MB04RD

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

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

Purpose

  To reduce a real matrix pair (A,B) in generalized real Schur form
  to a block-diagonal form using well-conditioned non-orthogonal
  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 MB04RD( JOBX, JOBY, SORT, N, PMAX, A, LDA, B, LDB, X,
     $                   LDX, Y, LDY, NBLCKS, BLSIZE, ALPHAR, ALPHAI,
     $                   BETA, TOL, IWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         JOBX, JOBY, SORT
      INTEGER           INFO, LDA, LDB, LDWORK, LDX, LDY, N, NBLCKS
      DOUBLE PRECISION  PMAX, TOL
C     .. Array Arguments ..
      INTEGER           BLSIZE(*), IWORK(*)
      DOUBLE PRECISION  A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
     $                  BETA(*), DWORK(*), 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 blocks of the
          generalized real Schur form are reordered, as follows:
          = 'N':  The diagonal blocks are not reordered;
          = 'S':  The diagonal blocks are reordered before each
                  step of reduction, so that clustered eigenvalues
                  appear in the same pair of blocks.
          = 'C':  The diagonal blocks are not reordered, but the
                  "closest-neighbour" strategy is used instead of
                  the standard "closest to the mean" strategy (see
                  METHOD);
          = 'B':  The diagonal blocks 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) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N upper quasi-triangular part
          of this array must contain the upper quasi-triangular
          matrix A in the generalized real Schur form, as returned
          by the LAPACK Library routine DGGES. The lower triangular
          part below the Schur matrix is used as workspace.
          On exit, the leading N-by-N upper quasi-triangular part of
          this array contains the computed block-diagonal matrix, in
          real Schur canonical form, 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) DOUBLE PRECISION 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 real Schur form, as returned by the LAPACK
          Library routine DGGES. The diagonal elements of B are
          non-negative. 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 non-negative.

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

  X       (input/output) DOUBLE PRECISION 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 DGGES.
          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-orthogonal 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) DOUBLE PRECISION 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 DGGES.
          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-orthogonal 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.

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
  BETA    (output) DOUBLE PRECISION array, dimension (N)
          On exit, if INFO = 0, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
          j = 1, ..., N, will be the generalized eigenvalues.
          ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j = 1, ..., N, are
          the diagonals of the complex Schur form (S,T) that would
          result if the 2-by-2 diagonal blocks of the real Schur
          form of (A,B) were further reduced to triangular form
          using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.
          All BETA(j) are non-negative real numbers.
          The quotients ALPHAR(j)/BETA(j) and ALPHAI(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 DGGES,
          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 blocks of the upper triangular
          matrix pair.
          If the user sets TOL > 0, then the given value of TOL is
          used as an absolute tolerance: a pair of blocks i and a
          temporarily fixed pair of blocks 1 (the first pair of
          blocks of the current trailing pair of submatrices to be
          reduced) are considered to belong to the same cluster if
          their eigenvalues 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: a pair of blocks i and a
          temporarily fixed pair of blocks 1 are considered to
          belong to the same cluster if their eigenvalues 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+6)

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) contains the optimal value
          of LDWORK.
          On exit, if INFO = -21, DWORK(1) returns the minimum
          value of LDWORK. When LDWORK = 0 is set on entry, the
          routine will return this value for INFO, and also set
          DWORK(1), but no error message related to LDWORK is
          issued by XERBLA.

  LDWORK  INTEGER
          The dimension of the array DWORK.
          LDWORK >= 1,         if N <= 1;
          LDWORK >= 4*N + 16,  if N >  1.

          If LDWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          DWORK array, returns this value as the first entry of
          the DWORK array, and no error message related to LDWORK
          is issued by XERBLA.

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 real Schur form, where
  initially A   and B   are the first pair of diagonal blocks of
             11      11
  dimension 1-by-1 or 2-by-2. 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 the 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, a 1-by-1 (or 2-by-2)
  pair of diagonal blocks of (A  , B  ), whose eigenvalue(s) is
                               22   22
  (are) the closest to the mean of those of (A  , B  ) is selected,
                                              11   11
  and moved by orthogonal equivalence transformations in the leading
  position of (A  , B  ); the moved diagonal blocks in A and B are
                22   22
  then added to the blocks A   and B  , respectively, increasing
                            11      11
  their order by 1 (or 2). 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
                 11   22   11       22
  transformation 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 blocks of the generalized real 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 blocks. The blocks 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-orthogonal 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.

Example

Program Text

  None
Program Data
  None
Program Results
  None

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