## MB03LZ

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

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

Purpose

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

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

      The structured Schur form of the embedded real skew-Hamiltonian/
      skew-Hamiltonian pencil aB_S - bB_T, defined as

              (  Re(A)  -Im(A)  |  Re(D)  -Im(D)  )
              (                 |                 )
              (  Im(A)   Re(A)  |  Im(D)   Re(D)  )
              (                 |                 )
        B_S = (-----------------+-----------------) , and
              (                 |                 )
              (  Re(E)  -Im(E)  |  Re(A')  Im(A') )
              (                 |                 )
              (  Im(E)   Re(E)  | -Im(A')  Re(A') )
                                                                     (2)
              ( -Im(B)  -Re(B)  | -Im(F)  -Re(F)  )
              (                 |                 )
              (  Re(B)  -Im(B)  |  Re(F)  -Im(F)  )
              (                 |                 )
        B_T = (-----------------+-----------------) ,  T = i*H,
              (                 |                 )
              ( -Im(G)  -Re(G)  | -Im(B')  Re(B') )
              (                 |                 )
              (  Re(G)  -Im(G)  | -Re(B') -Im(B') )

      is determined and used to compute the eigenvalues. The notation M'
      denotes the conjugate transpose of the matrix M. Optionally,
      if COMPQ = 'C', an orthonormal basis of the right deflating
      subspace of the pencil aS - bH, corresponding to the eigenvalues
      with strictly negative real part, is computed. Namely, after
      transforming aB_S - bB_H by unitary matrices, we have

                 ( BA  BD  )              ( BB  BF  )
        B_Sout = (         ) and B_Hout = (         ),               (3)
                 (  0  BA' )              (  0 -BB' )

      and the eigenvalues with strictly negative real part of the
      complex pencil aB_Sout - bB_Hout are moved to the top. The
      embedding doubles the multiplicities of the eigenvalues of the
      pencil aS - bH.

Specification

          SUBROUTINE MB03LZ( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
         $                   LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
         $                   IWORK, DWORK, LDWORK, ZWORK, LZWORK, BWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ, ORTH
          INTEGER            INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
         $                   LZWORK, N, NEIG
    C     .. Array Arguments ..
          LOGICAL            BWORK( * )
          INTEGER            IWORK( * )
          DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
          COMPLEX*16         A( LDA, * ), B( LDB, * ), DE( LDDE, * ),
         $                   FG( LDFG, * ), Q( LDQ, * ), ZWORK( * )

Arguments

Mode Parameters

      COMPQ   CHARACTER*1
              Specifies whether to compute the deflating subspace
              corresponding to the eigenvalues of aS - bH with strictly
              negative real part.
              = 'N': do not compute the deflating subspace; compute the
                     eigenvalues only;
              = '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
              orthonormal 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) COMPLEX*16 array, dimension (LDA, N)
              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-by-N part of this
              array contains the upper triangular matrix BA in (3) (see
              also METHOD). The strictly lower triangular part is not
              zeroed; it is preserved in the leading N/2-by-N/2 part.
              If COMPQ = 'N', this array is unchanged on exit.

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

      DE      (input/output) COMPLEX*16 array, dimension (LDDE, N)
              On entry, the leading N/2-by-N/2 lower triangular part of
              this array must contain the lower triangular part of the
              skew-Hermitian 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-Hermitian matrix D.
              On exit, if COMPQ = 'C', the leading N-by-N part of this
              array contains the skew-Hermitian matrix BD in (3) (see
              also METHOD). The strictly lower triangular part of the
              input matrix is preserved.
              If COMPQ = 'N', this array is unchanged on exit.

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

      B       (input/output) COMPLEX*16 array, dimension (LDB, N)
              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-by-N part of this
              array contains the upper triangular matrix BB in (3) (see
              also METHOD). The strictly lower triangular part is not
              zeroed; the elements below the first subdiagonal of the
              input matrix are preserved.
              If COMPQ = 'N', this array is unchanged on exit.

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

      FG      (input/output) COMPLEX*16 array, dimension (LDFG, N)
              On entry, the leading N/2-by-N/2 lower triangular part of
              this array must contain the lower triangular part of the
              Hermitian 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
              Hermitian matrix F.
              On exit, if COMPQ = 'C', the leading N-by-N part of this
              array contains the Hermitian matrix BF in (3) (see also
              METHOD). The strictly lower triangular part of the input
              matrix is preserved. The diagonal elements might have tiny
              imaginary parts.
              If COMPQ = 'N', this array is unchanged on exit.

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

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

      Q       (output) COMPLEX*16 array, dimension (LDQ, 2*N)
              On exit, if COMPQ = 'C', the leading N-by-NEIG part of
              this array contains an orthonormal basis of the right
              deflating subspace corresponding to the eigenvalues of the
              pencil aS - bH with strictly negative real part.
              The remaining entries are meaningless.
              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)
              The real parts of each scalar alpha defining an eigenvalue
              of the pencil aS - bH.

      ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
              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)
              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.

Workspace

      IWORK   INTEGER array, dimension (N+1)

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
              On exit, if INFO = -20, DWORK(1) returns the minimum value
              of LDWORK.

      LDWORK  INTEGER
              The dimension of the array DWORK.
              LDWORK >= MAX( 4*N*N + 2*N + MAX(3,N) ), if COMPQ = 'N';
              LDWORK >= MAX( 1, 11*N*N + 2*N ),        if COMPQ = 'C'.
              For good performance LDWORK should be generally 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.

      ZWORK   COMPLEX*16 array, dimension (LZWORK)
              On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
              On exit, if INFO = -22, ZWORK(1) returns the minimum value
              of LZWORK.

      LZWORK  INTEGER
              The dimension of the array ZWORK.
              LZWORK >= 1,       if COMPQ = 'N';
              LZWORK >= 8*N + 4, if COMPQ = 'C'.
              For good performance LZWORK should be generally larger.

              If LZWORK = -1, then a workspace query is assumed;
              the routine only calculates the optimal size of the
              ZWORK array, returns this value as the first entry of
              the ZWORK array, and no error message related to LZWORK
              is issued by XERBLA.

      BWORK   LOGICAL array, dimension (LBWORK)
              LBWORK >= 0,     if COMPQ = 'N';
              LBWORK >= N - 1, if COMPQ = 'C'.

Error Indicator

      INFO    INTEGER
              = 0: succesful exit;
              < 0: if INFO = -i, the i-th argument had an illegal value;
              = 1: QZ iteration failed in the SLICOT Library routine
                   MB04FD (QZ iteration did not converge or computation
                   of the shifts failed);
              = 2: QZ iteration failed in the LAPACK routine ZHGEQZ when
                   trying to triangularize the 2-by-2 blocks;
              = 3: the singular value decomposition failed in the LAPACK
                   routine ZGESVD (for ORTH = 'S');
              = 4: warning: the pencil is numerically singular.

Method

      First, T = i*H is set. Then, the embeddings, B_S and B_T, of the
      matrices S and T, are determined and, subsequently, the SLICOT
      Library routine MB04FD is applied to compute the structured Schur
      form, i.e., the factorizations

      ~                     (  S11  S12  )
      B_S = J Q' J' B_S Q = (            ) and
                            (   0   S11' )

      ~                     (  T11  T12  )           (  0  I  )
      B_T = J Q' J' B_T Q = (            ), with J = (        ),
                            (   0   T11' )           ( -I  0  )

      where Q is real orthogonal, S11 is upper triangular, and T11 is
      upper quasi-triangular.

      Second, the SLICOT Library routine MB03JZ is applied, to compute a
                     ~
      unitary matrix Q, such that

                         ~    ~
        ~     ~   ~   (  S11  S12  )
      J Q' J' B_S Q = (       ~    ) =: B_Sout,
                      (   0   S11' )

        ~        ~    ~   (  H11  H12  )
      J Q' J'(-i*B_T) Q = (            ) =: B_Hout,
                          (   0  -H11' )
           ~                                               ~       ~
      with S11, H11 upper triangular, and such that Spec_-(B_S, -i*B_T)
      is contained in the spectrum of the 2*NEIG-by-2*NEIG leading
                           ~
      principal subpencil aS11 - bH11.

      Finally, the right deflating subspace is computed.
      See also page 22 in [1] for more details.

References

      [1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
          Numerical Computation of Deflating Subspaces of Embedded
          Hamiltonian Pencils.
          Tech. Rep. SFB393/99-15, Technical University Chemnitz,
          Germany, June 1999.

Numerical Aspects

                                                                3
      The algorithm is numerically backward stable and needs O(N )
      complex 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

    *     MB03LZ EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER            NIN, NOUT
          PARAMETER          ( NIN = 5, NOUT = 6 )
          INTEGER            NMAX
          PARAMETER          ( NMAX = 50 )
          INTEGER            LDA, LDB, LDDE, LDFG, LDQ, LDWORK, LZWORK
          PARAMETER          ( LDA = NMAX, LDB = NMAX, LDDE = NMAX,
         $                     LDFG = NMAX, LDQ = 2*NMAX,
         $                     LDWORK = 11*NMAX*NMAX + 2*NMAX,
         $                     LZWORK =  8*NMAX + 4 )
    *
    *     .. Local Scalars ..
          CHARACTER*1        COMPQ, ORTH
          INTEGER            I, INFO, J, N, NEIG
    *
    *     .. Local Arrays ..
          COMPLEX*16         A( LDA, NMAX ), B( LDB, NMAX ),
         $                   DE( LDDE, NMAX ), FG( LDFG, NMAX ),
         $                   Q( LDQ, 2*NMAX ), ZWORK( LZWORK )
          DOUBLE PRECISION   ALPHAI( NMAX ), ALPHAR( NMAX ), BETA( NMAX ),
         $                   DWORK( LDWORK )
          INTEGER            IWORK( NMAX + 1 )
          LOGICAL            BWORK( NMAX )
    *
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *
    *     .. External Subroutines ..
          EXTERNAL           MB03LZ
    *
    *     .. Intrinsic Functions ..
          INTRINSIC          MOD
    *
    *     .. Executable statements ..
    *
          WRITE( NOUT, FMT = 99999 )
    *
    *     Skip first line in data file.
    *
          READ( NIN, FMT = * )
          READ( NIN, FMT = * ) COMPQ, ORTH, N
          READ( NIN, FMT = * ) ( (  A( I, J ), J = 1, N/2 ),   I = 1, N/2 )
          READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, N/2+1 ), I = 1, N/2 )
          READ( NIN, FMT = * ) ( (  B( I, J ), J = 1, N/2 ),   I = 1, N/2 )
          READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, N/2+1 ), I = 1, N/2 )
          IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
             WRITE( NOUT, FMT = 99998 ) N
          ELSE
    *        Compute the eigenvalues and an orthogonal basis of the right
    *        deflating subspace of a complex skew-Hamiltonian/Hamiltonian
    *        pencil, corresponding to the eigenvalues with strictly negative
    *        real part.
             CALL MB03LZ( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
         $                LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
         $                DWORK, LDWORK, ZWORK, LZWORK, BWORK, INFO )
             IF( INFO.NE.0 ) THEN
                WRITE( NOUT, FMT = 99997 ) INFO
             ELSE
                IF( LSAME( COMPQ, 'C' ) ) THEN
                   WRITE( NOUT, FMT = 99996 )
                   DO 10 I = 1, N
                      WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, N )
       10          CONTINUE
                   WRITE( NOUT, FMT = 99994 )
                   DO 20 I = 1, N
                      WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, N )
       20          CONTINUE
                   WRITE( NOUT, FMT = 99993 )
                   DO 30 I = 1, N
                      WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N )
       30          CONTINUE
                   WRITE( NOUT, FMT = 99992 )
                   DO 40 I = 1, N
                      WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N )
       40          CONTINUE
                END IF
                WRITE( NOUT, FMT = 99991 )
                WRITE( NOUT, FMT = 99990 ) ( ALPHAR( I ), I = 1, N )
                WRITE( NOUT, FMT = 99989 )
                WRITE( NOUT, FMT = 99990 ) ( ALPHAI( I ), I = 1, N )
                WRITE( NOUT, FMT = 99988 )
                WRITE( NOUT, FMT = 99990 ) (   BETA( I ), I = 1, N )
                IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
                   WRITE( NOUT, FMT = 99987 )
                   DO 50 I = 1, N
                      WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
       50          CONTINUE
                END IF
                IF( LSAME( COMPQ, 'C' ) )
         $         WRITE( NOUT, FMT = 99986 ) NEIG
             END IF
          END IF
          STOP
    99999 FORMAT ( 'MB03LZ EXAMPLE PROGRAM RESULTS', 1X )
    99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
    99997 FORMAT ( 'INFO on exit from MB03LZ = ', I2 )
    99996 FORMAT (/'The matrix A on exit is ' )
    99995 FORMAT ( 20( 1X, F9.4, SP, F9.4, S, 'i ') )
    99994 FORMAT (/'The matrix D on exit is ' )
    99993 FORMAT (/'The matrix B on exit is ' )
    99992 FORMAT (/'The matrix F on exit is ' )
    99991 FORMAT ( 'The vector ALPHAR is ' )
    99990 FORMAT ( 50( 1X, F8.4 ) )
    99989 FORMAT (/'The vector ALPHAI is ' )
    99988 FORMAT (/'The vector BETA is ' )
    99987 FORMAT (/'The deflating subspace corresponding to the ',
         $         'eigenvalues with negative real part is ' )
    99986 FORMAT (/'The number of eigenvalues in the initial pencil with ',
         $         'negative real part is ', I2 )
          END

Program Data

    MB03LZ EXAMPLE PROGRAM DATA
       C   P   4
       (0.0604,0.6568)   (0.5268,0.2919)
       (0.3992,0.6279)   (0.4167,0.4316)
            (0,0.4896)        (0,0.9516)   (0.3724,0.0526)
       (0.9840,0.3394)        (0,0.9203)        (0,0.7378)
       (0.2691,0.4177)   (0.5478,0.3014)
       (0.4228,0.9830)   (0.9427,0.7010)
        0.6663            0.6981           (0.1781,0.8818)
       (0.5391,0.1711)    0.6665            0.1280

Program Results

    MB03LZ EXAMPLE PROGRAM RESULTS

    The matrix A on exit is
        0.7430  +0.0000i    -0.1431  -0.1304i    -0.4169  -0.0495i     0.0650  -0.0262i
        0.3992  +0.6279i     0.7398  -1.2647i    -0.0861  -0.1075i     0.2826  +0.7725i
        0.0000  +0.0000i     0.0000  +0.0000i     1.4799  +0.1442i    -0.1094  -0.1061i
        0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.6816  +0.2278i

    The matrix D on exit is
        0.0000  -0.6858i    -0.3122  -0.1018i    -0.7813  -0.4163i    -0.1343  +0.3259i
        0.9840  +0.3394i     0.0000  +0.1465i    -0.1678  +0.2971i    -0.0728  -0.6524i
        0.0000  +0.0000i     0.0000  +0.0000i    -0.0000  +0.2979i    -0.0728  +0.3971i
        0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.2414i

    The matrix B on exit is
       -1.5832  +0.5069i    -0.0819  -0.1073i     0.7749  -0.0519i     0.0635  -0.0052i
        0.0000  +0.0000i    -0.1916  -0.0106i    -0.0074  +0.0165i    -0.1546  -0.6817i
        0.0000  +0.0000i     0.0000  +0.0000i    -0.0716  -0.1811i     0.3146  +0.1558i
        0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i    -1.6078  -0.0203i

    The matrix F on exit is
        0.3382   0.0000i    -0.0622  +0.8488i     0.0042  +0.9053i    -0.1584  +0.0726i
        0.5391  +0.1711i    -0.5888  +0.0000i     0.4089  +0.2018i    -0.6913  -0.5011i
        0.0000  +0.0000i     0.0000  +0.0000i    -0.2712  +0.0000i     0.5114  +0.3726i
        0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.5218  +0.0000i
    The vector ALPHAR is
      -1.5832   1.5832  -0.0842   0.0842

    The vector ALPHAI is
       0.5069   0.5069  -0.1642  -0.1642

    The vector BETA is
       0.7430   0.7430   1.4085   1.4085

    The deflating subspace corresponding to the eigenvalues with negative real part is
       -0.0793  -0.1949i     0.4845  -0.5472i
        0.4349  +0.1710i    -0.2878  +0.0952i
       -0.1266  +0.1505i     0.1364  -0.4776i
       -0.5035  +0.6671i     0.1628  +0.3174i

    The number of eigenvalues in the initial pencil with negative real part is  2

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