Purpose
To compute orthogonal transformation matrices Q and Z which
reduce the regular pole pencil A-lambda*E of the descriptor system
(A-lambda*E,B,C) to the generalized real Schur form with ordered
generalized eigenvalues. The pair (A,E) is reduced to the form
( * * * * ) ( * * * * )
( ) ( )
( 0 A1 * * ) ( 0 E1 * * )
Q'*A*Z = ( ) , Q'*E*Z = ( ) ,
( 0 0 A2 * ) ( 0 0 E2 * )
( ) ( )
( 0 0 0 * ) ( 0 0 0 * )
where the subpencil A1-lambda*E1 contains the eigenvalues which
belong to a suitably defined domain of interest and the subpencil
A2-lambda*E2 contains the eigenvalues which are outside of the
domain of interest.
If JOBAE = 'S', the pair (A,E) is assumed to be already in a
generalized real Schur form and the reduction is performed only
on the subpencil A12 - lambda*E12 defined by rows and columns
NLOW to NSUP of A - lambda*E.
Specification
SUBROUTINE TG01PD( DICO, STDOM, JOBAE, COMPQ, COMPZ, N, M, P,
$ NLOW, NSUP, ALPHA, A, LDA, E, LDE, B, LDB,
$ C, LDC, Q, LDQ, Z, LDZ, NDIM, ALPHAR, ALPHAI,
$ BETA, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, DICO, JOBAE, STDOM
INTEGER INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M, N,
$ NDIM, NLOW, NSUP, P
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
$ BETA(*), C(LDC,*), DWORK(*), E(LDE,*),
$ Q(LDQ,*), Z(LDZ,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the descriptor system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
STDOM CHARACTER*1
Specifies whether the domain of interest is of stability
type (left part of complex plane or inside of a circle)
or of instability type (right part of complex plane or
outside of a circle) as follows:
= 'S': stability type domain;
= 'U': instability type domain.
JOBAE CHARACTER*1
Specifies the shape of the matrix pair (A,E) on entry
as follows:
= 'S': (A,E) is in a generalized real Schur form;
= 'G': A and E are general square dense matrices.
COMPQ CHARACTER*1
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
This option can not be used when JOBAE = 'G'.
COMPZ CHARACTER*1
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
This option can not be used when JOBAE = 'G'.
Input/Output Parameters
N (input) INTEGER
The number of rows of the matrix B, the number of columns
of the matrix C, and the order of the square matrices A
and E. N >= 0.
M (input) INTEGER
The number of columns of the matrix B. M >= 0.
P (input) INTEGER
The number of rows of the matrix C. P >= 0.
NLOW, (input) INTEGER
NSUP (input) INTEGER
NLOW and NSUP specify the boundary indices for the rows
and columns of the principal subpencil of A - lambda*E
whose diagonal blocks are to be reordered.
0 <= NLOW <= NSUP <= N, if JOBAE = 'S'.
NLOW = MIN( 1, N ), NSUP = N, if JOBAE = 'G'.
ALPHA (input) DOUBLE PRECISION
The boundary of the domain of interest for the generalized
eigenvalues of the pair (A,E). For a continuous-time
system (DICO = 'C'), ALPHA is the boundary value for the
real parts of the generalized eigenvalues, while for a
discrete-time system (DICO = 'D'), ALPHA >= 0 represents
the boundary value for the moduli of the generalized
eigenvalues.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the state dynamics matrix A.
If JOBAE = 'S' then A must be a matrix in real Schur form.
On exit, the leading N-by-N part of this array contains
the matrix Q'*A*Z in real Schur form, with the elements
below the first subdiagonal set to zero.
The leading NDIM-by-NDIM part of the principal subpencil
A12 - lambda*E12, defined by A12 := A(NLOW:NSUP,NLOW:NSUP)
and E12 := E(NLOW:NSUP,NLOW:NSUP), has generalized
eigenvalues in the domain of interest, and the trailing
part of this subpencil has generalized eigenvalues outside
the domain of interest.
The domain of interest for eig(A12,E12), the generalized
eigenvalues of the pair (A12,E12), is defined by the
parameters ALPHA, DICO and STDOM as follows:
For DICO = 'C':
Real(eig(A12,E12)) < ALPHA if STDOM = 'S';
Real(eig(A12,E12)) > ALPHA if STDOM = 'U'.
For DICO = 'D':
Abs(eig(A12,E12)) < ALPHA if STDOM = 'S';
Abs(eig(A12,E12)) > ALPHA if STDOM = 'U'.
LDA INTEGER
The leading dimension of the 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 descriptor matrix E.
If JOBAE = 'S', then E must be an upper triangular matrix.
On exit, the leading N-by-N part of this array contains an
upper triangular matrix Q'*E*Z, with the elements below
the diagonal set to zero.
The leading NDIM-by-NDIM part of the principal subpencil
A12 - lambda*E12 (see description of A) has generalized
eigenvalues in the domain of interest, and the trailing
part of this subpencil has generalized eigenvalues outside
the domain of interest.
LDE INTEGER
The leading dimension of the 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 the 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 the array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Q,
where Q' is the product of orthogonal
transformations which are applied to A,
E, and B on the left.
If COMPQ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Q1;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Q1*Q.
LDQ INTEGER
The leading dimension of the array Q. LDQ >= MAX(1,N).
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Z,
which is the product of orthogonal
transformations applied to A, E, and C
on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix
Z1;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Z1*Z.
LDZ INTEGER
The leading dimension of the array Z. LDZ >= MAX(1,N).
NDIM (output) INTEGER
The number of generalized eigenvalues of the principal
subpencil A12 - lambda*E12 (see description of A) lying
inside the domain of interest for eigenvalues.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N,
are the generalized eigenvalues.
ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j = 1,...,N, are the
diagonals of the complex Schur form (S,T) that would
result if the 2-by-2 diagonal blocks of the real Schur
form of (A,B) 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 length of the array DWORK.
LDWORK >= 8*N+16, if JOBAE = 'G';
LDWORK >= 4*N+16, if JOBAE = 'S'.
For optimum performance LDWORK should be larger.
If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the QZ algorithm failed to compute all generalized
eigenvalues of the pair (A,E);
= 2: a failure occured during the ordering of the
generalized real Schur form of the pair (A,E).
Method
If JOBAE = 'G', the pair (A,E) is reduced to an ordered generalized real Schur form using an orthogonal equivalence transformation A <-- Q'*A*Z and E <-- Q'*E*Z. This transformation is determined so that the leading diagonal blocks of the resulting pair (A,E) have generalized eigenvalues in a suitably defined domain of interest. Then, the transformations are applied to the matrices B and C: B <-- Q'*B and C <-- C*Z. If JOBAE = 'S', then the diagonal blocks of the subpencil A12 - lambda*E12, defined by A12 := A(NLOW:NSUP,NLOW:NSUP) and E12 := E(NLOW:NSUP,NLOW:NSUP), are reordered using orthogonal equivalence transformations, such that the leading blocks have generalized eigenvalues in a suitably defined domain of interest.Numerical Aspects
3 The algorithm requires about 25N floating point operations.Further Comments
NoneExample
Program Text
* TG01PD EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDE, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDE = NMAX, LDQ = NMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 8*NMAX+16 )
* .. Local Scalars ..
CHARACTER*1 COMPQ, COMPZ, DICO, JOBAE, STDOM
INTEGER I, INFO, J, M, N, NDIM, NLOW, NSUP, P
DOUBLE PRECISION ALPHA, TOL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALPHAI(NMAX), ALPHAR(NMAX),
$ B(LDB,MMAX), BETA(NMAX), C(LDC,NMAX),
$ DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
$ Z(LDZ,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TG01PD
* .. Intrinsic Functions ..
INTRINSIC DCMPLX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, DICO, STDOM, JOBAE, COMPQ, COMPZ,
$ NLOW, NSUP, ALPHA, TOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
IF ( LSAME( COMPQ, 'U' ) )
$ READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( COMPZ, 'U' ) )
$ READ ( NIN, FMT = * ) ( ( Z(I,J), J = 1,N ), I = 1,N )
* Find the reduced descriptor system
* (A-lambda E,B,C).
CALL TG01PD( DICO, STDOM, JOBAE, COMPQ, COMPZ, N, M, P,
$ NLOW, NSUP, ALPHA, A, LDA, E, LDE, B, LDB,
$ C, LDC, Q, LDQ, Z, LDZ, NDIM, ALPHAR,
$ ALPHAI, BETA, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) NDIM
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
30 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 40 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
40 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
50 CONTINUE
WRITE ( NOUT, FMT = 99990 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
60 CONTINUE
WRITE ( NOUT, FMT = 99985 )
DO 70 I = 1, N
IF ( BETA(I).EQ.ZERO .OR. ALPHAI(I).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99984 )
$ ALPHAR(I)/BETA(I)
ELSE
WRITE ( NOUT, FMT = 99984 )
$ DCMPLX( ALPHAR(I), ALPHAI(I) )/BETA(I)
END IF
70 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TG01PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01PD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Number of eigenvalues in the domain =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT (/' The finite generalized eigenvalues are '/
$ ' real part imag part ')
99984 FORMAT (1X,F9.4,SP,F9.4,S,'i ')
END
Program Data
TG01PD EXAMPLE PROGRAM DATA
4 2 2 C S G I I 1 4 -1.E-7 0.0
-1 0 0 3
0 0 1 2
1 1 0 4
0 0 0 0
1 2 0 0
0 1 0 1
3 9 6 3
0 0 2 0
1 0
0 0
0 1
1 1
-1 0 1 0
0 1 -1 1
Program Results
TG01PD EXAMPLE PROGRAM RESULTS
Number of eigenvalues in the domain = 1
The transformed state dynamics matrix Q'*A*Z is
-1.6311 2.1641 -3.6829 -0.3369
0.0000 0.4550 -1.9033 0.6425
0.0000 0.0000 2.6950 0.6882
0.0000 0.0000 0.0000 0.0000
The transformed descriptor matrix Q'*E*Z is
0.4484 9.6340 -1.2601 -5.6475
0.0000 3.3099 0.6641 -1.4869
0.0000 0.0000 0.0000 -1.3765
0.0000 0.0000 0.0000 2.0000
The transformed input/state matrix Q'*B is
0.0232 -0.9413
-0.7251 -0.2478
0.6882 -0.2294
1.0000 1.0000
The transformed state/output matrix C*Z is
-0.8621 0.3754 0.3405 1.0000
-0.1511 -1.1192 0.8513 -1.0000
The left transformation matrix Q is
0.0232 -0.7251 0.6882 0.0000
-0.3369 0.6425 0.6882 0.0000
-0.9413 -0.2478 -0.2294 0.0000
0.0000 0.0000 0.0000 1.0000
The right transformation matrix Z is
0.8621 -0.3754 -0.3405 0.0000
-0.4258 -0.9008 -0.0851 0.0000
0.0000 0.0000 0.0000 1.0000
0.2748 -0.2184 0.9364 0.0000
The finite generalized eigenvalues are
real part imag part
-3.6375
0.1375
Infinity
0.0000