## MB04UD

### Column echelon form by unitary transformations for a rectangular matrix (added functionality)

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

Purpose

      To compute orthogonal transformations Q and Z such that the
      transformed pencil Q'(sE-A)Z has the E matrix in column echelon
      form, where E and A are M-by-N matrices.

Specification

          SUBROUTINE MB04UD( JOBQ, JOBZ, M, N, A, LDA, E, LDE, Q, LDQ,
         $                   Z, LDZ, RANKE, ISTAIR, TOL, DWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         JOBQ, JOBZ
          INTEGER           INFO, LDA, LDE, LDQ, LDZ, M, N, RANKE
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          INTEGER           ISTAIR(*)
          DOUBLE PRECISION  A(LDA,*), DWORK(*), E(LDE,*), Q(LDQ,*), Z(LDZ,*)

Arguments

Mode Parameters

      JOBQ    CHARACTER*1
              Indicates whether the user wishes to accumulate in a
              matrix Q the unitary row permutations, as follows:
              = 'N':  Do not form Q;
              = 'I':  Q is initialized to the unit matrix and the
                      unitary row permutation matrix Q is returned;
              = 'U':  The given matrix Q is updated by the unitary
                      row permutations used in the reduction.

      JOBZ    CHARACTER*1
              Indicates whether the user wishes to accumulate in a
              matrix Z the unitary column transformations, as follows:
              = 'N':  Do not form Z;
              = 'I':  Z is initialized to the unit matrix and the
                      unitary transformation matrix Z is returned;
              = 'U':  The given matrix Z is updated by the unitary
                      transformations used in the reduction.

Input/Output Parameters

      M       (input) INTEGER
              The number of rows in the matrices A, E and the order of
              the matrix Q.  M >= 0.

      N       (input) INTEGER
              The number of columns in the matrices A, E and the order
              of the matrix Z.  N >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading M-by-N part of this array must
              contain the A matrix of the pencil sE-A.
              On exit, the leading M-by-N part of this array contains
              the unitary transformed matrix Q' * A * Z.

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

      E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
              On entry, the leading M-by-N part of this array must
              contain the E matrix of the pencil sE-A, to be reduced to
              column echelon form.
              On exit, the leading M-by-N part of this array contains
              the unitary transformed matrix Q' * E * Z, which is in
              column echelon form.

      LDE     INTEGER
              The leading dimension of array E.  LDE >= MAX(1,M).

      Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,*)
              On entry, if JOBQ = 'U', then the leading M-by-M part of
              this array must contain a given matrix Q (e.g. from a
              previous call to another SLICOT routine), and on exit, the
              leading M-by-M part of this array contains the product of
              the input matrix Q and the row permutation matrix used to
              transform the rows of matrix E.
              On exit, if JOBQ = 'I', then the leading M-by-M part of
              this array contains the matrix of accumulated unitary
              row transformations performed.
              If JOBQ = 'N', the array Q is not referenced and can be
              supplied as a dummy array (i.e. set parameter LDQ = 1 and
              declare this array to be Q(1,1) in the calling program).

      LDQ     INTEGER
              The leading dimension of array Q. If JOBQ = 'U' or
              JOBQ = 'I', LDQ >= MAX(1,M); if JOBQ = 'N', LDQ >= 1.

      Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,*)
              On entry, if JOBZ = 'U', then the leading N-by-N part of
              this array must contain a given matrix Z (e.g. from a
              previous call to another SLICOT routine), and on exit, the
              leading N-by-N part of this array contains the product of
              the input matrix Z and the column transformation matrix
              used to transform the columns of matrix E.
              On exit, if JOBZ = 'I', then the leading N-by-N part of
              this array contains the matrix of accumulated unitary
              column transformations performed.
              If JOBZ = 'N', the array Z is not referenced and can be
              supplied as a dummy array (i.e. set parameter LDZ = 1 and
              declare this array to be Z(1,1) in the calling program).

      LDZ     INTEGER
              The leading dimension of array Z. If JOBZ = 'U' or
              JOBZ = 'I', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.

      RANKE   (output) INTEGER
              The computed rank of the unitary transformed matrix E.

      ISTAIR  (output) INTEGER array, dimension (M)
              This array contains information on the column echelon form
              of the unitary transformed matrix E. Specifically,
              ISTAIR(i) = +j if the first non-zero element E(i,j)
              is a corner point and -j otherwise, for i = 1,2,...,M.

Tolerances

      TOL     DOUBLE PRECISION
              A tolerance below which matrix elements are considered
              to be zero. If the user sets TOL to be less than (or
              equal to) zero then the tolerance is taken as
              EPS * MAX(ABS(E(I,J))), where EPS is the machine
              precision (see LAPACK Library routine DLAMCH),
              I = 1,2,...,M and J = 1,2,...,N.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (MAX(M,N))

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      Given an M-by-N matrix pencil sE-A with E not necessarily regular,
      the routine computes a unitary transformed pencil Q'(sE-A)Z such
      that the matrix Q' * E * Z is in column echelon form (trapezoidal
      form).  Further details can be found in [1].

      [An M-by-N matrix E with rank(E) = r is said to be in column
      echelon form if the following conditions are satisfied:
      (a) the first (N - r) columns contain only zero elements; and
      (b) if E(i(k),k) is the last nonzero element in column k for
          k = N-r+1,...,N, i.e. E(i(k),k) <> 0 and E(j,k) = 0 for
          j > i(k), then 1 <= i(N-r+1) < i(N-r+2) < ... < i(N) <= M.]

References

      [1] Beelen, Th. and Van Dooren, P.
          An improved algorithm for the computation of Kronecker's
          canonical form of a singular pencil.
          Linear Algebra and Applications, 105, pp. 9-65, 1988.

Numerical Aspects

      It is shown in [1] that the algorithm is numerically backward
      stable. The operations count is proportional to (MAX(M,N))**3.

Further Comments

      None

Example

Program Text

    *     MB04UD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          MMAX, NMAX
          PARAMETER        ( MMAX = 20, NMAX = 20 )
          INTEGER          LDA, LDE, LDQ, LDZ
          PARAMETER        ( LDA = MMAX, LDE = MMAX, LDQ = MMAX,
         $                   LDZ = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX( NMAX,MMAX ) )
    *     PARAMETER        ( LDWORK = NMAX+MMAX )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          I, INFO, J, M, N, RANKE
          CHARACTER*1      JOBQ, JOBZ
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), E(LDE,NMAX),
         $                 Q(LDQ,MMAX), Z(LDZ,NMAX)
          INTEGER          ISTAIR(MMAX)
    *     .. External Subroutines ..
          EXTERNAL         MB04UD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) M, N, TOL
          IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
             WRITE ( NOUT, FMT = 99993 ) M
          ELSE 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,M )
             READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,M )
             JOBQ = 'N'
             JOBZ = 'N'
    *        Reduce E to column echelon form and compute Q'*A*Z.
             CALL MB04UD( JOBQ, JOBZ, M, N, A, LDA, E, LDE, Q, LDQ, Z, LDZ,
         $                RANKE, ISTAIR, TOL, DWORK, INFO )
    *
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                WRITE ( NOUT, FMT = 99991 )
                DO 10 I = 1, M
                   WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
       10       CONTINUE
                WRITE ( NOUT, FMT = 99997 )
                DO 100 I = 1, M
                   WRITE ( NOUT, FMT = 99996 ) ( E(I,J), J = 1,N )
      100       CONTINUE
                WRITE ( NOUT, FMT = 99995 ) RANKE
                WRITE ( NOUT, FMT = 99994 ) ( ISTAIR(I), I = 1,M )
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' MB04UD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB04UD = ',I2)
    99997 FORMAT (' The transformed matrix E is ')
    99996 FORMAT (20(1X,F8.4))
    99995 FORMAT (/' The computed rank of E = ',I2)
    99994 FORMAT (/' ISTAIR is ',/20(1X,I5))
    99993 FORMAT (/' M is out of range.',/' M = ',I5)
    99992 FORMAT (/' N is out of range.',/' N = ',I5)
    99991 FORMAT (' The transformed matrix A is ')
          END

Program Data

     MB04UD EXAMPLE PROGRAM DATA
       4     4     0.0
       2.0  0.0  2.0 -2.0
       0.0 -2.0  0.0  2.0
       2.0  0.0 -2.0  0.0
       2.0 -2.0  0.0  2.0
       1.0  0.0  1.0 -1.0
       0.0 -1.0  0.0  1.0
       1.0  0.0 -1.0  0.0
       1.0 -1.0  0.0  1.0

Program Results

     MB04UD EXAMPLE PROGRAM RESULTS

     The transformed matrix A is
       0.5164   1.0328   1.1547  -2.3094
       0.0000  -2.5820   0.0000  -1.1547
       0.0000   0.0000  -3.4641   0.0000
       0.0000   0.0000   0.0000  -3.4641
     The transformed matrix E is
       0.2582   0.5164   0.5774  -1.1547
       0.0000  -1.2910   0.0000  -0.5774
       0.0000   0.0000  -1.7321   0.0000
       0.0000   0.0000   0.0000  -1.7321

     The computed rank of E =  4

     ISTAIR is
         1     2     3     4

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