## MB01ZD

### Computing H := alpha op( T ) H, or H := alpha H op( T ), with H Hessenberg-like, T triangular

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

Purpose

```  To compute the matrix product

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

where alpha is a scalar, H is an m-by-n upper or lower
Hessenberg-like matrix (with l nonzero subdiagonals or
superdiagonals, respectively), T is a unit, or non-unit,
upper or lower triangular matrix, and op( T ) is one of

op( T ) = T   or   op( T ) = T'.

```
Specification
```      SUBROUTINE MB01ZD( SIDE, UPLO, TRANST, DIAG, M, N, L, ALPHA, T,
\$                   LDT, H, LDH, INFO )
C     .. Scalar Arguments ..
CHARACTER          DIAG, SIDE, TRANST, UPLO
INTEGER            INFO, L, LDH, LDT, M, N
DOUBLE PRECISION   ALPHA
C     .. Array Arguments ..
DOUBLE PRECISION   H( LDH, * ), T( LDT, * )

```
Arguments

Mode Parameters

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

UPLO    CHARACTER*1
Specifies the form of the matrices T and H, as follows:
= 'U':  the matrix T is upper triangular and the matrix H
is upper Hessenberg-like;
= 'L':  the matrix T is lower triangular and the matrix H
is lower Hessenberg-like.

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

DIAG    CHARACTER*1.
Specifies whether or not T is unit triangular, as follows:
= 'U':  the matrix T is assumed to be unit triangular;
= 'N':  the matrix T is not assumed to be unit triangular.

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

N       (input) INTEGER
The number of columns of H.  N >= 0.

L       (input) INTEGER
If UPLO = 'U', matrix H has L nonzero subdiagonals.
If UPLO = 'L', matrix H has L nonzero superdiagonals.
MAX(0,M-1) >= L >= 0, if UPLO = 'U';
MAX(0,N-1) >= L >= 0, if UPLO = 'L'.

ALPHA   (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then T is not
referenced and H 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'.
If UPLO = 'U', the leading k-by-k upper triangular part
of this array must contain the upper triangular matrix T
and the strictly lower triangular part is not referenced.
If UPLO = 'L', the leading k-by-k lower triangular part
of this array must contain the lower triangular matrix T
and the strictly upper triangular part is not referenced.
Note that when DIAG = 'U', the diagonal elements of T are
not referenced either, but are assumed to be unity.

LDT     INTEGER
The leading dimension of array T.
LDT >= MAX(1,M), if SIDE = 'L';
LDT >= MAX(1,N), if SIDE = 'R'.

H       (input/output) DOUBLE PRECISION array, dimension (LDH,N)
On entry, if UPLO = 'U', the leading M-by-N upper
Hessenberg part of this array must contain the upper
Hessenberg-like matrix H.
On entry, if UPLO = 'L', the leading M-by-N lower
Hessenberg part of this array must contain the lower
Hessenberg-like matrix H.
On exit, the leading M-by-N part of this array contains
the matrix product alpha*op( T )*H, if SIDE = 'L',
or alpha*H*op( T ), if SIDE = 'R'. If TRANST = 'N', this
product has the same pattern as the given matrix H;
the elements below the L-th subdiagonal (if UPLO = 'U'),
or above the L-th superdiagonal (if UPLO = 'L'), are not
referenced in this case. If TRANST = 'T', the elements
below the (N+L)-th row (if UPLO = 'U', SIDE = 'R', and
M > N+L), or at the right of the (M+L)-th column
(if UPLO = 'L', SIDE = 'L', and N > M+L), are not set to
zero nor referenced.

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

```
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 problem
structure into account.

```
```  The matrix H may have the following patterns, when m = 7, n = 6,
and l = 2 are used for illustration:

UPLO = 'U'                    UPLO = 'L'

[ x x x x x x ]               [ x x x 0 0 0 ]
[ x x x x x x ]               [ x x x x 0 0 ]
[ x x x x x x ]               [ x x x x x 0 ]
H = [ 0 x x x x x ],          H = [ x x x x x x ].
[ 0 0 x x x x ]               [ x x x x x x ]
[ 0 0 0 x x x ]               [ x x x x x x ]
[ 0 0 0 0 x x ]               [ x x x x x x ]

The products T*H or H*T have the same pattern as H, but the
products T'*H or H*T' may be full matrices.

If m = n, the matrix H 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 DTRMM.
BLAS 1 calls are used when appropriate, instead of in-line code,
in order to increase the efficiency. If the matrix H is full, or
its zero triangle has small order, an optimized DTRMM code could
be faster than MB01ZD.

```
Example

Program Text

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