## MB03CD

### Exchanging eigenvalues of a real 2-by-2, 3-by-3 or 4-by-4 block upper triangular pencil (factored version)

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

Purpose

      To compute orthogonal matrices Q1, Q2, Q3 for a real 2-by-2,
      3-by-3, or 4-by-4 regular block upper triangular pencil

                     ( A11 A12 ) ( B11 B12 )     ( D11 D12 )
        aAB - bD = a (         ) (         ) - b (         ),        (1)
                     (  0  A22 ) (  0  B22 )     (  0  D22 )

      such that the pencil a(Q3' A Q2 )(Q2' B Q1 ) - b(Q3' D Q1) is
      still in block upper triangular form, but the eigenvalues in
      Spec(A11 B11, D11), Spec(A22 B22, D22) are exchanged, where
      Spec(X,Y) denotes the spectrum of the matrix pencil (X,Y), and M'
      denotes the transpose of the matrix M.

      Optionally, to upper triangularize the real regular pencil in
      block lower triangular form

                   ( A11  0  ) ( B11  0  )     ( D11  0  )
      aAB - bD = a (         ) (         ) - b (         ),          (2)
                   ( A21 A22 ) ( B21 B22 )     ( D21 D22 )

      while keeping the eigenvalues in the same diagonal position.

Specification

          SUBROUTINE MB03CD( UPLO, N1, N2, PREC, A, LDA, B, LDB, D, LDD, Q1,
         $                   LDQ1, Q2, LDQ2, Q3, LDQ3, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          UPLO
          INTEGER            INFO, LDA, LDB, LDD, LDQ1, LDQ2, LDQ3, LDWORK,
         $                   N1, N2
          DOUBLE PRECISION   PREC
    C     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), D( LDD, * ),
         $                   DWORK( * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
         $                   Q3( LDQ3, * )

Arguments

Mode Parameters

      UPLO    CHARACTER*1
              Specifies if the pencil is in lower or upper block
              triangular form on entry, as follows:
              = 'U': Upper block triangular, eigenvalues are exchanged
                     on exit;
              = 'L': Lower block triangular, eigenvalues are not
                     exchanged on exit.

Input/Output Parameters

      N1      (input/output) INTEGER
              Size of the upper left block, N1 <= 2.
              If UPLO = 'U' and INFO = 0, or UPLO = 'L' and INFO <> 0,
              N1 and N2 are exchanged on exit; otherwise, N1 is
              unchanged on exit.

      N2      (input/output) INTEGER
              Size of the lower right block, N2 <= 2.
              If UPLO = 'U' and INFO = 0, or UPLO = 'L' and INFO <> 0,
              N1 and N2 are exchanged on exit; otherwise, N2 is
              unchanged on exit.

      PREC    (input) DOUBLE PRECISION
              The machine precision, (relative machine precision)*base.
              See the LAPACK Library routine DLAMCH.

      A       (input or input/output) DOUBLE PRECISION array, dimension
                 (LDA, N1+N2)
              On entry, the leading (N1+N2)-by-(N1+N2) part of this
              array must contain the matrix A of the pencil aAB - bD.
              The (2,1) block, if UPLO = 'U', or the (1,2) block, if
              UPLO = 'L', need not be set to zero.
              On exit, if N1 = N2 = 1, this array contains the matrix
                                [  0 1 ]
              J' A J, where J = [ -1 0 ]; otherwise, this array is
              unchanged on exit.

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= N1+N2.

      B       (input or input/output) DOUBLE PRECISION array, dimension
                 (LDB, N1+N2)
              On entry, the leading (N1+N2)-by-(N1+N2) part of this
              array must contain the matrix B of the pencil aAB - bD.
              The (2,1) block, if UPLO = 'U', or the (1,2) block, if
              UPLO = 'L', need not be set to zero.
              On exit, if N1 = N2 = 1, this array contains the matrix
              J' B J; otherwise, this array is unchanged on exit.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= N1+N2.

      D       (input/output) DOUBLE PRECISION array, dimension
                 (LDD, N1+N2)
              On entry, the leading (N1+N2)-by-(N1+N2) part of this
              array must contain the matrix D of the pencil aAB - bD.
              On exit, if N1 = 2 or N2 = 2, the leading
              (N1+N2)-by-(N1+N2) part of this array contains the
              transformed matrix D in real Schur form. If N1 = 1 and
              N2 = 1, this array contains the matrix J' D J.

      LDD     INTEGER
              The leading dimension of the array D.  LDD >= N1+N2.

      Q1      (output) DOUBLE PRECISION array, dimension (LDQ1, N1+N2)
              The leading (N1+N2)-by-(N1+N2) part of this array contains
              the first orthogonal transformation matrix.

      LDQ1    INTEGER
              The leading dimension of the array Q1.  LDQ1 >= N1+N2.

      Q2      (output) DOUBLE PRECISION array, dimension (LDQ2, N1+N2)
              The leading (N1+N2)-by-(N1+N2) part of this array contains
              the second orthogonal transformation matrix.

      LDQ2    INTEGER
              The leading dimension of the array Q2.  LDQ2 >= N1+N2.

      Q3      (output) DOUBLE PRECISION array, dimension (LDQ3, N1+N2)
              The leading (N1+N2)-by-(N1+N2) part of this array contains
              the third orthogonal transformation matrix.

      LDQ3    INTEGER
              The leading dimension of the array Q3.  LDQ3 >= N1+N2.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              If N1+N2 = 2 then DWORK is not referenced.

      LDWORK  INTEGER
              The dimension of the array DWORK.
              If N1+N2 = 2, then LDWORK = 0; otherwise,
              LDWORK >= 16*N1 + 10*N2 + 23, UPLO = 'U';
              LDWORK >= 10*N1 + 16*N2 + 23, UPLO = 'L'.

Error Indicator

      INFO    INTEGER
              = 0: succesful exit;
              = 1: the QZ iteration failed in the LAPACK routine DGGEV;
              = 2: another error occured while executing a routine in
                   DGGEV;
              = 3: the QZ iteration failed in the LAPACK routine DGGES;
              = 4: another error occured during execution of DGGES;
              = 5: reordering of aA*B - bD in the LAPACK routine DTGSEN
                   failed because the transformed matrix pencil
                   aA*B - bD would be too far from generalized Schur
                   form; the problem is very ill-conditioned.

Method

      The algorithm uses orthogonal transformations as described in [2]
      (page 21). The QZ algorithm is used for N1 = 2 or N2 = 2, but it
      always acts on an upper block triangular pencil.

References

      [1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
          Numerical computation of deflating subspaces of skew-
          Hamiltonian/Hamiltonian pencils.
          SIAM J. Matrix Anal. Appl., 24 (1), pp. 165-190, 2002.

      [2] 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

      The algorithm is numerically backward stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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