MB03RX

Reordering the diagonal blocks of a principal submatrix of a real Schur form matrix (the last block is moved in the leading position)

Purpose

  To reorder the diagonal blocks of the principal submatrix between
  the indices KL and KU (KU >= KL) of a real Schur form matrix A
  together with their eigenvalues, using orthogonal similarity
  transformations, such that the block specified by KU is moved in
  the position KL. The transformations are optionally postmultiplied
  in a given matrix X.

      SUBROUTINE MB03RX( JOBV, N, KL, KU, A, LDA, X, LDX, WR, WI,
     $                   DWORK )
C     .. Scalar Arguments ..
      CHARACTER         JOBV
      INTEGER           KL, KU, LDA, LDX, N
C     .. Array Arguments ..
      DOUBLE PRECISION  A(LDA,*), DWORK(*), WI(*), WR(*), X(LDX,*)

Mode Parameters

  JOBV    CHARACTER*1
          Specifies whether or not the transformations are
          accumulated, as follows:
          = 'N':  The transformations are not accumulated;
          = 'V':  The transformations are accumulated in X (the
                  given matrix X is updated).

  N       (input) INTEGER
          The order of the matrices A and X.  N >= 0.

  KL      (input) INTEGER
          The lower boundary index for the rows and columns of the
          principal submatrix of A whose diagonal blocks are to be
          reordered, and also the target position for the block to
          be moved.  1 <= KL <= KU <= N.

  KU      (input/output) INTEGER
          On entry, KU specifies the upper boundary index for the
          rows and columns of the principal submatrix of A whose
          diagonal blocks are to be reordered, and also the original
          position for the block to be moved.  1 <= KL <= KU <= N.
          On exit, KU specifies the upper boundary index for the
          rows and columns of the principal submatrix of A whose
          diagonal blocks have been reordered. The given value will
          be increased by 1 if the moved block was 2-by-2 and it has
          been replaced by two 1-by-1 blocks. Otherwise, its input
          value is preserved.

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

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

  X       (input/output) DOUBLE PRECISION array, dimension (LDX,N)
          On entry, if JOBV = 'V', the leading N-by-N part of this
          array must contain a given matrix X.
          On exit, if JOBV = 'V', the leading N-by-N part of this
          array contains the product of the given matrix X and the
          transformation matrix that performed the reordering of A.
          If JOBV = 'N', this array is not referenced.

  LDX     INTEGER
          The leading dimension of array X.
          LDX >= 1,        if JOBV = 'N';
          LDX >= MAX(1,N), if JOBV = 'V'.

  WR,     (input/output) DOUBLE PRECISION arrays, dimension (N)
  WI      On entry, these arrays must contain the real and imaginary
          parts, respectively, of the eigenvalues of the matrix A.
          On exit, these arrays contain the real and imaginary
          parts, respectively, of the eigenvalues of the matrix A,
          possibly reordered.

  DWORK   DOUBLE PRECISION array, dimension (N)

  An attempt is made to move the block in the position (KU,KU) to
  the position (KL,KL) by a sequence of orthogonal similarity
  transformations, each swapping two consecutive blocks. The
  standard algorithm [1], [2] usually succeeds to perform this
  reordering. A failure of this algorithm means that two consecutive
  blocks (one of them being the desired block possibly moved) are
  too close to swap. In such a case, the leading block of the two
  is tried to be moved in the position (KL,KL) and the procedure is
  repeated.

  [1] Stewart, G.W.
      HQR3 and EXCHQZ: FORTRAN subroutines for calculating and
      ordering the eigenvalues of a real upper Hessenberg matrix.
      ACM TOMS, 2, pp. 275-280, 1976.

  [2] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
      Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
      Ostrouchov, S., and Sorensen, D.
      LAPACK Users' Guide: Second Edition.
      SIAM, Philadelphia, 1995.

  The algorithm is numerically stable. If some eigenvalues are
  ill-conditioned, their returned values could differ much from
  their input values.

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Program Text

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