## 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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