## MB04GD

### RQ factorization with row pivoting of a matrix

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

Purpose

      To compute an RQ factorization with row pivoting of a
      real m-by-n matrix A: P*A = R*Q.

Specification

          SUBROUTINE MB04GD( M, N, A, LDA, JPVT, TAU, DWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, LDA, M, N
    C     .. Array Arguments ..
          INTEGER            JPVT( * )
          DOUBLE PRECISION   A( LDA, * ), DWORK( * ), TAU( * )

Arguments

Input/Output Parameters

      M       (input) INTEGER
              The number of rows of the matrix A.  M >= 0.

      N       (input) INTEGER
              The number of columns of the matrix A.  N >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the m-by-n matrix A.
              On exit,
              if m <= n, the upper triangle of the subarray
              A(1:m,n-m+1:n) contains the m-by-m upper triangular
              matrix R;
              if m >= n, the elements on and above the (m-n)-th
              subdiagonal contain the m-by-n upper trapezoidal matrix R;
              the remaining elements, with the array TAU, represent the
              orthogonal matrix Q as a product of min(m,n) elementary
              reflectors (see METHOD).

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

      JPVT    (input/output) INTEGER array, dimension (M)
              On entry, if JPVT(i) .ne. 0, the i-th row of A is permuted
              to the bottom of P*A (a trailing row); if JPVT(i) = 0,
              the i-th row of A is a free row.
              On exit, if JPVT(i) = k, then the i-th row of P*A
              was the k-th row of A.

      TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
              The scalar factors of the elementary reflectors.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (3*M)

Error Indicator

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

Method

      The matrix Q is represented as a product of elementary reflectors

         Q = H(1) H(2) . . . H(k), where k = min(m,n).

      Each H(i) has the form

         H = I - tau * v * v'

      where tau is a real scalar, and v is a real vector with
      v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit
      in A(m-k+i,1:n-k+i-1), and tau in TAU(i).

      The matrix P is represented in jpvt as follows: If
         jpvt(j) = i
      then the jth row of P is the ith canonical unit vector.

References

      [1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
          Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
          Ostrouchov, S., and Sorensen, D.
          LAPACK Users' Guide: Second Edition.
          SIAM, Philadelphia, 1995.

Numerical Aspects

      The algorithm is backward stable.

Further Comments

      None

Example

Program Text

    *     MB04GD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          DOUBLE PRECISION ZERO
          PARAMETER        ( ZERO = 0.0D0 )
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX
          PARAMETER        ( NMAX = 10, MMAX = 10 )
          INTEGER          LDA
          PARAMETER        ( LDA = MMAX )
          INTEGER          LDTAU
          PARAMETER        ( LDTAU = MIN(MMAX,NMAX) )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 3*MMAX )
    *     .. Local Scalars ..
          INTEGER          I, INFO, J, M, N
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), TAU(LDTAU)
          INTEGER          JPVT(MMAX)
    *     .. External Subroutines ..
          EXTERNAL         DLASET, MB04GD
    *     .. Intrinsic Functions ..
          INTRINSIC        MIN
    *     .. 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
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99972 ) N
          ELSE
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99971 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
                READ ( NIN, FMT = * ) ( JPVT(I), I = 1,M )
    *           RQ with row pivoting.
                CALL MB04GD( M, N, A, LDA, JPVT, TAU, DWORK, INFO )
    *
                IF ( INFO.NE.0 ) THEN
                   WRITE ( NOUT, FMT = 99998 ) INFO
                ELSE
                   WRITE ( NOUT, FMT = 99994 ) ( JPVT(I), I = 1,M )
                   WRITE ( NOUT, FMT = 99990 )
                   IF ( M.GE.N ) THEN
                      IF ( N.GT.1 )
         $               CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO,
         $                            A(M-N+2,1), LDA )
                    ELSE
                       CALL DLASET( 'Full', M, N-M-1, ZERO, ZERO, A, LDA )
                       CALL DLASET( 'Lower', M, M, ZERO, ZERO, A(1,N-M),
         $                          LDA )
                   END IF
                   DO 20 I = 1, M
                      WRITE ( NOUT, FMT = 99989 ) ( A(I,J), J = 1,N )
       20          CONTINUE
                END IF
             END IF
          END IF
    *
          STOP
    *
    99999 FORMAT (' MB04GD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB04GD = ',I2)
    99994 FORMAT (' Row permutations are ',/(20(I3,2X)))
    99990 FORMAT (/' The matrix A is ')
    99989 FORMAT (20(1X,F8.4))
    99972 FORMAT (/' N is out of range.',/' N = ',I5)
    99971 FORMAT (/' M is out of range.',/' M = ',I5)
          END

Program Data

     MB04GD EXAMPLE PROGRAM DATA
       6     5
       1.    2.    6.    3.    5.
      -2.   -1.   -1.    0.   -2.
       5.    5.    1.    5.    1.
      -2.   -1.   -1.    0.   -2.
       4.    8.    4.   20.    4.
      -2.   -1.   -1.    0.   -2.
       0     0     0     0     0     0

Program Results

     MB04GD EXAMPLE PROGRAM RESULTS

     Row permutations are
      2    4    6    3    1    5

     The matrix A is
       0.0000  -1.0517  -1.8646  -1.9712   1.2374
       0.0000  -1.0517  -1.8646  -1.9712   1.2374
       0.0000  -1.0517  -1.8646  -1.9712   1.2374
       0.0000   0.0000   4.6768   0.0466  -7.4246
       0.0000   0.0000   0.0000   6.7059  -5.4801
       0.0000   0.0000   0.0000   0.0000 -22.6274

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