## MB01WD

### Residuals of Lyapunov or Stein equations for Cholesky factored solutions

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

Purpose

```  To compute the matrix formula
_
R = alpha*( op( A )'*op( T )'*op( T ) + op( T )'*op( T )*op( A ) )
+ beta*R,                                                  (1)

if DICO = 'C', or
_
R = alpha*( op( A )'*op( T )'*op( T )*op( A ) -  op( T )'*op( T ))
+ beta*R,                                                  (2)
_
if DICO = 'D', where alpha and beta are scalars, R, and R are
symmetric matrices, T is a triangular matrix, A is a general or
Hessenberg matrix, and op( M ) is one of

op( M ) = M   or   op( M ) = M'.

The result is overwritten on R.

```
Specification
```      SUBROUTINE MB01WD( DICO, UPLO, TRANS, HESS, N, ALPHA, BETA, R,
\$                   LDR, A, LDA, T, LDT, INFO )
C     .. Scalar Arguments ..
CHARACTER         DICO, HESS, TRANS, UPLO
INTEGER           INFO, LDA, LDR, LDT, N
DOUBLE PRECISION  ALPHA, BETA
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), R(LDR,*), T(LDT,*)

```
Arguments

Mode Parameters

```  DICO    CHARACTER*1
Specifies the formula to be evaluated, as follows:
= 'C':  formula (1), "continuous-time" case;
= 'D':  formula (2), "discrete-time" case.

UPLO    CHARACTER*1
Specifies which triangles of the symmetric matrix R and
triangular matrix T are given, as follows:
= 'U':  the upper triangular parts of R and T are given;
= 'L':  the lower triangular parts of R and T are given;

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

HESS    CHARACTER*1
Specifies the form of the matrix A, as follows:
= 'F':  matrix A is full;
= 'H':  matrix A is Hessenberg (or Schur), either upper
(if UPLO = 'U'), or lower (if UPLO = 'L').

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrices R, A, and T.  N >= 0.

ALPHA   (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then the arrays A
and T are not referenced.

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

R       (input/output) DOUBLE PRECISION array, dimension (LDR,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 R.
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 R.
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 computed matrix R.

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

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A. If HESS = 'H' the elements below the
first subdiagonal, if UPLO = 'U', or above the first
superdiagonal, if UPLO = 'L', need not be set to zero,
and are not referenced if DICO = 'D'.
On exit, the leading N-by-N part of this array contains
the following matrix product
alpha*T'*T*A, if TRANS = 'N', or
alpha*A*T*T', otherwise,
if DICO = 'C', or
T*A, if TRANS = 'N', or
A*T, otherwise,
if DICO = 'D' (and in this case, these products have a
Hessenberg form, if HESS = 'H').

LDA     INTEGER
The leading dimension of array A.  LDA >= MAX(1,N).

T       (input) DOUBLE PRECISION array, dimension (LDT,N)
If UPLO = 'U', the leading N-by-N upper triangular part of
this array must contain the upper triangular matrix T and
the strictly lower triangular part need not be set to zero
(and it is not referenced).
If UPLO = 'L', the leading N-by-N lower triangular part of
this array must contain the lower triangular matrix T and
the strictly upper triangular part need not be set to zero
(and it is not referenced).

LDT     INTEGER
The leading dimension of array T.  LDT >= MAX(1,N).

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

```
Method
```  The matrix expression (1) or (2) is efficiently evaluated taking
the structure into account. BLAS 3 operations (DTRMM, DSYRK and
their specializations) are used throughout.

```
Numerical Aspects
```  If A is a full matrix, the algorithm requires approximately
3
N  operations, if DICO = 'C';
3
7/6 x N  operations, if DICO = 'D'.

```
```  None
```
Example

Program Text

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