## MB03LF

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

[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

                                    (  B  F  )      (  0  I  )
        S = T Z = J Z' J' Z and H = (        ), J = (        ),      (1)
                                    (  G -B' )      ( -I  0  )

      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. Optionally, if
      COMPU = 'C', an orthonormal basis of the companion subspace,
      range(P_U) [1], which corresponds to the eigenvalues with strictly
      negative real part, is computed.

Specification

          SUBROUTINE MB03LF( COMPQ, COMPU, ORTH, N, Z, LDZ, B, LDB, FG,
         $                   LDFG, NEIG, Q, LDQ, U, LDU, ALPHAR, ALPHAI,
         $                   BETA, IWORK, LIWORK, DWORK, LDWORK, BWORK,
         $                   IWARN, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ, COMPU, ORTH
          INTEGER            INFO, IWARN, LDB, LDFG, LDQ, LDU, LDWORK, LDZ,
         $                   LIWORK, N, NEIG
    C     .. Array Arguments ..
          LOGICAL            BWORK( * )
          INTEGER            IWORK( * )
          DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
         $                   BETA( * ), DWORK( * ), FG( LDFG, * ),
         $                   Q( LDQ, * ), U( LDU, * ), Z( LDZ, * )

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.

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

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

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 non-trivial factor Z in the factorization
              S = J Z' J' Z of the skew-Hamiltonian matrix S.
              On exit, if COMPQ = 'C' or COMPU = 'C', the leading
              N-by-N part of this array contains the transformed upper
                                ~
              triangular matrix Z11 (see METHOD), after moving the
              eigenvalues with strictly negative real part to the top
              of the pencil (3). The strictly lower triangular part is
              not zeroed.
              If COMPQ = 'N' and COMPU = 'N', the leading N-by-N part of
              this array contains the matrix Z obtained by the SLICOT
              Library routine MB04AD 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).

      B       (input) 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.

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

      FG      (input) 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.

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

      NEIG    (output) INTEGER
              If COMPQ = 'C' or COMPU = '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
              aS - bH 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'.

      U       (output) DOUBLE PRECISION array, dimension (LDU, 2*N)
              On exit, if COMPU = 'C', the leading N-by-NEIG part of
              this array contains an orthogonal basis of the companion
              subspace corresponding to the eigenvalues of aS - bH with
              strictly negative real part. The remaining part of this
              array is used as workspace.
              If COMPU = 'N', this array is not referenced.

      LDU     INTEGER
              The leading dimension of the array U.
              LDU >= 1,         if COMPU = 'N';
              LDU >= MAX(1, N), if COMPU = '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.
              As a consequence, pairs of complex eigenvalues, stored in
              consecutive locations, are not complex conjugate.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)
              On exit, if INFO = -20, IWORK(1) returns the minimum value
              of LIWORK.

      LIWORK  INTEGER
              The dimension of the array IWORK.
              LIWORK >= N + 18,      if COMPQ = 'N' and COMPU = 'N';
              LIWORK >= MAX( 2*N+1, 48 ), otherwise.

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.
              On exit, if INFO = -22, DWORK(1) returns the minimum value
              of LDWORK.

      LDWORK  INTEGER
              The dimension of the array DWORK.
              LDWORK >= b*N*N + 3*N*N/2 + MAX( 6*N, 54 ),
                                        if COMPQ = 'N' and COMPU = 'N';
              LDWORK >= d*N*N + MAX( N/2+252, 432 ), otherwise, where
                        b = a,   d = c,   if COMPU = 'N',
                        b = a+1, d = c+1, if COMPU = 'C', and
                        a = 2,   c = 7,   if COMPQ = 'N',
                        a = 4,   c = 10,  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)

Warning Indicator

      IWARN   INTEGER
              = 0: no warning;
              = 1: some eigenvalues might be unreliable. More details
                   can be obtained by running the SLICOT routine MB04AD.

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 MB04AD, MB04CD or MB03BB (QZ iteration did
                   not converge or computation of the shifts failed);
              = 2: standard QZ iteration failed in the SLICOT Library
                   routines MB04CD or MB03CD (called by MB03ID);
              = 3: a numerically singular matrix was found in the SLICOT
                   Library routine MB03GD (called by MB03ID);
              = 4: the singular value decomposition failed in the LAPACK
                   routine DGESVD (for ORTH = 'S').

Method

      First, the decompositions of S and H are computed via orthogonal
      matrices Q1 and Q2 and orthogonal symplectic matrices U1 and U2,
      such that

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

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

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

      where T11, T22', Z11, Z22', H11 are upper triangular and H22' is
      upper quasi-triangular.

      Then, orthogonal matrices Q3, Q4 and U3 are found, for the
      matrices

        ~     ( T22'  0  )  ~     ( T11'  0  )  ~   (   0   H11 )
        Z11 = (          ), Z22 = (          ), H = (           ),
              (  0   Z11 )        (  0   Z22 )      ( -H22'  0  )

                ~          ~       ~          ~
      such that Z11 := U3' Z11 Q4, Z22 := U3' Z22 Q3 are upper
                     ~          ~
      triangular and H11 := Q3' H Q4 is upper quasi-triangular. The
      following matrices are computed:

        ~          ( -T12'  0  )        ~          (  0   H12 )
        Z12 := U3' (           ) Q3 and H12 := Q3' (          ) Q3.
                   (  0    Z12 )                   ( H12'  0  )

      Then, an orthogonal matrix Q and an orthogonal symplectic matrix U
      are found such that the eigenvalues with strictly negative real
      parts of the pencil

              ~    ~          ~    ~           ~    ~
            ( Z11  Z12 )'   ( Z11  Z12 )     ( H11  H12  )
        a J (      ~   ) J' (      ~   ) - b (      ~    )           (3)
            (  0   Z22 )    (  0   Z22 )     (  0  -H11' )

      are moved to the top of this pencil.

      Finally, an orthogonal basis of the right deflating subspace
      and an orthogonal basis of the companion subspace corresponding to
      the eigenvalues with strictly negative real part are computed.
      See also page 11 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

    *     MB03LF EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER            NIN, NOUT
          PARAMETER          ( NIN = 5, NOUT = 6 )
          INTEGER            NMAX
          PARAMETER          ( NMAX = 50 )
          INTEGER            LDB, LDFG, LDQ, LDU, LDWORK, LDZ, LIWORK
          PARAMETER          ( LDB = NMAX/2, LDFG = NMAX/2, LDQ = 2*NMAX,
         $                     LDU = NMAX,   LDZ  = NMAX,
         $                     LDWORK = 10*NMAX*NMAX +
         $                              MAX( NMAX*NMAX +
         $                                   MAX( NMAX/2 + 252, 432 ),
         $                                   MAX( 8*NMAX +  48, 171 ) ),
         $                     LIWORK = MAX( NMAX + 18, NMAX/2 + 48,
         $                                   5*NMAX/2 + 1 ) )
    *
    *     .. Local Scalars ..
          CHARACTER          COMPQ, COMPU, ORTH
          INTEGER            I, INFO, IWARN, J, M, N, NEIG
    *
    *     .. Local Arrays ..
          LOGICAL            BWORK( NMAX/2 )
          INTEGER            IWORK( LIWORK )
          DOUBLE PRECISION   ALPHAI( NMAX/2 ),  ALPHAR( NMAX/2 ),
         $                   B( LDB, NMAX/2 ),    BETA( NMAX/2 ),
         $                   DWORK( LDWORK ), FG( LDFG, NMAX/2+1 ),
         $                   Q( LDQ, 2*NMAX ),  U( LDU, 2*NMAX ),
         $                   Z( LDZ, NMAX )
    *
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *
    *     .. External Subroutines ..
          EXTERNAL           MB03LF
    *
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MOD
    *
    *     .. 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, COMPU, ORTH, N
          IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
             WRITE( NOUT, FMT = 99998 ) N
          ELSE
             M = N/2
             READ( NIN, FMT = * ) ( (  Z( I, J ), J = 1, N   ), I = 1, N )
             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 orthogonal bases of the right
    *        deflating subspace and companion subspace of a real
    *        skew-Hamiltonian/Hamiltonian pencil, corresponding to the
    *        eigenvalues with strictly negative real part.
             CALL MB03LF( COMPQ, COMPU, ORTH, N, Z, LDZ, B, LDB, FG, LDFG,
         $                NEIG, Q, LDQ, U, LDU, ALPHAR, ALPHAI, BETA, IWORK,
         $                LIWORK, DWORK, LDWORK, BWORK, IWARN, INFO )
    *
             IF( INFO.NE.0 ) THEN
                WRITE( NOUT, FMT = 99997 ) INFO
             ELSE
                WRITE( NOUT, FMT = 99996 )
                DO 10 I = 1, N
                   WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N )
       10       CONTINUE
                WRITE( NOUT, FMT = 99994 )
                WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
                WRITE( NOUT, FMT = 99993 )
                WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
                WRITE( NOUT, FMT = 99992 )
                WRITE( NOUT, FMT = 99995 ) (   BETA( I ), I = 1, M )
                IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
                   WRITE( NOUT, FMT = 99991 )
                   DO 20 I = 1, N
                      WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
       20          CONTINUE
                END IF
                IF( LSAME( COMPU, 'C' ) .AND. NEIG.GT.0 ) THEN
                   WRITE( NOUT, FMT = 99990 )
                   DO 30 I = 1, N
                      WRITE( NOUT, FMT = 99995 ) ( U( I, J ), J = 1, NEIG )
       30          CONTINUE
                END IF
                IF( LSAME( COMPQ, 'C' ) .OR. LSAME( COMPU, 'C' ) )
         $         WRITE( NOUT, FMT = 99989 ) NEIG
             END IF
          END IF
          STOP
    *
    99999 FORMAT ( 'MB03LF EXAMPLE PROGRAM RESULTS', 1X )
    99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
    99997 FORMAT ( 'INFO on exit from MB03LF = ', I2 )
    99996 FORMAT (/'The matrix Z on exit is ' )
    99995 FORMAT ( 50( 1X, F8.4 ) )
    99994 FORMAT (/'The vector ALPHAR is ' )
    99993 FORMAT (/'The vector ALPHAI is ' )
    99992 FORMAT (/'The vector BETA is ' )
    99991 FORMAT (/'The deflating subspace corresponding to the ',
         $         'eigenvalues with negative real part is ' )
    99990 FORMAT (/'The companion subspace corresponding to the ',
         $         'eigenvalues with negative real part is ' )
    99989 FORMAT (/'The number of eigenvalues in the initial pencil with ',
         $         'negative real part is ', I2 )
          END

Program Data

    MB03LF EXAMPLE PROGRAM DATA
       C   C   P   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
       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

    MB03LF EXAMPLE PROGRAM RESULTS

    The matrix Z on exit is
       4.4128   0.1059  -1.8709   1.2963  -4.3448   2.7633   2.3580   2.1931
       0.0000  10.0337  -1.9797   1.8052  -1.0112   1.1335   1.2374   0.3107
       0.0000   0.0000   8.9476   1.8523  -1.8578  -0.5807  -1.4157   1.3007
       0.0000   0.0000   0.0000  -7.0889  -2.1193  -2.1634  -2.4393   0.1148
       0.0765   1.0139   0.0000  -1.5390  -8.3187  -5.0172   0.7738  -2.8626
       1.1884  -0.9225   0.0000   0.2905   0.0000   6.4090   2.1994  -2.5933
      -0.5931   0.1981   0.0000  -0.5280   0.0000   0.0000   4.7155   2.3817
       1.8591  -1.8416   0.0000  -0.0807   0.0000   0.0000   0.0000  -5.3153

    The vector ALPHAR is
       0.7353   0.0000   0.5168  -0.5168

    The vector ALPHAI is
       0.0000   0.7190   0.5610   0.5610

    The vector BETA is
       2.0000   2.8284  11.3137  11.3137

    The deflating subspace corresponding to the eigenvalues with negative real part is
      -0.2509   0.3670   0.0416
      -0.3267  -0.7968  -0.1019
       0.0263   0.0338  -0.5795
      -0.0139  -0.0491  -0.5217
      -0.4637   0.2992  -0.4403
      -0.1345   0.3071  -0.0917
      -0.1364   0.2013   0.3447
      -0.7601  -0.0495   0.2426

    The companion subspace corresponding to the eigenvalues with negative real part is
      -0.3219   0.6590   0.1693
      -0.5216  -0.1829  -0.0689
      -0.0413  -0.4664  -0.1359
       0.1310  -0.1702   0.4543
      -0.3598   0.2660   0.3355
      -0.5082  -0.0512  -0.6035
      -0.3582  -0.4513   0.4649
       0.2991   0.0932  -0.2207

    The number of eigenvalues in the initial pencil with negative real part is  3

Return to index