Purpose
To compute the stable and unstable invariant subspaces for a Hamiltonian matrix with no eigenvalues on the imaginary axis, using the output of the SLICOT Library routine MB03XD.Specification
SUBROUTINE MB03ZD( WHICH, METH, STAB, BALANC, ORTBAL, SELECT, N,
$ MM, ILO, SCALE, S, LDS, T, LDT, G, LDG, U1,
$ LDU1, U2, LDU2, V1, LDV1, V2, LDV2, M, WR, WI,
$ US, LDUS, UU, LDUU, LWORK, IWORK, DWORK,
$ LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER BALANC, METH, ORTBAL, STAB, WHICH
INTEGER ILO, INFO, LDG, LDS, LDT, LDU1, LDU2, LDUS,
$ LDUU, LDV1, LDV2, LDWORK, M, MM, N
C .. Array Arguments ..
LOGICAL LWORK(*), SELECT(*)
INTEGER IWORK(*)
DOUBLE PRECISION DWORK(*), G(LDG,*), S(LDS,*), SCALE(*),
$ T(LDT,*), U1(LDU1,*), U2(LDU2,*), US(LDUS,*),
$ UU(LDUU,*), V1(LDV1,*), V2(LDV2,*), WI(*),
$ WR(*)
Arguments
Mode Parameters
WHICH CHARACTER*1
Specifies the cluster of eigenvalues for which the
invariant subspaces are computed:
= 'A': select all n eigenvalues;
= 'S': select a cluster of eigenvalues specified by
SELECT.
METH CHARACTER*1
If WHICH = 'A' this parameter specifies the method to be
used for computing bases of the invariant subspaces:
= 'S': compute the n-dimensional basis from a set of
n vectors;
= 'L': compute the n-dimensional basis from a set of
2*n vectors;
= 'Q': quick return of the set of n vectors;
= 'R': quick return of the set of 2*n vectors.
When in doubt, use METH = 'S'. In some cases, METH = 'L'
may result in more accurately computed invariant
subspaces, see [1].
Options METH = 'Q' or METH = 'R' return the range vectors
Y = [ Y1; Y2 ], where Y1 and Y2 have 2*n rows and n or 2*n
columns, respectively, which can be directly used, e.g.,
for finding the (stabilizing) solution of a Riccati
equation, by solving X*Y1 = Y2. Note that Y1 might be
singular when METH = 'Q'.
STAB CHARACTER*1
Specifies the type of invariant subspaces to be computed:
= 'S': compute the stable invariant subspace, i.e., the
invariant subspace belonging to those selected
eigenvalues that have negative real part;
= 'U': compute the unstable invariant subspace, i.e.,
the invariant subspace belonging to those
selected eigenvalues that have positive real
part;
= 'B': compute both the stable and unstable invariant
subspaces.
BALANC CHARACTER*1
Specifies the type of inverse balancing transformation
required:
= 'N': do nothing;
= 'P': do inverse transformation for permutation only;
= 'S': do inverse transformation for scaling only;
= 'B': do inverse transformations for both permutation
and scaling.
BALANC must be the same as the argument BALANC supplied to
MB03XD. Note that if the data is further post-processed,
e.g., for solving an algebraic Riccati equation, it is
recommended to delay inverse balancing (in particular the
scaling part) and apply it to the final result only,
see [2]. Inverse balancing is not used by this routine
if METH = 'Q' or METH = 'R'.
ORTBAL CHARACTER*1
If BALANC <> 'N', this option specifies how inverse
balancing is applied to the computed invariant subspaces:
= 'B': apply inverse balancing before orthogonal bases
for the invariant subspaces are computed;
= 'A': apply inverse balancing after orthogonal bases
for the invariant subspaces have been computed;
this may yield non-orthogonal bases if
BALANC = 'S' or BALANC = 'B'.
SELECT (input) LOGICAL array, dimension (N)
If WHICH = 'S', SELECT specifies the eigenvalues
corresponding to the positive and negative square
roots of the eigenvalues of S*T in the selected cluster.
To select a real eigenvalue w(j), SELECT(j) must be set
to .TRUE.. To select a complex conjugate pair of
eigenvalues w(j) and w(j+1), corresponding to a 2-by-2
diagonal block, both SELECT(j) and SELECT(j+1) must be set
to .TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
This array is not referenced if WHICH = 'A'.
Input/Output Parameters
N (input) INTEGER
The order of the matrices S, T and G. N >= 0.
MM (input) INTEGER
The number of columns in the arrays US and/or UU.
If WHICH = 'A' and (METH = 'S' or METH = 'Q'), MM = N;
if WHICH = 'A' and (METH = 'L' or METH = 'R'), MM = 2*N;
if WHICH = 'S', MM = M.
The values above for MM give the numbers of vectors to be
returned, if METH = 'Q' or METH = 'R', or the numbers of
vectors to be used for computing a basis for the invariant
subspace(s), if METH = 'S' or METH = 'L', or WHICH = 'S'.
ILO (input) INTEGER
If BALANC <> 'N', then ILO is the integer returned by
MB03XD. 1 <= ILO <= N+1.
SCALE (input) DOUBLE PRECISION array, dimension (N)
If BALANC <> 'N', the leading N elements of this array
must contain details of the permutation and scaling
factors, as returned by MB03XD.
This array is not referenced if BALANC = 'N'.
S (input/output) DOUBLE PRECISION array, dimension (LDS,N)
On entry, the leading N-by-N part of this array must
contain the matrix S in real Schur form.
On exit, the leading N-by-N part of this array is
overwritten.
LDS INTEGER
The leading dimension of the array S. LDS >= max(1,N).
T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular matrix T.
On exit, the leading N-by-N part of this array is
overwritten.
LDT INTEGER
The leading dimension of the array T. LDT >= max(1,N).
G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
On entry, if METH = 'L' or METH = 'R', the leading N-by-N
part of this array must contain a general matrix G.
On exit, if METH = 'L' or METH = 'R', the leading N-by-N
part of this array is overwritten.
This array is not referenced if METH = 'S' or METH = 'Q'.
LDG INTEGER
The leading dimension of the array G. LDG >= 1.
LDG >= max(1,N) if METH = 'L' or METH = 'R'.
U1 (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
On entry, the leading N-by-N part of this array must
contain the (1,1) block of an orthogonal symplectic
matrix U.
On exit, this array is overwritten.
LDU1 INTEGER
The leading dimension of the array U1. LDU1 >= MAX(1,N).
U2 (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
On entry, the leading N-by-N part of this array must
contain the (2,1) block of an orthogonal symplectic
matrix U.
On exit, this array is overwritten.
LDU2 INTEGER
The leading dimension of the array U2. LDU2 >= MAX(1,N).
V1 (input/output) DOUBLE PRECISION array, dimension (LDV1,N)
On entry, the leading N-by-N part of this array must
contain the (1,1) block of an orthogonal symplectic
matrix V.
On exit, this array is overwritten.
LDV1 INTEGER
The leading dimension of the array V1. LDV1 >= MAX(1,N).
V2 (input/output) DOUBLE PRECISION array, dimension (LDV1,N)
On entry, the leading N-by-N part of this array must
contain the (2,1) block of an orthogonal symplectic
matrix V.
On exit, this array is overwritten.
LDV2 INTEGER
The leading dimension of the array V2. LDV2 >= MAX(1,N).
M (output) INTEGER
The number of selected eigenvalues.
WR (output) DOUBLE PRECISION array, dimension (M)
WI (output) DOUBLE PRECISION array, dimension (M)
On exit, the leading M elements of WR and WI contain the
real and imaginary parts, respectively, of the selected
eigenvalues that have nonpositive real part. Complex
conjugate pairs of eigenvalues with real part not equal
to zero will appear consecutively with the eigenvalue
having the positive imaginary part first. Note that, due
to roundoff errors, these numbers may differ from the
eigenvalues computed by MB03XD.
US (output) DOUBLE PRECISION array, dimension (LDUS,MM)
On exit, if STAB = 'S' or STAB = 'B', the leading
2*N-by-MM part of this array contains a basis for the
stable invariant subspace belonging to the selected
eigenvalues, if METH = 'S' or METH = 'L', or the range
vectors Y, if METH = 'Q' or METH = 'R' (see parameter
METH). This basis is orthogonal unless ORTBAL = 'A'.
LDUS INTEGER
The leading dimension of the array US. LDUS >= 1.
If STAB = 'S' or STAB = 'B', LDUS >= 2*N.
UU (output) DOUBLE PRECISION array, dimension (LDUU,MM)
On exit, if STAB = 'U' or STAB = 'B', the leading
2*N-by-MM part of this array contains a basis for the
unstable invariant subspace belonging to the selected
eigenvalues, if METH = 'S' or METH = 'L', or the range
vectors Y, if METH = 'Q' or METH = 'R' (see parameter
METH). This basis is orthogonal unless ORTBAL = 'A'.
LDUU INTEGER
The leading dimension of the array UU. LDUU >= 1.
If STAB = 'U' or STAB = 'B', LDUU >= 2*N.
Workspace
LWORK LOGICAL array, dimension (2*N)
This array is only referenced if WHICH = 'A' and
(METH = 'L' or METH = 'R').
IWORK INTEGER array, dimension (LIWORK)
LIWORK = 2*N, if WHICH = 'A' and METH = 'L';
LIWORK = N, if WHICH = 'A' and METH = 'S';
LIWORK = 0, if WHICH = 'A' and METH = 'Q' or METH = 'R';
LIWORK = M, if WHICH = 'S'.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal
value of LDWORK.
On exit, if INFO = -35, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
If WHICH = 'S' or METH = 'S' or METH = 'Q':
LDWORK >= MAX( 1, 4*M*M + MAX( 8*M, 4*N ) ).
If WHICH = 'A' and (METH = 'L' or METH = 'R') and
( STAB = 'U' or STAB = 'S' ):
LDWORK >= MAX( 1, 2*N*N + 2*N, 8*N ).
If WHICH = 'A' and (METH = 'L' or METH = 'R') and
STAB = 'B':
LDWORK >= 8*N + 1.
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: some of the selected eigenvalues are on or too close
to the imaginary axis;
= 2: reordering of the product S*T in routine MB03ZA
failed because some eigenvalues are too close to
separate;
= 3: the QR algorithm failed to compute some Schur form
in MB03ZA;
= 4: reordering of the Hamiltonian Schur form in routine
MB03TD failed because some eigenvalues are too close
to separate;
= 5: the computed stable invariant subspace for
METH = 'S' is inaccurate. This may be taken as a
warning and a suggestion to try METH = 'L';
= 6: the computed unstable invariant subspace for
METH = 'S' is inaccurate. This may be taken as a
warning and a suggestion to try METH = 'L'.
Method
This is an implementation of Algorithm 1 in [1].Numerical Aspects
The method is strongly backward stable for an embedded (skew-)Hamiltonian matrix, see [1]. Although good results have been reported if the eigenvalues are not too close to the imaginary axis, the method is not backward stable for the original Hamiltonian matrix itself.References
[1] Benner, P., Mehrmann, V., and Xu, H.
A new method for computing the stable invariant subspace of a
real Hamiltonian matrix, J. Comput. Appl. Math., 86,
pp. 17-43, 1997.
[2] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
Further Comments
NoneExample
Program Text
* MB03ZD EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 200 )
INTEGER LDG, LDRES, LDS, LDT, LDU1, LDU2, LDUS, LDUU,
$ LDV1, LDV2, LDWORK
PARAMETER ( LDG = NMAX, LDRES = 2*NMAX, LDS = NMAX,
$ LDT = NMAX, LDU1 = NMAX, LDU2 = NMAX,
$ LDUS = 2*NMAX, LDUU = 2*NMAX, LDV1 = NMAX,
$ LDV2 = NMAX, LDWORK = 3*NMAX*NMAX + 7*NMAX )
* .. Local Scalars ..
CHARACTER*1 BALANC, METH, ORTBAL, STAB, WHICH
INTEGER I, ILO, INFO, J, M, N
* .. Local Arrays ..
LOGICAL LWORK(2*NMAX), SELECT(NMAX)
INTEGER IWORK(2*NMAX)
DOUBLE PRECISION DWORK(LDWORK), G(LDG, NMAX), RES(LDRES,NMAX),
$ S(LDS, NMAX), SCALE(NMAX), T(LDT,NMAX),
$ U1(LDU1,NMAX), U2(LDU2, NMAX), US(LDUS,2*NMAX),
$ UU(LDUU,2*NMAX), V1(LDV1,NMAX), V2(LDV2, NMAX),
$ WI(NMAX), WR(NMAX)
* .. External Functions ..
EXTERNAL DLANGE, LSAME
LOGICAL LSAME
DOUBLE PRECISION DLANGE
* .. External Subroutines ..
EXTERNAL MB03ZD
* .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, ILO, WHICH, METH, STAB, BALANC, ORTBAL
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
*
IF ( LSAME( WHICH, 'S' ) )
$ READ ( NIN, FMT = * ) ( SELECT(I), I = 1,N )
READ ( NIN, FMT = * ) ( ( S(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( WHICH, 'A' ).AND.LSAME( METH, 'L' ) )
$ READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( BALANC, 'P' ).OR.LSAME( BALANC, 'S' ).OR.
$ LSAME( BALANC, 'B' ) )
$ READ ( NIN, FMT = * ) ( SCALE(I), I = 1,N )
READ ( NIN, FMT = * ) ( ( U1(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( U2(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( V1(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( V2(I,J), J = 1,N ), I = 1,N )
*
CALL MB03ZD( WHICH, METH, STAB, BALANC, ORTBAL, SELECT, N, 2*N,
$ ILO, SCALE, S, LDS, T, LDT, G, LDG, U1, LDU1, U2,
$ LDU2, V1, LDV1, V2, LDV2, M, WR, WI, US, LDUS,
$ UU, LDUU, LWORK, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) I, WR(I), WI(I)
20 CONTINUE
*
IF ( LSAME( STAB, 'S' ).OR.LSAME( STAB, 'B' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, 2*N
WRITE ( NOUT, FMT = 99993 ) ( US(I,J), J = 1,M )
30 CONTINUE
IF ( LSAME( ORTBAL, 'B' ).OR.LSAME( BALANC, 'N' ).OR.
$ LSAME( BALANC, 'P' ) ) THEN
CALL DGEMM( 'Transpose', 'No Transpose', M, M, 2*N,
$ ONE, US, LDUS, US, LDUS, ZERO, RES,
$ LDRES )
DO 40 I = 1, M
RES(I,I) = RES(I,I) - ONE
40 CONTINUE
WRITE ( NOUT, FMT = 99991 ) DLANGE( 'Frobenius', M, M,
$ RES, LDRES, DWORK )
END IF
CALL DGEMM( 'Transpose', 'No Transpose', M, M, N, ONE,
$ US, LDUS, US(N+1,1), LDUS, ZERO, RES, LDRES )
CALL DGEMM( 'Transpose', 'No Transpose', M, M, N, -ONE,
$ US(N+1,1), LDUS, US, LDUS, ONE, RES, LDRES )
WRITE ( NOUT, FMT = 99990 ) DLANGE( 'Frobenius', M, M,
$ RES, LDRES, DWORK )
END IF
*
IF ( LSAME( STAB, 'U' ).OR.LSAME( STAB, 'B' ) ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 50 I = 1, 2*N
WRITE ( NOUT, FMT = 99993 ) ( UU(I,J), J = 1,M )
50 CONTINUE
IF ( LSAME( ORTBAL, 'B' ).OR.LSAME( BALANC, 'N' ).OR.
$ LSAME( BALANC, 'P' ) ) THEN
CALL DGEMM( 'Transpose', 'No Transpose', M, M, 2*N,
$ ONE, UU, LDUU, UU, LDUU, ZERO, RES,
$ LDRES )
DO 60 I = 1, M
RES(I,I) = RES(I,I) - ONE
60 CONTINUE
WRITE ( NOUT, FMT = 99989 ) DLANGE( 'Frobenius', M, M,
$ RES, LDRES, DWORK )
END IF
CALL DGEMM( 'Transpose', 'No Transpose', M, M, N, ONE,
$ UU, LDUU, UU(N+1,1), LDUU, ZERO, RES, LDRES )
CALL DGEMM( 'Transpose', 'No Transpose', M, M, N, -ONE,
$ UU(N+1,1), LDUU, UU, LDUU, ONE, RES, LDRES )
WRITE ( NOUT, FMT = 99988 ) DLANGE( 'Frobenius', M, M,
$ RES, LDRES, DWORK )
END IF
END IF
END IF
*
99999 FORMAT (' MB03ZD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03ZD = ',I2)
99997 FORMAT (' The stable eigenvalues are',//' i',6X,
$ 'WR(i)',6X,'WI(i)',/)
99996 FORMAT (I4,3X,F8.4,3X,F8.4)
99995 FORMAT (/' A basis for the stable invariant subspace is')
99994 FORMAT (/' A basis for the unstable invariant subspace is')
99993 FORMAT (20(1X,F9.3))
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (/' Orthogonality of US: || US''*US - I ||_F = ',G7.2)
99990 FORMAT (/' Symplecticity of US: || US''*J*US ||_F = ',G7.2)
99989 FORMAT (/' Orthogonality of UU: || UU''*UU - I ||_F = ',G7.2)
99988 FORMAT (/' Symplecticity of UU: || UU''*J*UU ||_F = ',G7.2)
END
Program Data
MB03ZD EXAMPLE PROGRAM DATA 5 1 A L B N B -3.1844761777714732 0.1612357243439331 -0.0628592203751138 0.2449004200921981 0.1974400149992579 0.0000000000000000 -0.1510667773167784 0.4260444411622838 -0.1775026035208615 0.3447278421198472 0.0000000000000000 -0.1386140422054264 -0.3006779624777515 0.2944143257134196 0.3456440339120323 0.0000000000000000 0.0000000000000000 0.0000000000000000 -0.2710128384740570 0.0933189808067138 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.4844146572359603 0.2004347508746697 3.2038208121776366 0.1805955192510651 0.2466389119377561 -0.2539149302433368 -0.0359238844381195 0.0000000000000000 -0.7196686433290816 0.0000000000000000 0.2428659121580384 -0.0594190100670832 0.0000000000000000 0.0000000000000000 -0.1891741194498107 -0.3309578443491266 -0.0303520731950515 0.0000000000000000 0.0000000000000000 0.0000000000000000 -0.4361574461961550 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.1530894573304220 -0.0370982242678464 0.0917788436945724 -0.0560402416315252 0.1345152517579192 0.0256668227276700 0.0652183678916931 -0.0700457231988297 0.0350041175858839 -0.2233868768749268 -0.1171980260782843 -0.0626428681377119 0.2327575351902772 -0.1251515732208170 -0.0177816046663201 0.3696921118421182 0.0746042309265599 -0.0828007611045140 0.0217427473546043 -0.1157775118548851 -0.3161183681200527 0.1374372236164812 0.1002727885506992 0.4021556774753973 -0.0431072263235579 0.1067394572547867 0.3806883009357247 -0.0347810363019649 -0.5014665065895758 0.5389691288472394 0.2685446895251367 0.4642712665555326 -0.5942766860716395 0.4781179763952615 0.2334370556238151 0.0166790369048933 0.2772789197782788 -0.0130145392695876 -0.2123817030594055 -0.2550292626960107 -0.5049268366774490 0.4209268575081796 0.1499593172661228 -0.1925590746592156 -0.5472292877802402 0.4543329704184054 0.3969669479129449 0.6321903535930828 0.3329156356041961 0.0163533225344433 -0.2638879466190024 -0.1795922007470742 0.1908329820840911 0.0868799433942070 0.3114741142062388 -0.2579907627915167 -0.2447897730222852 -0.1028403314750045 -0.1157840914576285 -0.1873268885694406 0.1700708002861580 -0.2243335325285328 0.3180998613802520 0.3315380214794822 0.1977859924739963 0.5072476567310013 -0.2128397588651423 -0.2740560593051881 0.1941418870268881 -0.3096684962457369 -0.0581576193198714 -0.2002027567371932 -0.0040094115506855 -0.3979373387545264 0.1520881534833910 -0.2010804514091372 0.4447147692018334 -0.6830166755147440 -0.0002576861753487 0.5781954611783305 -0.0375091627893805 0.5121756358795817 0.0297197140254773 0.4332229148788766 -0.3240527006890552 0.5330850295256511 0.3664711365265602 0.3288511296455119 0.0588396016404451 0.1134221597062257 0.1047567336850078 0.4535357098437908 0.1062866148880792 -0.3964092656837774 -0.2211800890450674 0.0350667323996222 0.4450432900616097 0.2950206358263853 -0.1617837757183893 -0.0376369332204927 -0.6746752660482623 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0299719306696789 -0.2322624725320701 -0.0280846899680325 -0.3044255686880000 -0.1077641482535519 -0.0069083614679702 0.3351358347080056 -0.4922707032978891 0.4293545450291714 0.4372821269062001 0.0167847133528843 0.2843629278945327 0.5958979805231146 0.3097336757510886 -0.2086733033047188 0.0248567764822071 -0.2810759958040470 -0.1653113624869834 -0.3528780198620412 -0.0254898556119252Program Results
MB03ZD EXAMPLE PROGRAM RESULTS
The stable eigenvalues are
i WR(i) WI(i)
1 -3.1941 0.0000
2 -0.1350 0.3179
3 -0.1350 -0.3179
4 -0.0595 0.2793
5 -0.0595 -0.2793
A basis for the stable invariant subspace is
-0.102 -0.116 0.627 0.118 -0.605
-0.100 -0.510 -0.266 0.504 0.124
-0.179 0.015 -0.112 -0.142 0.413
-0.055 0.252 0.182 -0.134 0.100
-0.078 0.576 -0.271 -0.252 -0.177
0.340 -0.135 0.053 -0.234 -0.110
0.528 0.108 -0.205 0.219 -0.096
0.397 -0.429 0.161 -0.598 0.199
0.444 0.342 0.447 0.406 0.440
0.434 0.014 -0.383 0.072 -0.391
Orthogonality of US: || US'*US - I ||_F = .62E-15
Symplecticity of US: || US'*J*US ||_F = .23E-14
A basis for the unstable invariant subspace is
-0.428 0.383 0.048 0.105 0.187
-0.506 -0.100 0.541 0.245 0.223
-0.334 -0.524 -0.044 -0.153 0.126
-0.453 0.076 0.103 -0.525 -0.268
-0.436 0.098 -0.752 0.209 -0.251
-0.093 -0.089 0.258 -0.114 -0.725
-0.112 -0.196 -0.186 -0.302 0.394
-0.120 -0.286 0.027 0.680 -0.119
-0.102 0.630 0.079 0.040 0.127
-0.091 -0.171 -0.136 -0.136 0.231
Orthogonality of UU: || UU'*UU - I ||_F = .69E-15
Symplecticity of UU: || UU'*J*UU ||_F = .10E-13