## MB03LD

### Eigenvalues and right deflating subspace of a real skew-Hamiltonian/Hamiltonian pencil

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

Purpose

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

            (  A  D  )         (  B  F  )
        S = (        ) and H = (        ),                           (1)
            (  E  A' )         (  G -B' )

      where the notation M' denotes the transpose of the matrix M.
      Optionally, if COMPQ = 'C', an orthogonal basis of the right
      deflating subspace of aS - bH corresponding to the eigenvalues
      with strictly negative real part is computed.

Specification

          SUBROUTINE MB03LD( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
         $                   LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
         $                   IWORK, LIWORK, DWORK, LDWORK, BWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ, ORTH
          INTEGER            INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
         $                   LIWORK, N, NEIG
    C     .. Array Arguments ..
          LOGICAL            BWORK( * )
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
         $                   B( LDB, * ), BETA( * ), DE( LDDE, * ),
         $                   DWORK( * ), FG( LDFG, * ), Q( LDQ, * )

Arguments

Mode Parameters

      COMPQ   CHARACTER*1
              Specifies whether to compute the right deflating subspace
              corresponding to the eigenvalues of aS - bH with strictly
              negative real part.
              = 'N':  do not compute the deflating subspace;
              = 'C':  compute the deflating subspace and store it in the
                      leading subarray of Q.

      ORTH    CHARACTER*1
              If COMPQ = 'C', specifies the technique for computing an
              orthogonal basis of the deflating subspace, as follows:
              = 'P':  QR factorization with column pivoting;
              = 'S':  singular value decomposition.
              If COMPQ = 'N', the ORTH value is not used.

Input/Output Parameters

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

      A       (input/output) DOUBLE PRECISION array, dimension
                             (LDA, N/2)
              On entry, the leading N/2-by-N/2 part of this array must
              contain the matrix A.
              On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
              this array contains the upper triangular matrix Aout
              (see METHOD); otherwise, it contains the upper triangular
              matrix A obtained just before the application of the
              periodic QZ algorithm (see SLICOT Library routine MB04BD).

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= MAX(1, N/2).

      DE      (input/output) DOUBLE PRECISION array, dimension
                             (LDDE, N/2+1)
              On entry, the leading N/2-by-N/2 lower triangular part of
              this array must contain the lower triangular part of the
              skew-symmetric matrix E, and the N/2-by-N/2 upper
              triangular part of the submatrix in the columns 2 to N/2+1
              of this array must contain the upper triangular part of the
              skew-symmetric matrix D.
              The entries on the diagonal and the first superdiagonal of
              this array need not be set, but are assumed to be zero.
              On exit, if COMPQ = 'C', the leading N/2-by-N/2 lower
              triangular part and the first superdiagonal contain the
              transpose of the upper quasi-triangular matrix C2out (see
              METHOD), and the (N/2-1)-by-(N/2-1) upper triangular part
              of the submatrix in the columns 3 to N/2+1 of this array
              contains the strictly upper triangular part of the
              skew-symmetric matrix Dout (see METHOD), without the main
              diagonal, which is zero.
              On exit, if COMPQ = 'N', the leading N/2-by-N/2 lower
              triangular part and the first superdiagonal contain the
              transpose of the upper Hessenberg matrix C2, and the
              (N/2-1)-by-(N/2-1) upper triangular part of the submatrix
              in the columns 3 to N/2+1 of this array contains the
              strictly upper triangular part of the skew-symmetric
              matrix D (without the main diagonal) just before the
              application of the periodic QZ algorithm.

      LDDE    INTEGER
              The leading dimension of the array DE.
              LDDE >= MAX(1, N/2).

      B       (input/output) DOUBLE PRECISION array, dimension
                             (LDB, N/2)
              On entry, the leading N/2-by-N/2 part of this array must
              contain the matrix B.
              On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
              this array contains the upper triangular matrix C1out
              (see METHOD); otherwise, it contains the upper triangular
              matrix C1 obtained just before the application of the
              periodic QZ algorithm.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= MAX(1, N/2).

      FG      (input/output) DOUBLE PRECISION array, dimension
                             (LDFG, N/2+1)
              On entry, the leading N/2-by-N/2 lower triangular part of
              this array must contain the lower triangular part of the
              symmetric matrix G, and the N/2-by-N/2 upper triangular
              part of the submatrix in the columns 2 to N/2+1 of this
              array must contain the upper triangular part of the
              symmetric matrix F.
              On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
              the submatrix in the columns 2 to N/2+1 of this array
              contains the matrix Vout (see METHOD); otherwise, it
              contains the matrix V obtained just before the application
              of the periodic QZ algorithm.

      LDFG    INTEGER
              The leading dimension of the array FG.
              LDFG >= MAX(1, N/2).

      NEIG    (output) INTEGER
              If COMPQ = 'C', the number of eigenvalues in aS - bH with
              strictly negative real part.

      Q       (output) DOUBLE PRECISION array, dimension (LDQ, 2*N)
              On exit, if COMPQ = 'C', the leading N-by-NEIG part of
              this array contains an orthogonal basis of the right
              deflating subspace corresponding to the eigenvalues of
              aA - bB with strictly negative real part. The remaining
              part of this array is used as workspace.
              If COMPQ = 'N', this array is not referenced.

      LDQ     INTEGER
              The leading dimension of the array Q.
              LDQ >= 1,           if COMPQ = 'N';
              LDQ >= MAX(1, 2*N), if COMPQ = '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 that define the eigenvalues of the pencil
              aS - bH.
              Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
              beta = BETA(j) represent 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, for every eigenvalue lambda, -lambda is also an
              eigenvalue, and thus it has only to be saved once in
              ALPHAR, ALPHAI and BETA.
              Specifically, only eigenvalues with imaginary parts
              greater than or equal to zero are stored; their conjugate
              eigenvalues are not stored. If imaginary parts are zero
              (i.e., for real eigenvalues), only positive eigenvalues
              are stored. The remaining eigenvalues have opposite signs.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)
              On exit, if INFO = -19, IWORK(1) returns the minimum value
              of LIWORK.

      LIWORK  INTEGER
              The dimension of the array IWORK.  LIWORK = 1, if N = 0,
              LIWORK >= MAX( N + 12, 2*N + 3 ),     if COMPQ = 'N',
              LIWORK >= MAX( 32, 2*N + 3 ),         if COMPQ = 'C'.

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
              On exit, if INFO = -21, DWORK(1) returns the minimum value
              of LDWORK.

      LDWORK  INTEGER
              The dimension of the array DWORK.  LDWORK = 1, if N = 0,
              LDWORK >= 3*(N/2)**2 + N**2 + MAX( L, 36 ),
                                                         if COMPQ = 'N',
              where     L = 4*N + 4, if N/2 is even, and
                        L = 4*N    , if N/2 is odd;
              LDWORK >= 8*N**2 + MAX( 8*N + 32, 272 ),   if COMPQ = 'C'.
              For good performance LDWORK should be generally larger.

              If LDWORK = -1  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 is issued by XERBLA.

      BWORK   LOGICAL array, dimension (N/2)

Error Indicator

      INFO    INTEGER
              = 0: succesful exit;
              < 0: if INFO = -i, the i-th argument had an illegal value;
              = 1: periodic QZ iteration failed in the SLICOT Library
                   routines MB04BD or MB04HD (QZ iteration did not
                   converge or computation of the shifts failed);
              = 2: standard QZ iteration failed in the SLICOT Library
                   routines MB04HD or MB03DD (called by MB03JD);
              = 3: a numerically singular matrix was found in the SLICOT
                   Library routine MB03HD (called by MB03JD);
              = 4: the singular value decomposition failed in the LAPACK
                   routine DGESVD (for ORTH = 'S');
              = 5: some eigenvalues might be inaccurate. This is a
                   warning.

Method

      First, the decompositions of S and H are computed via orthogonal
      transformations Q1 and Q2 as follows:

                        (  Aout  Dout  )
        Q1' S J Q1 J' = (              ),
                        (   0    Aout' )

                        (  Bout  Fout  )
        J' Q2' J S Q2 = (              ) =: T,                       (2)
                        (   0    Bout' )

                   (  C1out  Vout  )            (  0  I  )
        Q1' H Q2 = (               ), where J = (        ),
                   (  0     C2out' )            ( -I  0  )

      and Aout, Bout, C1out are upper triangular, C2out is upper quasi-
      triangular and Dout and Fout are skew-symmetric.

      Then, orthogonal matrices Q3 and Q4 are found, for the extended
      matrices

             (  Aout   0  )          (    0   C1out )
        Se = (            ) and He = (              ),
             (   0   Bout )          ( -C2out   0   )

      such that S11 := Q4' Se Q3 is upper triangular and
      H11 := Q4' He Q3 is upper quasi-triangular. The following matrices
      are computed:

                   (  Dout   0  )                   (   0   Vout )
        S12 := Q4' (            ) Q4 and H12 := Q4' (            ) Q4.
                   (   0   Fout )                   ( Vout'   0  )

      Then, an orthogonal matrix Q is found such that the eigenvalues
      with strictly negative real parts of the pencil

          (  S11  S12  )     (  H11  H12  )
        a (            ) - b (            )
          (   0   S11' )     (   0  -H11' )

      are moved to the top of this pencil.

      Finally, an orthogonal basis of the right deflating subspace
      corresponding to the eigenvalues with strictly negative real part
      is computed. See also page 12 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 )
      floating point operations.

Further Comments

      This routine does not perform any scaling of the matrices. Scaling
      might sometimes be useful, and it should be done externally.

Example

Program Text

    *     MB03LD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER            NIN, NOUT
          PARAMETER          ( NIN = 5, NOUT = 6 )
          INTEGER            NMAX
          PARAMETER          ( NMAX = 50 )
          INTEGER            LDA, LDB, LDDE, LDFG, LDQ, LDWORK, LIWORK
          PARAMETER          (  LDA = NMAX/2, LDB = NMAX/2, LDDE = NMAX/2,
         $                     LDFG = NMAX/2, LDQ = 2*NMAX,
         $                     LDWORK = 8*NMAX*NMAX +
         $                              MAX( 8*NMAX + 32, NMAX/2 + 168,
         $                                   272 ),
         $                     LIWORK = MAX( 32, NMAX + 12, NMAX*2 + 3 ) )
    *
    *     .. Local Scalars ..
          CHARACTER          COMPQ, ORTH
          INTEGER            I, INFO, J, M, N, NEIG
    *
    *     .. Local Arrays ..
          LOGICAL            BWORK( NMAX/2 )
          INTEGER            IWORK( LIWORK )
          DOUBLE PRECISION   A( LDA, NMAX/2 ),  ALPHAI( NMAX/2 ),
         $                   ALPHAR( NMAX/2 ),  B( LDB, NMAX/2 ),
         $                   BETA( NMAX/2 ),  DE( LDDE, NMAX/2+1 ),
         $                   DWORK( LDWORK ), FG( LDFG, NMAX/2+1 ),
         $                   Q( LDQ, 2*NMAX )
    *
    *     .. External Subroutines ..
          EXTERNAL           MB03LD
    *
    *     .. 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 = * ) COMPQ, ORTH, N
          IF( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE( NOUT, FMT = 99998 ) N
          ELSE
             M = N/2
             READ( NIN, FMT = * ) ( (  A( I, J ), J = 1, M   ), I = 1, M )
             READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, M+1 ), I = 1, M )
             READ( NIN, FMT = * ) ( (  B( I, J ), J = 1, M   ), I = 1, M )
             READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, M+1 ), I = 1, M )
    *        Compute the eigenvalues and an orthogonal basis of the right
    *        deflating subspace of a real skew-Hamiltonian/Hamiltonian
    *        pencil, corresponding to the eigenvalues with strictly negative
    *        real part.
             CALL MB03LD( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
         $                LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
         $                LIWORK, DWORK, LDWORK, BWORK, INFO )
    *
             IF( INFO.NE.0 ) THEN
                WRITE( NOUT, FMT = 99997 ) INFO
             ELSE
                WRITE( NOUT, FMT = 99996 )
                DO 10 I = 1, M
                   WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, M )
       10       CONTINUE
                WRITE( NOUT, FMT = 99994 )
                DO 20 I = 1, M
                   WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, M+1 )
       20       CONTINUE
                WRITE( NOUT, FMT = 99993 )
                DO 30 I = 1, M
                   WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, M )
       30       CONTINUE
                WRITE( NOUT, FMT = 99992 )
                DO 40 I = 1, M
                   WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 2, M+1 )
       40       CONTINUE
                WRITE( NOUT, FMT = 99991 )
                WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
                WRITE( NOUT, FMT = 99990 )
                WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
                WRITE( NOUT, FMT = 99989 )
                WRITE( NOUT, FMT = 99995 ) (   BETA( I ), I = 1, M )
                IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
                   WRITE( NOUT, FMT = 99988 )
                   DO 50 I = 1, N
                      WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
       50          CONTINUE
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT( 'MB03LD EXAMPLE PROGRAM RESULTS', 1X )
    99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
    99997 FORMAT( 'INFO on exit from MB03LD = ', I2 )
    99996 FORMAT( 'The matrix A on exit is ' )
    99995 FORMAT( 50( 1X, F8.4 ) )
    99994 FORMAT( 'The matrix DE on exit is ' )
    99993 FORMAT( 'The matrix C1 on exit is ' )
    99992 FORMAT( 'The matrix V on exit is ' )
    99991 FORMAT( 'The vector ALPHAR is ' )
    99990 FORMAT( 'The vector ALPHAI is ' )
    99989 FORMAT( 'The vector BETA is ' )
    99988 FORMAT( 'The matrix Q is ' )
          END

Program Data

    MB03LD EXAMPLE PROGRAM DATA
       C   P   8
       3.1472   1.3236   4.5751   4.5717
       4.0579  -4.0246   4.6489  -0.1462
      -3.7301  -2.2150  -3.4239   3.0028
       4.1338   0.4688   4.7059  -3.5811
       0.0000   0.0000  -1.5510  -4.5974  -2.5127
       3.5071   0.0000   0.0000   1.5961   2.4490
      -3.1428   2.5648   0.0000   0.0000  -0.0596
       3.0340   2.4892  -1.1604   0.0000   0.0000
       0.6882  -3.3782  -3.3435   1.8921
      -0.3061   2.9428   1.0198   2.4815
      -4.8810  -1.8878  -2.3703  -0.4946
      -1.6288   0.2853   1.5408  -4.1618
      -2.4013  -2.7102   0.3834  -3.9335   3.1730
      -3.1815  -2.3620   4.9613   4.6190   3.6869
       3.6929   0.7970   0.4986  -4.9537  -4.1556
       3.5303   1.2206  -1.4905   0.1325  -1.0022

Program Results

    MB03LD EXAMPLE PROGRAM RESULTS
    The matrix A on exit is
      -4.7460   4.1855   3.2696  -0.2244
       0.0000   6.4157   2.8287   1.4553
       0.0000   0.0000   7.4626   1.5726
       0.0000   0.0000   0.0000   8.8702
    The matrix DE on exit is
      -5.4562   2.5550  -1.3137  -6.3615  -0.8940
      -2.1348  -7.9616   0.0000   1.0704  -0.0659
       4.9694   1.1516   4.8504   0.0000  -0.6922
      -2.2744   3.4912   0.5046   4.4394   0.0000
    The matrix C1 on exit is
       6.9525  -4.9881   2.3661   4.2188
       0.0000   8.5009   0.7182   5.5533
       0.0000   0.0000  -4.6650  -2.8177
       0.0000   0.0000   0.0000   1.5124
    The matrix V on exit is
       0.9136   4.1106  -0.0079   3.5789
      -1.1553  -1.4785  -1.5155  -0.8018
      -2.2167   4.8029   1.3645   2.5202
      -1.0994  -0.6144   0.3970   2.0730
    The vector ALPHAR is
       0.8314  -0.8314   0.8131   0.0000
    The vector ALPHAI is
       0.4372   0.4372   0.0000   0.9164
    The vector BETA is
       0.7071   0.7071   1.4142   2.8284
    The matrix Q is
      -0.5844  -0.2949   0.1692
       0.5470  -0.2324  -0.4524
       0.0382   0.4673  -0.3092
      -0.0378   0.0904  -0.4451
      -0.0255   0.1497  -0.1929
      -0.1286  -0.6067  -0.2275
      -0.2260   0.4901   0.0951
       0.5367  -0.0430   0.6123

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