MB04RZ

Reduction of a complex matrix pencil in generalized complex Schur form to a block-diagonal form

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

Purpose

  To reduce a complex matrix pair (A,B) in generalized complex
  Schur form to a block-diagonal form using well-conditioned
  non-unitary equivalence transformations. The condition numbers
  of the left and right transformations used for the reduction
  are roughly bounded by PMAX, where PMAX is a given value.
  The transformations are optionally postmultiplied in the given
  matrices X and Y. The generalized Schur form is optionally
  ordered, so that clustered eigenvalues are grouped in the same
  pair of blocks.

Specification
      SUBROUTINE MB04RZ( JOBX, JOBY, SORT, N, PMAX, A, LDA, B, LDB, X,
     $                   LDX, Y, LDY, NBLCKS, BLSIZE, ALPHA, BETA,
     $                   TOL, IWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         JOBX, JOBY, SORT
      INTEGER           INFO, LDA, LDB, LDX, LDY, N, NBLCKS
      DOUBLE PRECISION  PMAX, TOL
C     .. Array Arguments ..
      INTEGER           BLSIZE(*), IWORK(*)
      COMPLEX*16        A(LDA,*), ALPHA(*), B(LDB,*), BETA(*), X(LDX,*),
     $                  Y(LDY,*)

Arguments

Mode Parameters

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

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

  SORT    CHARACTER*1
          Specifies whether or not the diagonal elements of the
          generalized 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 pair of blocks.
          = '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, B, X and Y.  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 and FURTHER COMMENTS).  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 in the
          generalized complex Schur form, as returned by the LAPACK
          Library routine ZGGES. The strictly lower triangular part
          is used as workspace.
          On exit, the leading N-by-N upper triangular part of
          this array contains the computed block-diagonal matrix,
          corresponding to the given matrix A. The remaining part
          is set to zero.

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

  B       (input/output) COMPLEX*16 array, dimension (LDB,N)
          On entry, the leading N-by-N upper triangular part of this
          array must contain the upper triangular matrix B in the
          generalized complex Schur form, as returned by the LAPACK
          Library routine ZGGES. The diagonal elements of B are real
          non-negative, hence the imaginary parts of these elements
          are zero. The strictly lower triangular part is used as
          workspace. The matrix B is assumed nonzero.
          On exit, the leading N-by-N upper triangular part of this
          array contains the computed upper triangular block-
          diagonal matrix, corresponding to the given matrix B. The
          remaining part is set to zero. The diagonal elements of B
          are real non-negative.

  LDB     INTEGER
          The leading dimension of the array B.  LDB >= 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, for instance the left
          transformation matrix VSL returned by the LAPACK Library
          routine ZGGES.
          On exit, if JOBX = 'U', the leading N-by-N part of this
          array contains the product of the given matrix X and the
          left transformation matrix that reduced (A,B) to block-
          diagonal form. The local transformation matrix is itself a
          product of non-unitary equivalence 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 the array X.
          LDX >= 1,        if JOBX = 'N';
          LDX >= MAX(1,N), if JOBX = 'U'.

  Y       (input/output) COMPLEX*16 array, dimension (LDY,*)
          On entry, if JOBY = 'U', the leading N-by-N part of this
          array must contain a given matrix Y, for instance the
          right transformation matrix VSR returned by the LAPACK
          Library routine ZGGES.
          On exit, if JOBY = 'U', the leading N-by-N part of this
          array contains the product of the given matrix Y and the
          right transformation matrix that reduced (A,B) to block-
          diagonal form. The local transformation matrix is itself a
          product of non-unitary equivalence transformations having
          elements with magnitude less than or equal to PMAX.
          If JOBY = 'N', this array is not referenced.

  LDY     INTEGER
          The leading dimension of the array Y.
          LDY >= 1,        if JOBY = 'N';
          LDY >= MAX(1,N), if JOBY = 'U'.

  NBLCKS  (output) INTEGER
          The number of diagonal blocks of the matrices A and B.

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

  ALPHA   (output) COMPLEX*16 array, dimension (N)
  BETA    (output) COMPLEX*16 array, dimension (N)
          If INFO = 0, then ALPHA(j)/BETA(j), j = 1, ..., N, are
          the generalized eigenvalues of the matrix pair (A, B)
          (the diagonals of the complex Schur form).
          All BETA(j) are non-negative real numbers.
          The quotients ALPHA(j)/BETA(j) may easily over- or
          underflow, and BETA(j) may even be zero. Thus, the user
          should avoid naively computing the ratio.
          If A and B are obtained from general matrices using ZGGES,
          ALPHA will be always less than and usually comparable with
          norm(A) in magnitude, and BETA always less than and
          usually comparable with norm(B).

Tolerances
  TOL     DOUBLE PRECISION
          If SORT = 'S' or 'B', the tolerance to be used in the
          ordering of the diagonal elements of the upper triangular
          matrix pair.
          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 (represented by the first
          element of the current trailing pair of submatrices to be
          reduced) are considered to belong to the same cluster if
          they satisfy the following "distance" condition

            | 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 finite lambda_j,
          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).
          The approximate symmetric chordal metric is used as
          "distance" of two complex, possibly infinite numbers, x
          and y. This metric is given by the formula

            d(x,y) = min( |x-y|, |1/x-1/y| ),

          taking into account the special cases of infinite or NaN
          values.
          If SORT = 'N' or 'C', this parameter is not referenced.

Workspace
  IWORK   INTEGER array, dimension (N+2)

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  the matrix pencil defined by A and B is singular.

Method
  Consider first that SORT = 'N'. Let

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

  be the given matrix pair in generalized Schur form, where
  initially A   and B   are the first diagonal elements.
             11      11
  An attempt is made to compute the transformation matrices X and Y
  of the form

         ( I   V )       ( I   W )
     X = (       ),  Y = (       )                               (1)
         ( 0   I )       ( 0   I )

  (partitioned as A and B ), so that (' denotes conjugate-transpose)

              ( A     0  )            ( B     0  )
              (  11      )            (  11      )
     X' A Y = (          ),  X' B Y = (          ),
              ( 0    A   )            ( 0    B   )
              (       22 )            (       22 )

  and the elements of V and W do not exceed the value PMAX in
  magnitude. An adaptation of the standard method for solving
  generalized Sylvester equations [1], which controls the magnitude
  of the individual elements of the computed solution [2], is used
  to obtain V and W. When this attempt fails, an eigenvalue of
  (A  , B  ) closest to the mean of those of (A  , B  ) is selected,
    22   22                                    11   11
  and moved by unitary equivalence transformations in the leading
  position of (A  , B  ); the moved diagonal elements in A and B are
                22   22
  then added to the blocks A   and B  , respectively, increasing
                            11      11
  their order by 1. Another attempt is made to compute suitable
  transformation matrices X and Y with the new definitions of the

  blocks A  , A  , B  , and B  . After successful transformation
          11   22   11       22
  matrices X and Y have been obtained, they postmultiply the current
  transformation matrices (if JOBX = 'U' and/or JOBY = 'U') and the
  whole procedure is repeated for the new blocks A   and B  .
                                                  22      22

  When SORT = 'S', the diagonal elements of the generalized Schur
  form are reordered before each step of the reduction, so that each
  cluster of generalized 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 and Y 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 blocks of A   and B
                                                      22      22
  whose eigenvalue(s) is (are) the closest to those of (A  , B  )
                                                         11   11
  (not to their mean) are selected and moved to the leading position
  of A   and B  . This is called the "closest-neighbour" strategy.
      22      22

References
  [1] Kagstrom, B. and Westin, L.
      Generalized Schur Methods with Condition Estimators for
      Solving the Generalized Sylvester Equation.
      IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.

  [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 pencil cannot be
  diagonalized by well-conditioned transformations.

Further Comments
  The individual non-unitary transformation matrices used in the
  reduction of A and B 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 generalized eigenvalues.
  For efficiency reasons, the "absolute value" (or "magnitude") of
  the complex elements x of the transformation matrices is computed
  as |real(x)| + |imag(x)|.

Example

Program Text

  None
Program Data
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Program Results
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