MB03TS

Swapping two diagonal blocks of a matrix in (skew-)Hamiltonian canonical Schur form

Purpose

  To swap diagonal blocks A11 and A22 of order 1 or 2 in the upper
  quasi-triangular matrix A contained in a skew-Hamiltonian matrix

                [  A   G  ]          T
          X  =  [       T ],   G = -G,
                [  0   A  ]

  or in a Hamiltonian matrix

                [  A   G  ]          T
          X  =  [       T ],   G =  G.
                [  0  -A  ]

  This routine is a modified version of the LAPACK subroutine
  DLAEX2.

  The matrix A must be in Schur canonical form (as returned by the
  LAPACK routine DHSEQR), that is, block upper triangular with
  1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has
  its diagonal elements equal and its off-diagonal elements of
  opposite sign.

      SUBROUTINE MB03TS( ISHAM, WANTU, N, A, LDA, G, LDG, U1, LDU1, U2,
     $                   LDU2, J1, N1, N2, DWORK, INFO )
C     .. Scalar Arguments ..
      LOGICAL            ISHAM, WANTU
      INTEGER            INFO, J1, LDA, LDG, LDU1, LDU2, N, N1, N2
C     .. Array Arguments ..
      DOUBLE PRECISION   A(LDA,*), DWORK(*), G(LDG,*), U1(LDU1,*),
     $                   U2(LDU2,*)

Mode Parameters

  ISHAM   LOGIGAL
          Specifies the type of X:
          = .TRUE.:   X is a Hamiltonian matrix;
          = .FALSE.:  X is a skew-Hamiltonian matrix.

  WANTU   LOGIGAL
          = .TRUE.:   update the matrices U1 and U2 containing the
                      Schur vectors;
          = .FALSE.:  do not update U1 and U2.

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the upper quasi-triangular matrix A, in Schur
          canonical form.
          On exit, the leading N-by-N part of this array contains
          the reordered matrix A, again in Schur canonical form.

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

  G       (input/output) DOUBLE PRECISION array, dimension (LDG,N)
          On entry, the leading N-by-N part of this array must
          contain the upper triangular part of the symmetric
          matrix G, if ISHAM = .TRUE., or the strictly upper
          triangular part of the skew-symmetric matrix G, otherwise.
          The rest of this array is not referenced.
          On exit, the leading N-by-N part of this array contains
          the upper or strictly upper triangular part of the
          symmetric or skew-symmetric matrix G, respectively,
          updated by the orthogonal transformation which reorders A.

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

  U1      (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
          On entry, if WANTU = .TRUE., the leading N-by-N part of
          this array must contain the matrix U1.
          On exit, if WANTU = .TRUE., the leading N-by-N part of
          this array contains U1, postmultiplied by the orthogonal
          transformation which reorders A. See the description in
          the SLICOT subroutine MB03TD for further details.
          If WANTU = .FALSE., this array is not referenced.

  LDU1    INTEGER
          The leading dimension of the array U1.
          LDU1 >= MAX(1,N),  if WANTU = .TRUE.;
          LDU1 >= 1,         otherwise.

  U2      (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
          On entry, if WANTU = .TRUE., the leading N-by-N part of
          this array must contain the matrix U2.
          On exit, if WANTU = .TRUE., the leading N-by-N part of
          this array contains U2, postmultiplied by the orthogonal
          transformation which reorders A.
          If WANTU = .FALSE., this array is not referenced.

  LDU2    INTEGER
          The leading dimension of the array U2.
          LDU2 >= MAX(1,N),  if WANTU = .TRUE.;
          LDU2 >= 1,         otherwise.

  J1      (input) INTEGER
          The index of the first row of the first block A11.
          If J1+N1 < N, then A11 is swapped with the block starting
          at (J1+N1+1)-th diagonal element.
          If J1+N1 > N, then A11 is the last block in A and swapped
          with -A11', if ISHAM = .TRUE.,
          or    A11', if ISHAM = .FALSE..

  N1      (input) INTEGER
          The order of the first block A11. N1 = 0, 1 or 2.

  N2      (input) INTEGER
          The order of the second block A22. N2 = 0, 1 or 2.

  DWORK   DOUBLE PRECISION array, dimension (N)

  INFO    INTEGER
          = 0:  successful exit;
          = 1:  the transformed matrix A would be too far from Schur
                form; the blocks are not swapped and A, G, U1 and
                U2 are unchanged.

  [1] Bai, Z., and Demmel, J.W.
     On swapping diagonal blocks in real Schur form.
     Linear Algebra Appl., 186, pp. 73-95, 1993.

  [2] Benner, P., Kressner, D., and Mehrmann, V.
      Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory,
      Algorithms and Applications. Techn. Report, TU Berlin, 2003.

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Program Text

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