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<HTML>
<HEAD><TITLE>TG01WD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="TG01WD">TG01WD</A></H2>
<H3>
Reduction of the descriptor dynamics matrix pair to generalized real Schur form
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
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.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
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,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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.
</PRE>
<B>Workspace</B>
<PRE>
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.
</PRE>
<B>Error Indicator</B>
<PRE>
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.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 3
The algorithm requires about 25N floating point operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
<HR>
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