## MB04HD

### Reducing a special real block (anti-)diagonal skew-Hamiltonian/Hamiltonian pencil to generalized Schur form

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

Purpose

      To compute the transformed matrices A and B, using orthogonal
      matrices Q1 and Q2 for a real N-by-N regular pencil

                    ( A11   0  )     (  0   B12 )
        aA - bB = a (          ) - b (          ),                   (1)
                    (  0   A22 )     ( B21   0  )

      where A11, A22 and B12 are upper triangular, B21 is upper
      quasi-triangular and the generalized matrix product
         -1        -1
      A11   B12 A22   B21 is in periodic Schur form, such that the
      matrix Q2' A Q1 is upper triangular, Q2' B Q1 is upper
      quasi-triangular and the transformed pencil
      a(Q2' A Q1) - b(Q2' B Q1) is in generalized Schur form. The
      notation M' denotes the transpose of the matrix M.

Specification

          SUBROUTINE MB04HD( COMPQ1, COMPQ2, N, A, LDA, B, LDB, Q1, LDQ1,
         $                   Q2, LDQ2, IWORK, LIWORK, DWORK, LDWORK, BWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ1, COMPQ2
          INTEGER            INFO, LDA, LDB, LDQ1, LDQ2, LDWORK, LIWORK, N
    C     .. Array Arguments ..
          LOGICAL            BWORK( * )
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DWORK( * ),
         $                   Q1( LDQ1, * ), Q2( LDQ2, * )

Arguments

Mode Parameters

      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 Q01 on
                      entry, and the matrix Q01*Q1 is returned, where Q1
                      is the product of the orthogonal transformations
                      that are applied on the right to the pencil
                      aA - bB in (1).

      COMPQ2  CHARACTER*1
              Specifies whether to compute the orthogonal transformation
              matrix Q2, as follows:
              = 'N':  Q2 is not computed;
              = 'I':  the array Q2 is initialized internally to the unit
                      matrix, and the orthogonal matrix Q2 is returned;
              = 'U':  the array Q2 contains an orthogonal matrix Q02 on
                      entry, and the matrix Q02*Q2 is returned, where Q2
                      is the product of the orthogonal transformations
                      that are applied on the left to the pencil aA - bB
                      in (1).

Input/Output Parameters

      N       (input) INTEGER
              Order of the pencil aA - bB, N >= 0, even.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
              On entry, the leading N-by-N block diagonal part of this
              array must contain the matrix A in (1). The off-diagonal
              blocks need not be set to zero.
              On exit, the leading N-by-N part of this array contains
              the transformed upper triangular matrix.

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

      B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
              On entry, the leading N-by-N block anti-diagonal part of
              this array must contain the matrix B in (1). The diagonal
              blocks need not be set to zero.
              On exit, the leading N-by-N part of this array contains
              the transformed upper quasi-triangular matrix.

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

      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 Q01, and on exit,
              the leading N-by-N part of this array contains the product
              of the input matrix Q01 and the transformation matrix Q1
              used to transform the matrices A and B.
              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      (input/output) DOUBLE PRECISION array, dimension (LDQ2, N)
              On entry, if COMPQ2 = 'U', then the leading N-by-N part of
              this array must contain a given matrix Q02, and on exit,
              the leading N-by-N part of this array contains the product
              of the input matrix Q02 and the transformation matrix Q2
              used to transform the matrices A and B.
              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'.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)

      LIWORK  INTEGER
              The dimension of the array IWORK.
              LIWORK >= MAX( N/2+1, 32 ).

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
              On exit, if INFO = -16, DWORK(1) returns the minimum value
              of LDWORK.

      LDWORK  INTEGER
              The dimension of the array DWORK.
              LDWORK >= 2*N*N + MAX( N/2 + 168, 272 ).
              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: the periodic QZ algorithm failed to reorder the
                   eigenvalues (the problem is very ill-conditioned) in
                   the SLICOT Library routine MB03KD;
              = 2: the standard QZ algorithm failed in the LAPACK
                   routines DGGES or DHGEQZ, called by the SLICOT
                   routines MB03DD or MB03FD;
              = 3: the eigenvalue reordering failed in the LAPACK
                   routine DTGEX2, called by the SLICOT routine MB03FD;
              = 4: the standard QZ algorithm failed to reorder the
                   eigenvalues in the LAPACK routine DTGSEN, called by
                   the SLICOT routine MB03DD.

Method

      First, the periodic QZ algorithm (see also [2] and [3]) is applied
                                      -1        -1
      to the formal matrix product A11   B12 A22   B21 to reorder the
      eigenvalues, i.e., orthogonal matrices V1, V2, V3 and V4 are
      computed such that V2' A11 V1, V2' B12 V3, V4' A22 V3 and
      V4' B21 V1 keep the triangular form, but they can be partitioned
      into 2-by-2 block forms and the last diagonal blocks correspond to
      all nonpositive real eigenvalues of the formal product, and the
      first diagonal blocks correspond to the remaining eigenvalues.

      Second, Q1 = diag(V1, V3), Q2 = diag(V2, V4) and

                       ( AA11 AA12   0    0  )
                       (                     )
                       (   0  AA22   0    0  )
      A := Q2' A Q1 =: (                     ),
                       (   0    0  AA33 AA34 )
                       (                     )
                       (   0    0    0  AA44 )

                       (   0    0  BB13 BB14 )
                       (                     )
                       (   0    0    0  BB24 )
      B := Q2' B Q1 =: (                     ),
                       ( BB31 BB32   0    0  )
                       (                     )
                       (   0  BB42   0    0  )

                             -1          -1
      are set, such that AA22   BB24 AA44   BB42 has only nonpositive
      real eigenvalues.

      Third, the permutation matrix

          (  I  0  0  0  )
          (              )
          (  0  0  I  0  )
      P = (              ),
          (  0  I  0  0  )
          (              )
          (  0  0  0  I  )

      where I denotes the identity matrix of appropriate size, is used
      to transform aA - bB to block upper triangular form

                    ( AA11   0  | AA12   0  )
                    (           |           )
                    (   0  AA33 |   0  AA34 )   ( AA1  *  )
      A := P' A P = (-----------+-----------) = (         ),
                    (   0    0  | AA22   0  )   (  0  AA2 )
                    (           |           )
                    (   0    0  |   0  AA44 )

                    (   0  BB13 |   0  BB14 )
                    (           |           )
                    ( BB31   0  | BB32   0  )   ( BB1  *  )
      B := P' B P = (-----------+-----------) = (         ).
                    (   0    0  |   0  BB24 )   (  0  BB2 )
                    (           |           )
                    (   0    0  | BB42   0  )

      Then, further orthogonal transformations that are provided by
      MB03FD and MB03DD are used to triangularize the subpencil
      aAA1 - bBB1.

      Finally, the subpencil aAA2 - bBB2 is triangularized by applying a
      special permutation matrix.

      See also page 31 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.

      [2] Bojanczyk, A., Golub, G. H. and Van Dooren, P.
          The periodic Schur decomposition: algorithms and applications.
          In F.T. Luk (editor), Advanced Signal Processing Algorithms,
          Architectures, and Implementations III, Proc. SPIE Conference,
          vol. 1770, pp. 31-42, 1992.

      [3] Hench, J. J. and Laub, A. J.
          Numerical Solution of the discrete-time periodic Riccati
          equation. IEEE Trans. Automat. Control, 39, 1197-1210, 1994.

Numerical Aspects

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

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

Return to Supporting Routines index