## MB03QD

### Reordering the diagonal blocks of a principal submatrix of an upper quasi-triangular matrix to have eigenvalues in a specified domain

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

Purpose

      To reorder the diagonal blocks of a principal submatrix of an
      upper quasi-triangular matrix A together with their eigenvalues by
      constructing an orthogonal similarity transformation UT.
      After reordering, the leading block of the selected submatrix of A
      has eigenvalues in a suitably defined domain of interest, usually
      related to stability/instability in a continuous- or discrete-time
      sense.

Specification

          SUBROUTINE MB03QD( DICO, STDOM, JOBU, N, NLOW, NSUP, ALPHA,
         $                   A, LDA, U, LDU, NDIM, DWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER        DICO, JOBU, STDOM
          INTEGER          INFO, LDA, LDU, N, NDIM, NLOW, NSUP
          DOUBLE PRECISION ALPHA
    C     .. Array Arguments ..
          DOUBLE PRECISION A(LDA,*), DWORK(*), U(LDU,*)

Arguments

Mode Parameters

      DICO    CHARACTER*1
              Specifies the type of the spectrum separation to be
              performed as follows:
              = 'C':  continuous-time sense;
              = 'D':  discrete-time sense.

      STDOM   CHARACTER*1
              Specifies whether the domain of interest is of stability
              type (left part of complex plane or inside of a circle)
              or of instability type (right part of complex plane or
              outside of a circle) as follows:
              = 'S':  stability type domain;
              = 'U':  instability type domain.

      JOBU    CHARACTER*1
              Indicates how the performed orthogonal transformations UT
              are accumulated, as follows:
              = 'I':  U is initialized to the unit matrix and the matrix
                      UT is returned in U;
              = 'U':  the given matrix U is updated and the matrix U*UT
                      is returned in U.

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrices A and U.  N >= 1.

      NLOW,   (input) INTEGER
      NSUP    NLOW and NSUP specify the boundary indices for the rows
              and columns of the principal submatrix of A whose diagonal
              blocks are to be reordered.  1 <= NLOW <= NSUP <= N.

      ALPHA   (input) DOUBLE PRECISION
              The boundary of the domain of interest for the eigenvalues
              of A. If DICO = 'C', ALPHA is the boundary value for the
              real parts of eigenvalues, while for DICO = 'D',
              ALPHA >= 0 represents the boundary value for the moduli of
              eigenvalues.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain a matrix in a real Schur form whose 1-by-1 and
              2-by-2 diagonal blocks between positions NLOW and NSUP
              are to be reordered.
              On exit, the leading N-by-N part contains the ordered
              real Schur matrix UT' * A * UT with the elements below the
              first subdiagonal set to zero.
              The leading NDIM-by-NDIM part of the principal submatrix
              D = A(NLOW:NSUP,NLOW:NSUP) has eigenvalues in the domain
              of interest and the trailing part of this submatrix has
              eigenvalues outside the domain of interest.
              The domain of interest for lambda(D), the eigenvalues of
              D, is defined by the parameters ALPHA, DICO and STDOM as
              follows:
                For DICO = 'C':
                   Real(lambda(D)) < ALPHA if STDOM = 'S';
                   Real(lambda(D)) > ALPHA if STDOM = 'U'.
                For DICO = 'D':
                   Abs(lambda(D)) < ALPHA if STDOM = 'S';
                   Abs(lambda(D)) > ALPHA if STDOM = 'U'.

      LDA     INTEGER
              The leading dimension of array A.  LDA >= N.

      U       (input/output) DOUBLE PRECISION array, dimension (LDU,N)
              On entry with JOBU = 'U', the leading N-by-N part of this
              array must contain a transformation matrix (e.g. from a
              previous call to this routine).
              On exit, if JOBU = 'U', the leading N-by-N part of this
              array contains the product of the input matrix U and the
              orthogonal matrix UT used to reorder the diagonal blocks
              of A.
              On exit, if JOBU = 'I', the leading N-by-N part of this
              array contains the matrix UT of the performed orthogonal
              transformations.
              Array U need not be set on entry if JOBU = 'I'.

      LDU     INTEGER
              The leading dimension of array U.  LDU >= N.

      NDIM    (output) INTEGER
              The number of eigenvalues of the selected principal
              submatrix lying inside the domain of interest.
              If NLOW = 1, NDIM is also the dimension of the invariant
              subspace corresponding to the eigenvalues of the leading
              NDIM-by-NDIM submatrix. In this case, if U is the
              orthogonal transformation matrix used to compute and
              reorder the real Schur form of A, its first NDIM columns
              form an orthonormal basis for the above invariant
              subspace.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (N)

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  A(NLOW,NLOW-1) is nonzero, i.e. A(NLOW,NLOW) is not
                    the leading element of a 1-by-1 or 2-by-2 diagonal
                    block of A, or A(NSUP+1,NSUP) is nonzero, i.e.
                    A(NSUP,NSUP) is not the bottom element of a 1-by-1
                    or 2-by-2 diagonal block of A;
              = 2:  two adjacent blocks are too close to swap (the
                    problem is very ill-conditioned).

Method

      Given an upper quasi-triangular matrix A with 1-by-1 or 2-by-2
      diagonal blocks, the routine reorders its diagonal blocks along
      with its eigenvalues by performing an orthogonal similarity
      transformation UT' * A * UT. The column transformation UT is also
      performed on the given (initial) transformation U (resulted from
      a possible previous step or initialized as the identity matrix).
      After reordering, the eigenvalues inside the region specified by
      the parameters ALPHA, DICO and STDOM appear at the top of
      the selected diagonal block between positions NLOW and NSUP.
      In other words, lambda(A(NLOW:NSUP,NLOW:NSUP)) are ordered such
      that lambda(A(NLOW:NLOW+NDIM-1,NLOW:NLOW+NDIM-1)) are inside and
      lambda(A(NLOW+NDIM:NSUP,NLOW+NDIM:NSUP)) are outside the domain
      of interest. If NLOW = 1, the first NDIM columns of U*UT span the
      corresponding invariant subspace of A.

References

      [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.

Numerical Aspects

                                          3
      The algorithm requires less than 4*N  operations.

Further Comments

      None

Example

Program Text

    *     MB03QD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 10 )
          INTEGER          LDA, LDU
          PARAMETER        ( LDA = NMAX, LDU = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 3*NMAX )
    *     .. Local Scalars ..
          CHARACTER*1      DICO, JOBU, STDOM
          INTEGER          I, INFO, J, N, NDIM, NLOW, NSUP
          DOUBLE PRECISION ALPHA
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), U(LDU,NMAX),
         $                 WI(NMAX), WR(NMAX)
          LOGICAL          BWORK(NMAX)
    *     .. External Functions ..
          LOGICAL          SELECT
    *     .. External Subroutines ..
          EXTERNAL         DGEES, MB03QD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, NLOW, NSUP, ALPHA, DICO, STDOM, JOBU
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99992 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
    *        Compute Schur form, eigenvalues and Schur vectors.
             CALL DGEES( 'Vectors', 'Not sorted', SELECT, N, A, LDA, NDIM,
         $               WR, WI, U, LDU, DWORK, LDWORK, BWORK, INFO )
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
    *           Block reordering.
                CALL MB03QD( DICO, STDOM, JOBU, N, NLOW, NSUP, ALPHA,
         $                   A, LDA, U, LDU, NDIM, DWORK, INFO )
                IF ( INFO.NE.0 ) THEN
                   WRITE ( NOUT, FMT = 99997 ) INFO
                ELSE
                   WRITE ( NOUT, FMT = 99996 ) NDIM
                   WRITE ( NOUT, FMT = 99994 )
                   DO 10 I = 1, N
                      WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
       10          CONTINUE
                   WRITE ( NOUT, FMT = 99993 )
                   DO 20 I = 1, N
                      WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
       20          CONTINUE
                END IF
             END IF
          END IF
    *
          STOP
    *
    99999 FORMAT (' MB03QD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from DGEES  = ',I2)
    99997 FORMAT (' INFO on exit from MB03QD = ',I2)
    99996 FORMAT (' The number of eigenvalues in the domain is ',I5)
    99995 FORMAT (8X,20(1X,F8.4))
    99994 FORMAT (/' The ordered Schur form matrix is ')
    99993 FORMAT (/' The transformation matrix is ')
    99992 FORMAT (/' N is out of range.',/' N = ',I5)
          END

Program Data

     MB03QD EXAMPLE PROGRAM DATA
       4     1     4      0.0      C      S      U
      -1.0  37.0 -12.0 -12.0
      -1.0 -10.0   0.0   4.0
       2.0  -4.0   7.0  -6.0
       2.0   2.0   7.0  -9.0

Program Results

     MB03QD EXAMPLE PROGRAM RESULTS

     The number of eigenvalues in the domain is     4

     The ordered Schur form matrix is
              -3.1300 -26.5066  27.2262 -16.2009
               0.9070  -3.1300  13.6254   8.9206
               0.0000   0.0000  -3.3700   0.3419
               0.0000   0.0000  -1.7879  -3.3700

     The transformation matrix is
               0.9611   0.1784   0.2064  -0.0440
              -0.1468  -0.2704   0.8116  -0.4965
              -0.2224   0.7675   0.4555   0.3924
              -0.0733   0.5531  -0.3018  -0.7730

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