MB04AD

Eigenvalues of a real skew-Hamiltonian/Hamiltonian pencil in factored form

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

Purpose

  To compute the eigenvalues of a real N-by-N skew-Hamiltonian/
  Hamiltonian pencil aS - bH with

                                   (  0  I  )
    S = T Z = J Z' J' Z, where J = (        ),                   (1)
                                   ( -I  0  )

  via generalized symplectic URV decomposition. That is, orthogonal
  matrices Q1 and Q2 and orthogonal symplectic matrices U1 and U2
  are computed such that

                                (  T11  T12 )
    Q1' T U1 = Q1' J Z' J' U1 = (           ) = Tout,
                                (   0   T22 )

               (  Z11  Z12 )
    U2' Z Q2 = (           ) = Zout,                             (2)
               (   0   Z22 )

               ( H11  H12 )
    Q1' H Q2 = (          ) = Hout,
               (  0   H22 )

  where T11, T22', Z11, Z22', H11 are upper triangular and H22' is
  upper quasi-triangular.
  Optionally, if COMPQ1 = 'C' the orthogonal transformation matrix
  Q1 will be computed.
  Optionally, if COMPQ2 = 'C' the orthogonal transformation matrix
  Q2 will be computed.
  Optionally, if COMPU1 = 'C' the orthogonal symplectic
  transformation matrix

         (  U11  U12  )
    U1 = (            )
         ( -U12  U11  )

  will be computed.
  Optionally, if COMPU2 = 'C' the orthogonal symplectic
  transformation matrix

         (  U21  U22  )
    U2 = (            )
         ( -U22  U21  )

  will be computed.

Specification
      SUBROUTINE MB04AD( JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N, Z, LDZ,
     $                   H, LDH, T, LDT, Q1, LDQ1, Q2, LDQ2, U11, LDU11,
     $                   U12, LDU12, U21, LDU21, U22, LDU22, ALPHAR,
     $                   ALPHAI, BETA, IWORK, LIWORK, DWORK, LDWORK,
     $                   INFO )
C     .. Scalar Arguments ..
      CHARACTER          COMPQ1, COMPQ2, COMPU1, COMPU2, JOB
      INTEGER            INFO, LDH, LDQ1, LDQ2, LDT, LDU11, LDU12,
     $                   LDU21, LDU22, LDWORK, LDZ, LIWORK, N
C     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), BETA( * ),
     $                   DWORK( * ), H( LDH, * ), Q1( LDQ1, * ),
     $                   Q2( LDQ2, * ), T( LDT, * ), U11( LDU11, * ),
     $                   U12( LDU12, * ), U21( LDU21, * ),
     $                   U22( LDU22, * ), Z( LDZ, * )

Arguments

Mode Parameters

  JOB     CHARACTER*1
          Specifies the computation to be performed, as follows:
          = 'E': compute the eigenvalues only; Z, T, and H will not
                 necessarily be put into the forms in (2); H22' is
                 upper Hessenberg;
          = 'T': put Z, T, and H into the forms in (2), and return
                 the eigenvalues in ALPHAR, ALPHAI and BETA.

  COMPQ1  CHARACTER*1
          Specifies whether to compute the orthogonal transformation
          matrix Q1, as follows:
          = 'N': Q1 is not computed;
          = 'C': compute the matrix Q1 of the orthogonal
                 transformations applied on the left to the pencil
                 aTZ - bH to reduce its matrices to the form (2).
                 The array Q1 is initialized internally to the
                 identity matrix.

  COMPQ2  CHARACTER*1
          Specifies whether to compute the orthogonal transformation
          matrix Q2, as follows:
          = 'N': Q2 is not computed;
          = 'C': compute the matrix Q2 of the orthogonal
                 transformations applied on the right to the pencil
                 aTZ - bH to reduce its matrices to the form (2).
                 The array Q2 is initialized internally to the
                 identity matrix.

  COMPU1  CHARACTER*1
          Specifies whether to compute the orthogonal symplectic
          transformation matrix U1, as follows:
          = 'N': U1 is not computed;
          = 'C': compute the matrices U11 and U12 of the orthogonal
                 symplectic transformations applied to the pencil
                 aTZ - bT to reduce its matrices to the form (2).
                 The arrays U11 and U12 are initialized internally
                 to correspond to an identity matrix U1.

  COMPU2  CHARACTER*1
          Specifies whether to compute the orthogonal symplectic
          transformation matrix U2, as follows:
          = 'N': U2 is not computed;
          = 'C': compute the matrices U21 and U22 of the orthogonal
                 symplectic transformations applied to the pencil
                 aTZ - bT to reduce its matrices to the form (2).
                 The arrays U21 and U22 are initialized internally
                 to correspond to an identity matrix U2.

Input/Output Parameters
  N       (input) INTEGER
          The order of the pencil aS - bH.  N >= 0, even.

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
          On entry, the leading N-by-N part of this array must
          contain the matrix Z.
          On exit, if JOB = 'T', the leading N-by-N part of this
          array contains the matrix Zout; otherwise, it contains the
          matrix Z obtained just before the application of the
          periodic QZ algorithm.
          The elements of the (2,1) block, i.e., in the rows N/2+1
          to N and in the columns 1 to N/2 are not set to zero, but
          are unchanged on exit.

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

  H       (input/output) DOUBLE PRECISION array, dimension (LDH, N)
          On entry, the leading N-by-N part of this array must
          contain the Hamiltonian matrix H (H22 = -H11', H12 = H12',
          H21 = H21').
          On exit, if JOB = 'T', the leading N-by-N part of this
          array contains the matrix Hout; otherwise, it contains the
          matrix H obtained just before the application of the
          periodic QZ algorithm.

  LDH     INTEGER
          The leading dimension of the array H.  LDH >= MAX(1, N).

  T       (output) DOUBLE PRECISION array, dimension (LDT, N)
          If JOB = 'T', the leading N-by-N part of this array
          contains the matrix Tout; otherwise, it contains the
          matrix T obtained just before the application of the
          periodic QZ algorithm.

  LDT     INTEGER
          The leading dimension of the array T.  LDT >= MAX(1, N).

  Q1      (output) DOUBLE PRECISION array, dimension (LDQ1, N)
          On exit, if COMPQ1 = 'C', the leading N-by-N part of this
          array contains the orthogonal transformation matrix Q1.
          If COMPQ1 = 'N', this array is not referenced.

  LDQ1    INTEGER
          The leading dimension of the array Q1.
          LDQ1 >= 1,         if COMPQ1 = 'N';
          LDQ1 >= MAX(1, N), if COMPQ1 = 'C'.

  Q2      (output) DOUBLE PRECISION array, dimension (LDQ2, N)
          On exit, if COMPQ2 = 'C', the leading N-by-N part of this
          array contains the orthogonal transformation matrix Q2.
          If COMPQ2 = 'N', this array is not referenced.

  LDQ2    INTEGER
          The leading dimension of the array Q2.
          LDQ2 >= 1,         if COMPQ2 = 'N';
          LDQ2 >= MAX(1, N), if COMPQ2 = 'C'.

  U11     (output) DOUBLE PRECISION array, dimension (LDU11, N/2)
          On exit, if COMPU1 = 'C', the leading N/2-by-N/2 part of
          this array contains the upper left block U11 of the
          orthogonal symplectic transformation matrix U1.
          If COMPU1 = 'N', this array is not referenced.

  LDU11   INTEGER
          The leading dimension of the array U11.
          LDU11 >= 1,           if COMPU1 = 'N';
          LDU11 >= MAX(1, N/2), if COMPU1 = 'C'.

  U12     (output) DOUBLE PRECISION array, dimension (LDU12, N/2)
          On exit, if COMPU1 = 'C', the leading N/2-by-N/2 part of
          this array contains the upper right block U12 of the
          orthogonal symplectic transformation matrix U1.
          If COMPU1 = 'N', this array is not referenced.

  LDU12   INTEGER
          The leading dimension of the array U12.
          LDU12 >= 1,           if COMPU1 = 'N';
          LDU12 >= MAX(1, N/2), if COMPU1 = 'C'.

  U21     (output) DOUBLE PRECISION array, dimension (LDU21, N/2)
          On exit, if COMPU2 = 'C', the leading N/2-by-N/2 part of
          this array contains the upper left block U21 of the
          orthogonal symplectic transformation matrix U2.
          If COMPU2 = 'N', this array is not referenced.

  LDU21   INTEGER
          The leading dimension of the array U21.
          LDU21 >= 1,           if COMPU2 = 'N';
          LDU21 >= MAX(1, N/2), if COMPU2 = 'C'.

  U22     (output) DOUBLE PRECISION array, dimension (LDU22, N/2)
          On exit, if COMPU2 = 'C', the leading N/2-by-N/2 part of
          this array contains the upper right block U22 of the
          orthogonal symplectic transformation matrix U2.
          If COMPU2 = 'N', this array is not referenced.

  LDU22   INTEGER
          The leading dimension of the array U22.
          LDU22 >= 1,           if COMPU2 = 'N';
          LDU22 >= MAX(1, N/2), if COMPU2 = 'C'.

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N/2)
          The real parts of each scalar alpha defining an eigenvalue
          of the pencil aS - bH.

  ALPHAI  (output) DOUBLE PRECISION array, dimension (N/2)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of the pencil aS - bH.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real.

  BETA    (output) DOUBLE PRECISION array, dimension (N/2)
          The scalars beta defining the eigenvalues of the pencil
          aS - bH.
          If INFO = 0, the quantities alpha = (ALPHAR(j),ALPHAI(j)),
          and beta = BETA(j) represent together the j-th eigenvalue
          of the pencil aS - bH, in the form lambda = alpha/beta.
          Since lambda may overflow, the ratios should not, in
          general, be computed. Due to the skew-Hamiltonian/
          Hamiltonian structure of the pencil, only half of the
          spectrum is saved in ALPHAR, ALPHAI and BETA.
          Specifically, the eigenvalues with positive real parts or
          with non-negative imaginary parts, when real parts are
          zero, are returned. The remaining eigenvalues have
          opposite signs.
          If INFO = 3, one or more BETA(j) is not representable, and
          the eigenvalues are returned as described below.

Workspace
  IWORK   INTEGER array, dimension (LIWORK)
          On exit, if INFO = 3, IWORK(1), ..., IWORK(N/2) return the
          scaling parameters for the eigenvalues of the pencil
          aS - bH (see INFO = 3).

  LIWORK  INTEGER
          The dimension of the array IWORK.
          LIWORK >= N/2+18.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal value
          of LDWORK, and DWORK(2) returns the machine base, b.
          On exit, if INFO = -31, DWORK(1) returns the minimum value
          of LDWORK.

  LDWORK  INTEGER
          The dimension of the array DWORK.
          If JOB = 'E' and COMPQ1 = 'N' and COMPQ2 = 'N' and
          COMPU1 = 'N' and COMPU2 = 'N', then
                LDWORK >= 3/2*N**2+MAX(N, 48);
          else, LDWORK >=   3*N**2+MAX(N, 48).
          For good performance LDWORK should generally 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: succesful exit.
          < 0: if INFO = -i, the i-th argument had an illegal value.
          = 1: the periodic QZ algorithm was not able to reveal
               information about the eigenvalues from the 2-by-2
               blocks in the SLICOT Library routine MB03BD;
          = 2: the periodic QZ algorithm did not converge in the
               SLICOT Library routine MB03BD;
          = 3: the eigenvalues will under- or overflow if evaluated;
               therefore, the j-th eigenvalue is represented by
               the quantities alpha = (ALPHAR(j),ALPHAI(j)),
               beta = BETA(j), and gamma = IWORK(j) in the form
               lambda = (alpha/beta) * b**gamma, where b is the
               machine base (often 2.0). This is not an error.

Method
  The algorithm uses Givens rotations and Householder reflections to
  annihilate elements in T, Z, and H such that T11, T22', Z11, Z22'
  and H11 are upper triangular and H22' is upper Hessenberg. Finally
  the periodic QZ algorithm is applied to transform H22' to upper
  quasi-triangular form while T11, T22', Z11, Z22', and H11 stay in
  upper triangular form.
  See also page 17 in [1] for more details.

References
  [1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
      Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
      Eigenproblems.
      Tech. Rep., Technical University Chemnitz, Germany,
      Nov. 2007.

Numerical Aspects
                                                            3
  The algorithm is numerically backward stable and needs O(N ) real
  floating point operations.

Further Comments
  None
Example

Program Text

*     MB04AD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2010 NICONET e.V.
*
*     .. Parameters ..
      INTEGER            NIN, NOUT
      PARAMETER          ( NIN = 5, NOUT = 6 )
      INTEGER            NMAX
      PARAMETER          ( NMAX = 50 )
      INTEGER            LDH, LDQ1, LDQ2, LDT, LDU11, LDU12, LDU21,
     $                   LDU22, LDWORK, LDZ, LIWORK
      PARAMETER          ( LDH = NMAX, LDQ1  = NMAX,   LDQ2 = NMAX,
     $                     LDT = NMAX, LDU11 = NMAX/2, LDU12 = NMAX/2,
     $                     LDU21  = NMAX/2, LDU22 = NMAX/2,
     $                     LDWORK = 3*NMAX*NMAX + MAX( NMAX, 48 ),
     $                     LDZ = NMAX,  LIWORK = NMAX/2 + 18 )
      DOUBLE PRECISION  ZERO
      PARAMETER         ( ZERO = 0.0D0 )
*
*     .. Local Scalars ..
      CHARACTER          COMPQ1, COMPQ2, COMPU1, COMPU2, JOB
      INTEGER            I, INFO, J, M, N
*
*     .. Local Arrays ..
      INTEGER            IWORK( LIWORK )
      DOUBLE PRECISION   ALPHAI( NMAX/2 ), ALPHAR( NMAX/2 ),
     $                   BETA( NMAX/2 ), DWORK( LDWORK ),
     $                   H( LDH, NMAX ), Q1( LDQ1, NMAX ),
     $                   Q2( LDQ2, NMAX ), T( LDT, NMAX ),
     $                   U11( LDU11, NMAX/2 ), U12( LDU12, NMAX/2 ),
     $                   U21( LDU21, NMAX/2 ), U22( LDU22, NMAX/2 ),
     $                   Z( LDZ, NMAX )
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*
*     .. External Subroutines ..
      EXTERNAL           DLASET, MB04AD
*
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*
*     .. Executable Statements ..
*
      WRITE( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read in the data.
      READ( NIN, FMT = * )
      READ( NIN, FMT = * ) JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N
      IF( N.LT.0 .OR. N.GT.NMAX ) THEN
         WRITE( NOUT, FMT = 99998 ) N
      ELSE
         READ( NIN, FMT = * ) ( ( Z( I, J ), J = 1, N ), I = 1, N )
         READ( NIN, FMT = * ) ( ( H( I, J ), J = 1, N ), I = 1, N )
*        Compute the eigenvalues of a real skew-Hamiltonian/Hamiltonian
*        pencil (factored version).
         CALL MB04AD( JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N, Z, LDZ, H,
     $                LDH, T, LDT, Q1, LDQ1, Q2, LDQ2, U11, LDU11, U12,
     $                LDU12, U21, LDU21, U22, LDU22, ALPHAR, ALPHAI,
     $                BETA, IWORK, LIWORK, DWORK, LDWORK, INFO )
*
         IF( INFO.NE.0 ) THEN
            WRITE( NOUT, FMT = 99997 ) INFO
         ELSE
            M = N/2
            CALL DLASET( 'Full', M, M, ZERO, ZERO, Z( M+1, 1 ), LDZ )
            WRITE( NOUT, FMT = 99996 )
            DO 10 I = 1, N
               WRITE( NOUT, FMT = 99995 ) ( T( I, J ), J = 1, N )
   10       CONTINUE
            WRITE( NOUT, FMT = 99994 )
            DO 20 I = 1, N
               WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N )
   20       CONTINUE
            WRITE( NOUT, FMT = 99993 )
            DO 30 I = 1, N
               WRITE( NOUT, FMT = 99995 ) ( H( I, J ), J = 1, N )
   30       CONTINUE
            IF( LSAME( COMPQ1, 'C' ) ) THEN
               WRITE( NOUT, FMT = 99992 )
               DO 40 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( Q1( I, J ), J = 1, N )
   40          CONTINUE
            END IF
            IF( LSAME( COMPQ2, 'C' ) ) THEN
               WRITE( NOUT, FMT = 99991 )
               DO 50 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( Q2( I, J ), J = 1, N )
   50          CONTINUE
            END IF
            IF( LSAME( COMPU1, 'C' ) ) THEN
               WRITE( NOUT, FMT = 99990 )
               DO 60 I = 1, M
                  WRITE( NOUT, FMT = 99995 ) ( U11( I, J ), J = 1, M )
   60          CONTINUE
               WRITE( NOUT, FMT = 99989 )
               DO 70 I = 1, M
                  WRITE( NOUT, FMT = 99995 ) ( U12( I, J ), J = 1, M )
   70          CONTINUE
            END IF
            IF( LSAME( COMPU2, 'C' ) ) THEN
               WRITE( NOUT, FMT = 99988 )
               DO 80 I = 1, M
                  WRITE( NOUT, FMT = 99995 ) ( U21( I, J ), J = 1, M )
   80          CONTINUE
               WRITE( NOUT, FMT = 99987 )
               DO 90 I = 1, M
                  WRITE( NOUT, FMT = 99995 ) ( U22( I, J ), J = 1, M )
   90          CONTINUE
            END IF
            WRITE( NOUT, FMT = 99986 )
            WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
            WRITE( NOUT, FMT = 99985 )
            WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
            WRITE( NOUT, FMT = 99984 )
            WRITE( NOUT, FMT = 99995 ) (   BETA( I ), I = 1, M )
         END IF
      END IF
      STOP
* 
99999 FORMAT( 'MB04AD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB04AD = ', I2 )
99996 FORMAT( 'The matrix T on exit is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The matrix Z on exit is ' )
99993 FORMAT( 'The matrix H is ' )
99992 FORMAT( 'The matrix Q1 is ' )
99991 FORMAT( 'The matrix Q2 is ' )
99990 FORMAT( 'The upper left block of the matrix U1 is ' )
99989 FORMAT( 'The upper right block of the matrix U1 is ' )
99988 FORMAT( 'The upper left block of the matrix U2 is ' )
99987 FORMAT( 'The upper right block of the matrix U2 is ' )
99986 FORMAT( 'The vector ALPHAR is ' )
99985 FORMAT( 'The vector ALPHAI is ' )
99984 FORMAT( 'The vector BETA is ' )
      END
Program Data
MB04AD EXAMPLE PROGRAM DATA
   T   C   C   C   C   8
   3.1472    4.5751   -0.7824    1.7874   -2.2308   -0.6126    2.0936    4.5974
   4.0579    4.6489    4.1574    2.5774   -4.5383   -1.1844    2.5469   -1.5961
  -3.7301   -3.4239    2.9221    2.4313   -4.0287    2.6552   -2.2397    0.8527
   4.1338    4.7059    4.5949   -1.0777    3.2346    2.9520    1.7970   -2.7619
   1.3236    4.5717    1.5574    1.5548    1.9483   -3.1313    1.5510    2.5127
  -4.0246   -0.1462   -4.6429   -3.2881   -1.8290   -0.1024   -3.3739   -2.4490
  -2.2150    3.0028    3.4913    2.0605    4.5022   -0.5441   -3.8100    0.0596
   0.4688   -3.5811    4.3399   -4.6817   -4.6555    1.4631   -0.0164    1.9908
   3.9090   -3.5071    3.1428   -3.0340   -1.4834    3.7401   -0.1715    0.4026
   4.5929   -2.4249   -2.5648   -2.4892    3.7401   -2.1416    1.6251    2.6645
   0.4722    3.4072    4.2926    1.1604   -0.1715    1.6251   -4.2415   -0.0602
  -3.6138   -2.4572   -1.5002   -0.2671    0.4026    2.6645   -0.0602   -3.7009
   0.6882   -1.8421   -4.1122    0.1317   -3.9090   -4.5929   -0.4722    3.6138
  -1.8421    2.9428   -0.4340    1.3834    3.5071    2.4249   -3.4072    2.4572
  -4.1122   -0.4340   -2.3703    0.5231   -3.1428    2.5648   -4.2926    1.5002
   0.1317    1.3834    0.5231   -4.1618    3.0340    2.4892   -1.1604    0.2671
Program Results
MB04AD EXAMPLE PROGRAM RESULTS
The matrix T on exit is 
  -3.9699   3.7658   5.5815  -1.7750  -0.8818  -0.0511  -4.2158   1.9054
   0.0000   5.3686  -5.9166   4.9163   1.3839   0.8870   3.9458  -4.9167
   0.0000   0.0000   5.9641   1.9432  -2.0680   2.4402  -1.4091   5.8512
   0.0000   0.0000   0.0000   5.9983  -3.8172   4.0147  -2.0739  -1.2570
   0.0000   0.0000   0.0000   0.0000   8.2005   0.0000   0.0000   0.0000
   0.0000   0.0000   0.0000   0.0000   1.5732   8.0098   0.0000   0.0000
   0.0000   0.0000   0.0000   0.0000   0.6017   2.4397   5.9751   0.0000
   0.0000   0.0000   0.0000   0.0000  -2.5869   0.5598   0.2544   5.2129
The matrix Z on exit is 
  -6.4705  -2.5511  -4.0551  -1.9895  -2.7642   0.7532  -4.1047  -2.2046
   0.0000   7.3589  -4.4480  -2.7491  -1.5465  -1.4345  -0.9272   1.3121
   0.0000   0.0000   4.9125  -0.4968   5.3574   3.8579   5.2547  -1.7324
   0.0000   0.0000   0.0000   9.0822   0.0460  -0.3382   3.9302   3.1084
   0.0000   0.0000   0.0000   0.0000   6.1869   0.0000   0.0000   0.0000
   0.0000   0.0000   0.0000   0.0000   5.5573   6.6549   0.0000   0.0000
   0.0000   0.0000   0.0000   0.0000   2.7456  -3.5789   4.3432   0.0000
   0.0000   0.0000   0.0000   0.0000   0.1549   3.5335   3.1346   4.1062
The matrix H is 
  -7.4834   0.4404   2.3558   1.6724  -0.4630   1.9533   1.5724  -2.7254
   0.0000  -7.3500   3.7414   3.7466   0.2837   0.6849   0.7727  -4.2140
   0.0000   0.0000  -2.3493  -3.7994  -0.6872   1.1773  -2.6901  -5.1494
   0.0000   0.0000   0.0000  -3.4719   5.3322   0.4182   1.9779   1.5175
   0.0000   0.0000   0.0000   0.0000  -6.1880   0.0000   0.0000   0.0000
   0.0000   0.0000   0.0000   0.0000  -3.3324   9.0833   0.0000   0.0000
   0.0000   0.0000   0.0000   0.0000  -1.8703   0.0799  -2.8180   0.0000
   0.0000   0.0000   0.0000   0.0000  -2.3477   3.3110   0.6561   0.7281
The matrix Q1 is 
  -0.2489  -0.1409   0.3615   0.6458   0.0113   0.6063  -0.0470   0.0238
  -0.2436   0.1294  -0.0874  -0.4103   0.3408   0.3628  -0.3267   0.6272
  -0.4316  -0.2352   0.5553  -0.2811  -0.2198  -0.2880  -0.4564  -0.1773
   0.1992  -0.2176  -0.5198   0.1561  -0.1523   0.1299  -0.7281  -0.2197
   0.0161   0.7390   0.1125  -0.2226  -0.1003   0.3608  -0.1118  -0.4886
  -0.5824   0.0984  -0.3052   0.1996   0.5889  -0.2442   0.0060  -0.3341
  -0.3246   0.4661  -0.1835   0.3523  -0.5153  -0.3034  -0.0865   0.3931
  -0.4559  -0.2961  -0.3790  -0.3127  -0.4356   0.3452   0.3642  -0.1467
The matrix Q2 is 
   0.0288  -0.1842  -0.6791  -0.2115  -0.4790   0.4212  -0.0417  -0.2253
  -0.0666  -0.0787  -0.3711   0.1737  -0.0482  -0.5770  -0.6785   0.1607
   0.1506   0.6328   0.0518  -0.6266   0.0652  -0.0790  -0.2854  -0.2994
  -0.2900  -0.2737  -0.0076  -0.3671  -0.2017  -0.6241   0.4521  -0.2675
   0.3353   0.4107   0.0326   0.1400  -0.6447  -0.2043   0.2561   0.4187
   0.0905  -0.1648  -0.2363  -0.5323   0.3180   0.0286   0.1252   0.7126
  -0.7246   0.0468   0.3328  -0.1794  -0.3639   0.2257  -0.2623   0.2786
   0.4922  -0.5353   0.4803  -0.2501  -0.2723   0.0199  -0.3194  -0.0371
The upper left block of the matrix U1 is 
   0.4144   0.2249   0.6015  -0.1964
  -0.0198   0.5131  -0.2823  -0.3058
  -0.6620   0.1508   0.2237   0.0240
  -0.0743  -0.4323  -0.0332  -0.7263
The upper right block of the matrix U1 is 
  -0.3474   0.1306  -0.3391  -0.3530
  -0.3760   0.1550   0.6087  -0.1646
   0.1707   0.6553  -0.1262  -0.1177
   0.3048  -0.0773   0.0767  -0.4173
The upper left block of the matrix U2 is 
   0.1403  -0.6447  -0.6536  -0.3707
   0.7069   0.2609  -0.0091  -0.1702
  -0.1218  -0.1120   0.3766  -0.5154
   0.0773   0.6349  -0.5070  -0.1810
The upper right block of the matrix U2 is 
   0.0000   0.0000   0.0000   0.0000
   0.1182   0.1587   0.1930  -0.5716
   0.6051  -0.2720   0.3364   0.1089
   0.2823  -0.0386  -0.1529   0.4434
The vector ALPHAR is 
   0.0000   0.7122   0.0000   0.7450
The vector ALPHAI is 
   0.7540   0.0000   0.7465   0.0000
The vector BETA is 
   4.0000   4.0000   8.0000  16.0000

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