## MB03QG

### Reorder diagonal blocks of a principal subpencil of an upper quasi-triangular matrix pencil

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

Purpose

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

Specification

          SUBROUTINE MB03QG( DICO, STDOM, JOBU, JOBV, N, NLOW, NSUP, ALPHA,
         $                   A, LDA, E, LDE, U, LDU, V, LDV, NDIM, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER        DICO, JOBU, JOBV, STDOM
          INTEGER          INFO, LDA, LDE, LDU, LDV, LDWORK, N, NDIM, NLOW,
         $                 NSUP
          DOUBLE PRECISION ALPHA
    C     .. Array Arguments ..
          DOUBLE PRECISION A(LDA,*), DWORK(*), E(LDE,*), U(LDU,*), V(LDV,*)

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.

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

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrices A, E, U, and V.  N >= 0.

      NLOW,   (input) INTEGER
      NSUP    (input) INTEGER
              NLOW and NSUP specify the boundary indices for the rows
              and columns of the principal subpencil  of A - lambda*E
              whose diagonal blocks are to be reordered.
              0 <= 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 the generalized eigenvalues, while for
              DICO = 'D', ALPHA >= 0 represents the boundary value for
              the moduli of the generalized 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 of this array contains
              a real Schur matrix UT' * A * VT, with the elements below
              the first subdiagonal set to zero.
              The leading NDIM-by-NDIM part of the principal subpencil
              B - lambda*C, defined with B := A(NLOW:NSUP,NLOW:NSUP),
              C := E(NLOW:NSUP,NLOW:NSUP), has generalized eigenvalues
              in the domain of interest and the trailing part of this
              subpencil has generalized eigenvalues outside the domain
              of interest.
              The domain of interest for eig(B,C), the generalized
              eigenvalues of the pair (B,C), is defined by the
              parameters ALPHA, DICO and STDOM as follows:
                For DICO = 'C':
                   Real(eig(B,C)) < ALPHA if STDOM = 'S';
                   Real(eig(B,C)) > ALPHA if STDOM = 'U'.
                For DICO = 'D':
                   Abs(eig(B,C)) < ALPHA if STDOM = 'S';
                   Abs(eig(B,C)) > ALPHA if STDOM = 'U'.

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

      E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
              On entry, the leading N-by-N part of this array must
              contain a matrix in an upper triangular form.
              On exit, the leading N-by-N part of this array contains an
              upper triangular matrix UT' * E * VT, with the elements
              below the diagonal set to zero.
              The leading NDIM-by-NDIM part of the principal subpencil
              B - lambda*C, defined with B := A(NLOW:NSUP,NLOW:NSUP)
              C := E(NLOW:NSUP,NLOW:NSUP) has generalized eigenvalues
              in the domain of interest and the trailing part of this
              subpencil has generalized eigenvalues outside the domain
              of interest (see description of A).

      LDE     INTEGER
              The leading dimension of the array E.  LDE >= MAX(1,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 - lambda*E.
              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 the array U.  LDU >= MAX(1,N).

      V       (input/output) DOUBLE PRECISION array, dimension (LDV,N)
              On entry with JOBV = '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 JOBV = 'U', the leading N-by-N part of this
              array contains the product of the input matrix V and the
              orthogonal matrix VT used to reorder the diagonal blocks
              of A - lambda*E.
              On exit, if JOBV = 'I', the leading N-by-N part of this
              array contains the matrix VT of the performed orthogonal
              transformations.
              Array V need not be set on entry if JOBV = 'I'.

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

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

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.  LDWORK >= 1, and if N > 1,
              LDWORK >= 4*N + 16.

              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:  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 pencil A - lambda*E with
      1-by-1 or 2-by-2 diagonal blocks, the routine reorders its
      diagonal blocks along with its eigenvalues by performing an
      orthogonal equivalence transformation UT'*(A - lambda*E)* VT.
      The column transformations UT and VT are also performed on the
      given (initial) transformations U and V (resulted from a
      possible previous step or initialized as identity matrices).
      After reordering, the generalized eigenvalues inside the region
      specified by the parameters ALPHA, DICO and STDOM appear at the
      top of the selected diagonal subpencil between positions NLOW and
      NSUP. In other words, lambda(A(Select,Select),E(Select,Select))
      are ordered such that lambda(A(Inside,Inside),E(Inside,Inside))
      are inside, and lambda(A(Outside,Outside),E(Outside,Outside)) are
      outside the domain of interest, where Select = NLOW:NSUP,
      Inside = NLOW:NLOW+NDIM-1, and Outside = NLOW+NDIM:NSUP.
      If NLOW = 1, the first NDIM columns of V*VT span the corresponding
      right deflating subspace of (A,E).

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

    *     MB03QG EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 10 )
          INTEGER          LDA, LDE, LDU, LDV
          PARAMETER        ( LDA = NMAX, LDE = NMAX, LDU = NMAX, LDV = NMAX)
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 8*NMAX + 16 )
    *     .. Local Scalars ..
          CHARACTER*1      DICO, JOBU, JOBV, STDOM
          INTEGER          I, INFO, J, N, NDIM, NLOW, NSUP
          DOUBLE PRECISION ALPHA
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), BETA(NMAX), DWORK(LDWORK),
         $                 E(LDE,NMAX), U(LDU,NMAX), V(LDV,NMAX), WI(NMAX),
         $                 WR(NMAX)
          LOGICAL          BWORK(NMAX)
    *     .. External Functions ..
          LOGICAL          DELCTG
    *     .. External Subroutines ..
          EXTERNAL         DGGES, MB03QG
    *     .. 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,
         $                      JOBV
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99990 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
    *        Compute Schur form, eigenvalues and Schur vectors.
             CALL DGGES( 'Vectors', 'Vectors', 'Not sorted', DELCTG, N,
         $               A, LDA, E, LDE, NDIM, WR, WI, BETA, U, LDU, V, LDV,
         $               DWORK, LDWORK, BWORK, INFO )
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
    *           Block reordering.
                CALL MB03QG( DICO, STDOM, JOBU, JOBV, N, NLOW, NSUP, ALPHA,
         $                   A, LDA, E, LDE, U, LDU, V, LDV, NDIM, DWORK,
         $                   LDWORK, 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 ) ( E(I,J), J = 1,N )
       20          CONTINUE
                   WRITE ( NOUT, FMT = 99992 )
                   DO 30 I = 1, N
                      WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
       30          CONTINUE
                   WRITE ( NOUT, FMT = 99991 )
                   DO 40 I = 1, N
                      WRITE ( NOUT, FMT = 99995 ) ( V(I,J), J = 1,N )
       40          CONTINUE
               END IF
             END IF
          END IF
    *
          STOP
    *
    99999 FORMAT (' MB03QG EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from DGEES  = ',I2)
    99997 FORMAT (' INFO on exit from MB03QG = ',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 ordered triangular matrix is ')
    99992 FORMAT (/' The transformation matrix U is ')
    99991 FORMAT (/' The transformation matrix V is ')
    99990 FORMAT (/' N is out of range.',/' N = ',I5)
          END

Program Data

     MB03QG EXAMPLE PROGRAM DATA
       4     1     4      0.0      C      S      U      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
       1.0   3.0   2.0  -1.0
      -2.0   5.0   3.0   2.0
       2.0   4.0   5.0   6.0
       3.0   7.0   6.0   9.0

Program Results

     MB03QG EXAMPLE PROGRAM RESULTS

     The number of eigenvalues in the domain is     2

     The ordered Schur form matrix is
              -1.4394   2.5550 -12.5655  -4.0714
               2.8887  -1.1242   9.2819  -2.6724
               0.0000   0.0000 -19.7785  36.4447
               0.0000   0.0000   0.0000   3.5537

     The ordered triangular matrix is
             -16.0178   0.0000   2.3850   4.7645
               0.0000   3.2809  -1.5640   1.9954
               0.0000   0.0000  -3.0652   0.3039
               0.0000   0.0000   0.0000   1.1671

     The transformation matrix U is
              -0.1518  -0.0737  -0.9856   0.0140
              -0.2865  -0.9466   0.1136  -0.0947
              -0.5442   0.0924   0.0887   0.8292
              -0.7738   0.3000   0.0890  -0.5508

     The transformation matrix V is
               0.2799   0.9041   0.2685   0.1794
               0.4009  -0.0714   0.3780  -0.8315
               0.7206  -0.4006   0.2628   0.5012
               0.4917   0.1306  -0.8462  -0.1588

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