## TG01WD

### Reduction of the descriptor dynamics matrix pair to generalized real Schur form

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

Purpose

```  To reduce the pair (A,E) to a real generalized Schur form
by using an orthogonal equivalence transformation
(A,E) <-- (Q'*A*Z,Q'*E*Z) and to apply the transformation
to the matrices B and C: B <-- Q'*B and C <-- C*Z.

```
Specification
```      SUBROUTINE TG01WD( N, M, P, A, LDA, E, LDE, B, LDB, C, LDC,
\$                   Q, LDQ, Z, LDZ, ALPHAR, ALPHAI, BETA, DWORK,
\$                   LDWORK, INFO )
C     .. Scalar Arguments ..
INTEGER           INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ,
\$                  M, N, P
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
\$                  BETA(*),  C(LDC,*),  DWORK(*),  E(LDE,*),
\$                  Q(LDQ,*), Z(LDZ,*)

```
Arguments

Input/Output Parameters

```  N       (input) INTEGER
The order of the original state-space representation,
i.e., the order of the matrices A and E.  N >= 0.

M       (input) INTEGER
The number of system inputs, or of columns of B.  M >= 0.

P       (input) INTEGER
The number of system outputs, or of rows of C.  P >= 0.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, the leading N-by-N part of this array contains
the matrix Q' * A * Z in an upper quasi-triangular form.
The elements below the first subdiagonal are set to zero.

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

E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the original descriptor matrix E.
On exit, the leading N-by-N part of this array contains
the matrix Q' * E * Z in an upper triangular form.
The elements below the diagonal are set to zero.

LDE     INTEGER
The leading dimension of array E.  LDE >= MAX(1,N).

B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed input matrix Q' * B.

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

C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output matrix C * Z.

LDC     INTEGER
The leading dimension of array C.  LDC >= MAX(1,P).

Q       (output) DOUBLE PRECISION array, dimension (LDQ,N)
The leading N-by-N part of this array contains the left
orthogonal transformation matrix used to reduce (A,E) to
the real generalized Schur form.
The columns of Q are the left generalized Schur vectors
of the pair (A,E).

LDQ     INTEGER
The leading dimension of array Q.  LDQ >= max(1,N).

Z       (output) DOUBLE PRECISION array, dimension (LDZ,N)
The leading N-by-N part of this array contains the right
orthogonal transformation matrix used to reduce (A,E) to
the real generalized Schur form.
The columns of Z are the right generalized Schur vectors
of the pair (A,E).

LDZ     INTEGER
The leading dimension of array Z.  LDZ >= max(1,N).

ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
BETA    (output) DOUBLE PRECISION array, dimension (N)
On exit, if INFO = 0, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
j=1,...,N, will be the generalized eigenvalues.
ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j=1,...,N, are the
diagonals of the complex Schur form that would result if
the 2-by-2 diagonal blocks of the real Schur form of
(A,E) were further reduced to triangular form using
2-by-2 complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real;
if positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.

LDWORK  INTEGER
The dimension of working array DWORK.  LDWORK >= 8*N+16.
For optimum performance LDWORK should be larger.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
> 0:  if INFO = i, the QZ algorithm failed to compute
the generalized real Schur form; elements i+1:N of
ALPHAR, ALPHAI, and BETA should be correct.

```
Method
```  The pair (A,E) is reduced to a real generalized Schur form using
an orthogonal equivalence transformation (A,E) <-- (Q'*A*Z,Q'*E*Z)
and the transformation is applied to the matrices B and C:
B <-- Q'*B and C <-- C*Z.

```
Numerical Aspects
```                                  3
The algorithm requires about 25N  floating point operations.

```
```  None
```
Example

Program Text

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