are not referenced,
except possibly for those in the first column, which
could be overwritten, but are restored on exit.

LDH     INTEGER
The leading dimension of array H.  LDH >= MAX(1,M).

B       (input) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading M-by-M part of this array must
contain the matrix B.

LDB     INTEGER
The leading dimension of array B.  LDB >= MAX(1,M).

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
LDWORK >= M, if  beta <> 0 and SIDE = 'L';
LDWORK >= 0, if  beta =  0 or  SIDE = 'R'.
This array is not referenced when beta = 0 or SIDE = 'R'.

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

```
Method
```  The matrix expression is efficiently evaluated taking the
Hessenberg/triangular structure into account. BLAS 2 operations
are used. A block algorithm can be constructed; it can use BLAS 3
GEMM operations for most computations, and calls of this BLAS 2
algorithm for computing the triangles.

```
```  The main application of this routine is when the result should
be a symmetric matrix, e.g., when B = X*op( H )', for (1), or
B = op( H )'*X, for (2), where B is already available and X = X'.

```
Example

Program Text

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