Purpose
To reorder the diagonal blocks of a principal subpencil of an upper quasi-triangular matrix pencil A-lambda*E together with their generalized eigenvalues, by constructing orthogonal similarity transformations UT and VT. After reordering, the leading block of the selected subpencil of A-lambda*E has generalized eigenvalues in a suitably defined domain of interest, usually related to stability/instability in a continuous- or discrete-time sense.Specification
SUBROUTINE MB03QG( DICO, STDOM, JOBU, JOBV, N, NLOW, NSUP, ALPHA,
$ A, LDA, E, LDE, U, LDU, V, LDV, NDIM, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, JOBU, JOBV, STDOM
INTEGER INFO, LDA, LDE, LDU, LDV, LDWORK, N, NDIM, NLOW,
$ NSUP
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), E(LDE,*), U(LDU,*), V(LDV,*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the spectrum separation to be
performed, as follows:
= 'C': continuous-time sense;
= 'D': discrete-time sense.
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.
JOBU CHARACTER*1
Indicates how the performed orthogonal transformations UT
are accumulated, as follows:
= 'I': U is initialized to the unit matrix and the matrix
UT is returned in U;
= 'U': the given matrix U is updated and the matrix U*UT
is returned in U.
JOBV CHARACTER*1
Indicates how the performed orthogonal transformations VT
are accumulated, as follows:
= 'I': V is initialized to the unit matrix and the matrix
VT is returned in V;
= 'U': the given matrix V is updated and the matrix V*VT
is returned in V.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, E, U, and V. N >= 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.
ALPHA (input) DOUBLE PRECISION
The boundary of the domain of interest for the eigenvalues
of A. If DICO = 'C', ALPHA is the boundary value for the
real parts of the generalized eigenvalues, while for
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 a matrix in a real Schur form whose 1-by-1 and
2-by-2 diagonal blocks between positions NLOW and NSUP
are to be reordered.
On exit, the leading N-by-N part of this array contains
a real Schur matrix UT' * A * VT, with the elements below
the first subdiagonal set to zero.
The leading NDIM-by-NDIM part of the principal subpencil
B - lambda*C, defined with B := A(NLOW:NSUP,NLOW:NSUP),
C := 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(B,C), the generalized
eigenvalues of the pair (B,C), is defined by the
parameters ALPHA, DICO and STDOM as follows:
For DICO = 'C':
Real(eig(B,C)) < ALPHA if STDOM = 'S';
Real(eig(B,C)) > ALPHA if STDOM = 'U'.
For DICO = 'D':
Abs(eig(B,C)) < ALPHA if STDOM = 'S';
Abs(eig(B,C)) > 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 a matrix in an upper triangular form.
On exit, the leading N-by-N part of this array contains an
upper triangular matrix UT' * E * VT, with the elements
below the diagonal set to zero.
The leading NDIM-by-NDIM part of the principal subpencil
B - lambda*C, defined with B := A(NLOW:NSUP,NLOW:NSUP)
C := 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 (see description of A).
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
On entry with JOBU = 'U', the leading N-by-N part of this
array must contain a transformation matrix (e.g., from a
previous call to this routine).
On exit, if JOBU = 'U', the leading N-by-N part of this
array contains the product of the input matrix U and the
orthogonal matrix UT used to reorder the diagonal blocks
of A - lambda*E.
On exit, if JOBU = 'I', the leading N-by-N part of this
array contains the matrix UT of the performed orthogonal
transformations.
Array U need not be set on entry if JOBU = 'I'.
LDU INTEGER
The leading dimension of the array U. LDU >= MAX(1,N).
V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
On entry with JOBV = 'U', the leading N-by-N part of this
array must contain a transformation matrix (e.g., from a
previous call to this routine).
On exit, if JOBV = 'U', the leading N-by-N part of this
array contains the product of the input matrix V and the
orthogonal matrix VT used to reorder the diagonal blocks
of A - lambda*E.
On exit, if JOBV = 'I', the leading N-by-N part of this
array contains the matrix VT of the performed orthogonal
transformations.
Array V need not be set on entry if JOBV = 'I'.
LDV INTEGER
The leading dimension of the array V. LDV >= MAX(1,N).
NDIM (output) INTEGER
The number of generalized eigenvalues of the selected
principal subpencil lying inside the domain of interest.
If NLOW = 1, NDIM is also the dimension of the deflating
subspace corresponding to the generalized eigenvalues of
the leading NDIM-by-NDIM subpencil. In this case, if U and
V are the orthogonal transformation matrices used to
compute and reorder the generalized real Schur form of the
pair (A,E), then the first NDIM columns of V form an
orthonormal basis for the above deflating subspace.
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 >= 1, and if N > 1,
LDWORK >= 4*N + 16.
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: A(NLOW,NLOW-1) is nonzero, i.e., A(NLOW,NLOW) is not
the leading element of a 1-by-1 or 2-by-2 diagonal
block of A, or A(NSUP+1,NSUP) is nonzero, i.e.,
A(NSUP,NSUP) is not the bottom element of a 1-by-1
or 2-by-2 diagonal block of A;
= 2: two adjacent blocks are too close to swap (the
problem is very ill-conditioned).
Method
Given an upper quasi-triangular matrix pencil A - lambda*E with 1-by-1 or 2-by-2 diagonal blocks, the routine reorders its diagonal blocks along with its eigenvalues by performing an orthogonal equivalence transformation UT'*(A - lambda*E)* VT. The column transformations UT and VT are also performed on the given (initial) transformations U and V (resulted from a possible previous step or initialized as identity matrices). After reordering, the generalized eigenvalues inside the region specified by the parameters ALPHA, DICO and STDOM appear at the top of the selected diagonal subpencil between positions NLOW and NSUP. In other words, lambda(A(Select,Select),E(Select,Select)) are ordered such that lambda(A(Inside,Inside),E(Inside,Inside)) are inside, and lambda(A(Outside,Outside),E(Outside,Outside)) are outside the domain of interest, where Select = NLOW:NSUP, Inside = NLOW:NLOW+NDIM-1, and Outside = NLOW+NDIM:NSUP. If NLOW = 1, the first NDIM columns of V*VT span the corresponding right deflating subspace of (A,E).References
[1] Stewart, G.W.
HQR3 and EXCHQZ: FORTRAN subroutines for calculating and
ordering the eigenvalues of a real upper Hessenberg matrix.
ACM TOMS, 2, pp. 275-280, 1976.
Numerical Aspects
3 The algorithm requires less than 4*N operations.Further Comments
NoneExample
Program Text
* MB03QG EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 10 )
INTEGER LDA, LDE, LDU, LDV
PARAMETER ( LDA = NMAX, LDE = NMAX, LDU = NMAX, LDV = NMAX)
INTEGER LDWORK
PARAMETER ( LDWORK = 8*NMAX + 16 )
* .. Local Scalars ..
CHARACTER*1 DICO, JOBU, JOBV, STDOM
INTEGER I, INFO, J, N, NDIM, NLOW, NSUP
DOUBLE PRECISION ALPHA
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), BETA(NMAX), DWORK(LDWORK),
$ E(LDE,NMAX), U(LDU,NMAX), V(LDV,NMAX), WI(NMAX),
$ WR(NMAX)
LOGICAL BWORK(NMAX)
* .. External Functions ..
LOGICAL DELCTG
* .. External Subroutines ..
EXTERNAL DGGES, MB03QG
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, NLOW, NSUP, ALPHA, DICO, STDOM, JOBU,
$ JOBV
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) 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 )
* Compute Schur form, eigenvalues and Schur vectors.
CALL DGGES( 'Vectors', 'Vectors', 'Not sorted', DELCTG, N,
$ A, LDA, E, LDE, NDIM, WR, WI, BETA, U, LDU, V, LDV,
$ DWORK, LDWORK, BWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
* Block reordering.
CALL MB03QG( DICO, STDOM, JOBU, JOBV, N, NLOW, NSUP, ALPHA,
$ A, LDA, E, LDE, U, LDU, V, LDV, NDIM, DWORK,
$ LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 ) NDIM
WRITE ( NOUT, FMT = 99994 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 30 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
30 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( V(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB03QG EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DGEES = ',I2)
99997 FORMAT (' INFO on exit from MB03QG = ',I2)
99996 FORMAT (' The number of eigenvalues in the domain is ',I5)
99995 FORMAT (8X,20(1X,F8.4))
99994 FORMAT (/' The ordered Schur form matrix is ')
99993 FORMAT (/' The ordered triangular matrix is ')
99992 FORMAT (/' The transformation matrix U is ')
99991 FORMAT (/' The transformation matrix V is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
END
Program Data
MB03QG EXAMPLE PROGRAM DATA 4 1 4 0.0 C S U U -1.0 37.0 -12.0 -12.0 -1.0 -10.0 0.0 4.0 2.0 -4.0 7.0 -6.0 2.0 2.0 7.0 -9.0 1.0 3.0 2.0 -1.0 -2.0 5.0 3.0 2.0 2.0 4.0 5.0 6.0 3.0 7.0 6.0 9.0Program Results
MB03QG EXAMPLE PROGRAM RESULTS
The number of eigenvalues in the domain is 2
The ordered Schur form matrix is
-1.4394 2.5550 -12.5655 -4.0714
2.8887 -1.1242 9.2819 -2.6724
0.0000 0.0000 -19.7785 36.4447
0.0000 0.0000 0.0000 3.5537
The ordered triangular matrix is
-16.0178 0.0000 2.3850 4.7645
0.0000 3.2809 -1.5640 1.9954
0.0000 0.0000 -3.0652 0.3039
0.0000 0.0000 0.0000 1.1671
The transformation matrix U is
-0.1518 -0.0737 -0.9856 0.0140
-0.2865 -0.9466 0.1136 -0.0947
-0.5442 0.0924 0.0887 0.8292
-0.7738 0.3000 0.0890 -0.5508
The transformation matrix V is
0.2799 0.9041 0.2685 0.1794
0.4009 -0.0714 0.3780 -0.8315
0.7206 -0.4006 0.2628 0.5012
0.4917 0.1306 -0.8462 -0.1588