## MB04CD

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

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

Purpose

      To compute the transformed matrices A, B and D, using orthogonal
      matrices Q1, Q2 and Q3 for a real N-by-N regular pencil

                      ( A11   0  ) ( B11   0  )     (  0   D12 )
        aA*B - bD = a (          ) (          ) - b (          ),    (1)
                      (  0   A22 ) (  0   B22 )     ( D21   0  )

      where A11, A22, B11, B22 and D12 are upper triangular, D21 is
      upper quasi-triangular and the generalized matrix product
         -1        -1    -1        -1
      A11   D12 B22   A22   D21 B11   is upper quasi-triangular, such
      that Q3' A Q2, Q2' B Q1 are upper triangular, Q3' D Q1 is upper
      quasi-triangular and the transformed pencil
      a(Q3' A B Q1) - b(Q3' D Q1) is in generalized Schur form. The
      notation M' denotes the transpose of the matrix M.

Specification

          SUBROUTINE MB04CD( COMPQ1, COMPQ2, COMPQ3, N, A, LDA, B, LDB, D,
         $                   LDD, Q1, LDQ1, Q2, LDQ2, Q3, LDQ3, IWORK,
         $                   LIWORK, DWORK, LDWORK, BWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ1, COMPQ2, COMPQ3
          INTEGER            INFO, LDA, LDB, LDD, LDQ1, LDQ2, LDQ3, LDWORK,
         $                   LIWORK, N
    C     .. Array Arguments ..
          LOGICAL            BWORK( * )
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), D( LDD, * ),
         $                   DWORK( * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
         $                   Q3( LDQ3, * )

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*B - bD 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*B - bD in (1).

      COMPQ3  CHARACTER*1
              Specifies whether to compute the orthogonal transformation
              matrix Q3, as follows:
              = 'N':  Q3 is not computed;
              = 'I':  the array Q3 is initialized internally to the unit
                      matrix, and the orthogonal matrix Q3 is returned;
              = 'U':  the array Q3 contains an orthogonal matrix Q01 on
                      entry, and the matrix Q03*Q3 is returned, where Q3
                      is the product of the orthogonal transformations
                      that are applied on the right to the pencil
                      aA*B - bD in (1).

Input/Output Parameters

      N       (input) INTEGER
              Order of the pencil aA*B - bD.  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 diagonal part of this
              array must contain the matrix B 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.

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

      D       (input/output) DOUBLE PRECISION array, dimension (LDD, N)
              On entry, the leading N-by-N block anti-diagonal part of
              this array must contain the matrix D 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.

      LDD     INTEGER
              The leading dimension of the array D.  LDD >= 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, B, and D.
              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
              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, B, and D.
              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'.

      Q3      (input/output) DOUBLE PRECISION array, dimension (LDQ3, N)
              On entry, if COMPQ3 = 'U', then the leading N-by-N part of
              this array must contain a given matrix Q03, and on exit,
              the leading N-by-N part of this array contains the product
              of the input matrix Q03 and the transformation matrix Q3
              used to transform the matrices A, B and D.
              On exit, if COMPQ3 = 'I', then the leading N-by-N part of
              this array contains the orthogonal transformation matrix
              Q3.
              If COMPQ3 = 'N' this array is not referenced.

      LDQ3    INTEGER
              The leading dimension of the array Q3.
              LDQ3 >= 1,         if COMPQ3 = 'N';
              LDQ3 >= MAX(1, N), if COMPQ3 = 'I' or COMPQ3 = 'U'.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)

      LIWORK  INTEGER
              The dimension of the array IWORK.
              LIWORK >= MAX( N/2+1, 48 ).

      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 >= 3*N*N + MAX( N/2 + 252, 432 ).
              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
                   routine DGGEV, called by the SLICOT routine MB03CD;
              = 3: the standard QZ algorithm failed in the LAPACK
                   routines DGGES, called by the SLICOT routines MB03CD
                   or MB03ED;
              = 4: the standard QZ algorithm failed to reorder the
                   eigenvalues in the LAPACK routine DTGSEN, called by
                   the SLICOT routine MB03CD.

Method

      First, the periodic QZ algorithm (see also [2] and [3]) is applied
                                      -1        -1    -1        -1
      to the formal matrix product A11   D12 B22   A22   D21 B11   to
      reorder the eigenvalues, i.e., orthogonal matrices V1, V2, V3, V4,
      V5 and V6 are computed such that V2' A11 V1, V2' D12 V3,
      V4' B22 V3, V5' A22 V4, V5' D21 V6 and V1' B11 V6 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(V6, V3), Q2 = diag(V1, V4), Q3 = diag(V2, V5)
      and

                       ( AA11 AA12   0    0  )
                       (                     )
                       (   0  AA22   0    0  )
      A := Q3' A Q2 =: (                     ),
                       (   0    0  AA33 AA34 )
                       (                     )
                       (   0    0    0  AA44 )

                       ( BB11 BB12   0    0  )
                       (                     )
                       (   0  BB22   0    0  )
      B := Q2' B Q1 =: (                     ),
                       (   0    0  BB33 BB34 )
                       (                     )
                       (   0    0    0  BB44 )

                       (   0    0  DD13 DD14 )
                       (                     )
                       (   0    0    0  DD24 )
      D := Q3' D Q1 =: (                     ),
                       ( DD31 DD32   0    0  )
                       (                     )
                       (   0  DD42   0    0  )

                             -1          -1     -1          -1
      are set, such that AA22   DD24 BB44   AA44   DD42 BB22   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*B - bD 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 )

                    ( BB11   0  | BB12   0  )
                    (           |           )
                    (   0  BB33 |   0  BB34 )   ( BB1  *  )
      B := P' B P = (-----------+-----------) = (         ),
                    (   0    0  | BB22   0  )   (  0  BB2 )
                    (           |           )
                    (   0    0  |   0  BB44 )

                    (   0  DD13 |   0  DD14 )
                    (           |           )
                    ( DD31   0  | DD32   0  )   ( DD1  *  )
      D := P' D P = (-----------+-----------) = (         ).
                    (   0    0  |   0  DD24 )   (  0  DD2 )
                    (           |           )
                    (   0    0  | DD42   0  )

      Then, further orthogonal transformations that are provided by the
      SLICOT Library routines MB03ED and MB03CD are used to
      triangularize the subpencil aAA1 BB1 - bDD1.

      Finally, the subpencil aAA2 BB2 - bDD2 is triangularized by
      applying a special permutation matrix.

      See also page 22 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

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