MB03RZ

Reduction of a complex Schur form matrix to a block-diagonal form

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

Purpose

  To reduce an upper triangular complex matrix A (Schur form) to a
  block-diagonal form using well-conditioned non-unitary similarity
  transformations. The condition numbers of the transformations used
  for reduction are roughly bounded by PMAX, where PMAX is a given
  value. The transformations are optionally postmultiplied in a
  given matrix X. The Schur form is optionally ordered, so that
  clustered eigenvalues are grouped in the same block.

Specification
      SUBROUTINE MB03RZ( JOBX, SORT, N, PMAX, A, LDA, X, LDX, NBLCKS,
     $                   BLSIZE, W, TOL, INFO )C     .. Scalar Arguments ..
      CHARACTER         JOBX, SORT
      INTEGER           INFO, LDA, LDX, N, NBLCKS
      DOUBLE PRECISION  PMAX, TOL
C     .. Array Arguments ..
      INTEGER           BLSIZE(*)
      COMPLEX*16        A(LDA,*), W(*), X(LDX,*)

Arguments

Mode Parameters

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

  SORT    CHARACTER*1
          Specifies whether or not the diagonal elements of the
          Schur form are reordered, as follows:
          = 'N':  The diagonal elements are not reordered;
          = 'S':  The diagonal elements are reordered before each
                  step of reduction, so that clustered eigenvalues
                  appear in the same block;
          = 'C':  The diagonal elements are not reordered, but the
                  "closest-neighbour" strategy is used instead of
                  the standard "closest to the mean" strategy (see
                  METHOD);
          = 'B':  The diagonal elements are reordered before each
                  step of reduction, and the "closest-neighbour"
                  strategy is used (see METHOD).

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

  PMAX    (input) DOUBLE PRECISION
          An upper bound for the absolute value of the elements of
          the individual transformations used for reduction
          (see METHOD). PMAX >= 1.0D0.

  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
          On entry, the leading N-by-N upper triangular part of this
          array must contain the upper triangular matrix A to be
          block-diagonalized.
          On exit, the leading N-by-N upper triangular part of this
          array contains the computed block-diagonal matrix, in
          Schur form.
          The strictly lower triangular part is used as workspace,
          but it is set to zero before exit.

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

  X       (input/output) COMPLEX*16 array, dimension (LDX,*)
          On entry, if JOBX = 'U', the leading N-by-N part of this
          array must contain a given matrix X.
          On exit, if JOBX = 'U', the leading N-by-N part of this
          array contains the product of the given matrix X and the
          transformation matrix that reduced A to block-diagonal
          form. The transformation matrix is itself a product of
          non-unitary similarity transformations having elements
          with magnitude less than or equal to PMAX.
          If JOBX = 'N', this array is not referenced.

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

  NBLCKS  (output) INTEGER
          The number of diagonal blocks of the matrix A.

  BLSIZE  (output) INTEGER array, dimension (N)
          The first NBLCKS elements of this array contain the orders
          of the resulting diagonal blocks of the matrix A.

  W       (output) COMPLEX*16 array, dimension (N)
          This array contains the eigenvalues of the matrix A.

Tolerances
  TOL     DOUBLE PRECISION
          The tolerance to be used in the ordering of the diagonal
          elements of the upper triangular matrix.
          If the user sets TOL > 0, then the given value of TOL is
          used as an absolute tolerance: an eigenvalue i and a
          temporarily fixed eigenvalue 1 (the first element of the
          current trailing submatrix to be reduced) are considered
          to belong to the same cluster if they satisfy

            | lambda_1 - lambda_i | <= TOL.

          If the user sets TOL < 0, then the given value of TOL is
          used as a relative tolerance: an eigenvalue i and a
          temporarily fixed eigenvalue 1 are considered to belong to
          the same cluster if they satisfy, for j = 1, ..., N,

            | lambda_1 - lambda_i | <= | TOL | * max | lambda_j |.

          If the user sets TOL = 0, then an implicitly computed,
          default tolerance, defined by TOL = SQRT( SQRT( EPS ) )
          is used instead, as a relative tolerance, where EPS is
          the machine precision (see LAPACK Library routine DLAMCH).
          If SORT = 'N' or 'C', this parameter is not referenced.

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value.

Method
  Consider first that SORT = 'N'. Let

         ( A    A   )
         (  11   12 )
     A = (          ),
         ( 0    A   )
         (       22 )

  be the given matrix in Schur form, where initially A   is the
                                                      11
  first diagonal element. An attempt is made to compute a
  transformation matrix X of the form

         ( I   P )
     X = (       )                                               (1)
         ( 0   I )

  (partitioned as A), so that

              ( A     0  )
      -1      (  11      )
     X  A X = (          ),
              ( 0    A   )
              (       22 )

  and the elements of P do not exceed the value PMAX in magnitude.
  An adaptation of the standard method for solving Sylvester
  equations [1], which controls the magnitude of the individual
  elements of the computed solution [2], is used to obtain matrix P.
  When this attempt failed, a diagonal element of A  , closest to
                                                   22
  the mean of those of A   is selected, and moved by unitary
                        11
  similarity transformations in the leading position of A  ; the
                                                         22
  moved diagonal element is then added to the block A  , increasing
                                                     11
  its order by 1. Another attempt is made to compute a suitable
  transformation matrix X with the new definitions of the blocks A
                                                                  11
  and A  . After a successful transformation matrix X has been
       22
  obtained, it postmultiplies the current transformation matrix
  (if JOBX = 'U'), and the whole procedure is repeated for the
  block A  .
         22

  When SORT = 'S', the diagonal elements of the Schur form are
  reordered before each step of the reduction, so that each cluster
  of eigenvalues, defined as specified in the definition of TOL,
  appears in adjacent elements. The elements for each cluster are
  merged together, and the procedure described above is applied to
  the larger blocks. Using the option SORT = 'S' will usually
  provide better efficiency than the standard option (SORT = 'N'),
  proposed in [2], because there could be no or few unsuccessful
  attempts to compute individual transformation matrices X of the
  form (1). However, the resulting dimensions of the blocks are
  usually larger; this could make subsequent calculations less
  efficient.

  When SORT = 'C' or 'B', the procedure is similar to that for
  SORT = 'N' or 'S', respectively, but the block of A   whose
                                                     22
  eigenvalue(s) is (are) the closest to those of A   (not to their
                                                  11
  mean) is selected and moved to the leading position of A  . This
                                                          22
  is called the "closest-neighbour" strategy.

References
  [1] Bartels, R.H. and Stewart, G.W.  T
      Solution of the matrix equation A X + XB = C.
      Comm. A.C.M., 15, pp. 820-826, 1972.

  [2] Bavely, C. and Stewart, G.W.
      An Algorithm for Computing Reducing Subspaces by Block
      Diagonalization.
      SIAM J. Numer. Anal., 16, pp. 359-367, 1979.

  [3] Demmel, J.
      The Condition Number of Equivalence Transformations that
      Block Diagonalize Matrix Pencils.
      SIAM J. Numer. Anal., 20, pp. 599-610, 1983.

Numerical Aspects
                                    3                     4
  The algorithm usually requires 0(N ) operations, but 0(N ) are
  possible in the worst case, when the matrix cannot be diagonalized
  by well-conditioned transformations.

Further Comments
  The individual non-unitary transformation matrices used in the
  reduction of A to a block-diagonal form have condition numbers of
  the order PMAX. This does not guarantee that their product is
  well-conditioned enough. The routine can be easily modified to
  provide estimates for the condition numbers of the clusters of
  eigenvalues.

Example

Program Text

  None
Program Data
  None
Program Results
  None

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