## MB04BP

### Eigenvalues of a real skew-Hamiltonian/Hamiltonian pencil (applying transformations on panels of columns)

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

Purpose

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

            (  A  D  )         (  C  V  )
        S = (        ) and H = (        ).                           (1)
            (  E  A' )         (  W -C' )

      Optionally, if JOB = 'T', decompositions of S and H will be
      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. The notation M'
      denotes the transpose of the matrix M.
      Optionally, if COMPQ1 = 'I' or COMPQ1 = 'U', then the orthogonal
      transformation matrix Q1 will be computed.
      Optionally, if COMPQ2 = 'I' or COMPQ2 = 'U', then the orthogonal
      transformation matrix Q2 will be computed.

Specification

          SUBROUTINE MB04BP( JOB, COMPQ1, COMPQ2, N, A, LDA, DE, LDDE, C1,
         $                   LDC1, VW, LDVW, Q1, LDQ1, Q2, LDQ2, B, LDB, F,
         $                   LDF, C2, LDC2, ALPHAR, ALPHAI, BETA, IWORK,
         $                   LIWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ1, COMPQ2, JOB
          INTEGER            INFO, LDA, LDB, LDC1, LDC2, LDDE, LDF, LDQ1,
         $                   LDQ2, LDVW, LDWORK, LIWORK, N
    C     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
         $                   B( LDB, * ), BETA( * ), C1( LDC1, * ),
         $                   C2( LDC2, * ), DE( LDDE, * ), DWORK( * ),
         $                   F( LDF, * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
         $                   VW( LDVW, * )

Arguments

Mode Parameters

      JOB     CHARACTER*1
              Specifies the computation to be performed, as follows:
              = 'E': compute the eigenvalues only; S and H will not
                     necessarily be transformed as in (2).
              = 'T': put S 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;
              = 'I':  the array Q1 is initialized internally to the unit
                      matrix, and the orthogonal matrix Q1 is returned;
              = 'U':  the array Q1 contains an orthogonal matrix Q on
                      entry, and the product Q*Q1 is returned, where Q1
                      is the product of the orthogonal transformations
                      that are applied to the pencil aS - bH to reduce
                      S and H to the forms in (2), for COMPQ1 = 'I'.

      COMPQ2  CHARACTER*1
              Specifies whether to compute the orthogonal transformation
              matrix Q2, as follows:
              = 'N':  Q2 is not computed;
              = 'I':  on exit, the array Q2 contains the orthogonal
                      matrix Q2;
              = 'U':  on exit, the array Q2 contains the matrix product
                      J*Q*J'*Q2, where Q2 is the product of the
                      orthogonal transformations that are applied to
                      the pencil aS - bH to reduce S and H to the forms
                      in (2), for COMPQ2 = 'I'.
                      Setting COMPQ2 <> 'N' assumes COMPQ2 = COMPQ1.

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 JOB = 'T', the leading N/2-by-N/2 part of this
              array contains the matrix Aout; otherwise, it contains the
              upper triangular matrix A obtained just before the
              application of the periodic QZ algorithm.

      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 strictly lower triangular
              part of this array must contain the strictly lower
              triangular part of the skew-symmetric matrix E, and the
              N/2-by-N/2 strictly upper triangular part of the submatrix
              in the columns 2 to N/2+1 of this array must contain the
              strictly 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 JOB = 'T', the leading N/2-by-N/2 strictly
              upper triangular part of the submatrix in the columns 2 to
              N/2+1 of this array contains the strictly upper triangular
              part of the skew-symmetric matrix Dout.
              If JOB = 'E', the leading N/2-by-N/2 strictly upper
              triangular part of the submatrix in the columns 2 to N/2+1
              of this array contains the strictly upper triangular part
              of the skew-symmetric matrix D just before the application
              of the periodic QZ algorithm. The remaining entries are
              meaningless.

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

      C1      (input/output) DOUBLE PRECISION array, dimension
                             (LDC1, N/2)
              On entry, the leading N/2-by-N/2 part of this array must
              contain the matrix C1 = C.
              On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
              array contains the matrix C1out; otherwise, it contains
              the upper triangular matrix C1 obtained just before the
              application of the periodic QZ algorithm.

      LDC1    INTEGER
              The leading dimension of the array C1.
              LDC1 >= MAX(1, N/2).

      VW      (input/output) DOUBLE PRECISION array, dimension
                             (LDVW, 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 W, 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 V.
              On exit, if JOB = 'T', the N/2-by-N/2 part in the columns
              2 to N/2+1 of this array contains the matrix Vout.
              If JOB = 'E', the N/2-by-N/2 part in the columns 2 to
              N/2+1 of this array contains the matrix V just before the
              application of the periodic QZ algorithm.

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

      Q1      (input/output) DOUBLE PRECISION array, dimension (LDQ1, N)
              On entry, if COMPQ1 = 'U', then the leading N-by-N part of
              this array must contain a given matrix Q, and on exit,
              the leading N-by-N part of this array contains the product
              of the input matrix Q and the transformation matrix Q1
              used to transform the matrices S and H.
              On exit, if COMPQ1 = 'I', then 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 = 'I' or COMPQ1 = 'U'.

      Q2      (output) DOUBLE PRECISION array, dimension (LDQ2, N)
              On exit, if COMPQ2 = 'U', then the leading N-by-N part of
              this array contains the product of the matrix J*Q*J' and
              the transformation matrix Q2 used to transform the
              matrices S and H.
              On exit, if COMPQ2 = 'I', then 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 = 'I' or COMPQ2 = 'U'.

      B       (output) DOUBLE PRECISION array, dimension (LDB, N/2)
              On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
              array contains the matrix Bout; otherwise, it contains the
              upper triangular matrix B obtained just before the
              application of the periodic QZ algorithm.

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

      F       (output) DOUBLE PRECISION array, dimension (LDF, N/2)
              On exit, if JOB = 'T', the leading N/2-by-N/2 strictly
              upper triangular part of this array contains the strictly
              upper triangular part of the skew-symmetric matrix Fout.
              If JOB = 'E', the leading N/2-by-N/2 strictly upper
              triangular part of this array contains the strictly upper
              triangular part of the skew-symmetric matrix F just before
              the application of the periodic QZ algorithm.
              The entries on the leading N/2-by-N/2 lower triangular
              part of this array are not referenced.

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

      C2      (output) DOUBLE PRECISION array, dimension (LDC2, N/2)
              On exit, if JOB = 'T', the leading N/2-by-N/2 part of this
              array contains the matrix C2out; otherwise, it contains
              the upper Hessenberg matrix C2 obtained just before the
              application of the periodic QZ algorithm.

      LDC2    INTEGER
              The leading dimension of the array C2.
              LDC2 >= MAX(1, N/2).

      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 = 3, IWORK(1) contains the number of
              (pairs of) possibly inaccurate eigenvalues, q <= N/2, and
              IWORK(2), ..., IWORK(q+1) indicate their indices.
              Specifically, a positive value is an index of a real or
              purely imaginary eigenvalue, corresponding to a 1-by-1
              block, while the absolute value of a negative entry in
              IWORK is an index to the first eigenvalue in a pair of
              consecutively stored eigenvalues, corresponding to a
              2-by-2 block. A 2-by-2 block may have two complex, two
              real, two purely imaginary, or one real and one purely
              imaginary eigenvalue.
              For i = q+2, ..., 2*q+1, IWORK(i) contains a pointer to
              the starting location in DWORK of the i-th quadruple of
              1-by-1 blocks, if IWORK(i-q) > 0, or 2-by-2 blocks,
              if IWORK(i-q) < 0, defining unreliable eigenvalues.
              IWORK(2*q+2) contains the number of the 1-by-1 blocks, and
              IWORK(2*q+3) contains the number of the 2-by-2 blocks,
              corresponding to unreliable eigenvalues. IWORK(2*q+4)
              contains the total number t of the 2-by-2 blocks.
              If INFO = 0, then q = 0, therefore IWORK(1) = 0.

      LIWORK  INTEGER
              The dimension of the array IWORK.  LIWORK >= N+12.

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
              optimal LDWORK, and DWORK(2), ..., DWORK(5) contain the
              Frobenius norms of the factors of the formal matrix
              product used by the algorithm. In addition, DWORK(6), ...,
              DWORK(5+4*s) contain the s quadruple values corresponding
              to the 1-by-1 blocks. Their eigenvalues are real or purely
              imaginary. Such an eigenvalue is obtained from
              -i*sqrt(a1*a3/a2/a4), but always taking a positive sign,
              where a1, ..., a4 are the corresponding quadruple values.
              Moreover, DWORK(6+4*s), ..., DWORK(5+4*s+16*t) contain the
              t groups of quadruple 2-by-2 matrices corresponding to the
              2-by-2 blocks. Their eigenvalue pairs are either complex,
              or placed on the real and imaginary axes. Such an
              eigenvalue pair is obtained as -1i*sqrt(ev), but taking
              positive imaginary parts, where ev are the eigenvalues of
              the product A1*inv(A2)*A3*inv(A4), where A1, ..., A4
              define the corresponding 2-by-2 matrix quadruple.
              On exit, if INFO = -27, 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',
                 LDWORK >= N**2 + MAX(L,36);
              if JOB = 'T' or COMPQ1 <> 'N' or COMPQ2 <> 'N',
                 LDWORK >= 2*N**2 + MAX(L,36);
              where
                 L = 4*N + 4, if N/2 is even, and
                 L = 4*N,     if N/2 is odd.
              For good performance LDWORK should generally be larger.

Error Indicator

      INFO    INTEGER
              = 0: succesful exit;
              < 0: if INFO = -i, the i-th argument had an illegal value;
              = 1: problem during computation of the eigenvalues;
              = 2: periodic QZ algorithm did not converge in the SLICOT
                   Library subroutine MB03BD;
              = 3: some eigenvalues might be inaccurate, and details can
                   be found in IWORK and DWORK. This is a warning.

Method

      The algorithm uses Givens rotations and Householder reflections to
      annihilate elements in S, T, and H such that A, B, and C1 are
      upper triangular and C2 is upper Hessenberg. Finally, the periodic
      QZ algorithm is applied to transform C2 to upper quasi-triangular
      form while A, B, and C1 stay in upper triangular form.
      See also page 27 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

      For large values of N, the routine applies the transformations
      for reducing T on panels of columns. The user may specify in INFO
      the desired number of columns. If on entry INFO < 0, then the
      routine estimates a suitable value of this number. If INFO = 0,
      the routine MB04BD is directly called.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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