dgeqpf.f

          SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO )
    *
    *  -- LAPACK deprecated driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, LDA, M, N
    *     ..
    *     .. Array Arguments ..
          INTEGER            JPVT( * )
          DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  This routine is deprecated and has been replaced by routine DGEQP3.
    *
    *  DGEQPF computes a QR factorization with column pivoting of a
    *  real M-by-N matrix A: A*P = Q*R.
    *
    *  Arguments
    *  =========
    *
    *  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, the upper triangle of the array contains the
    *          min(M,N)-by-N upper triangular matrix R; the elements
    *          below the diagonal, together with the array TAU,
    *          represent the orthogonal matrix Q as a product of
    *          min(m,n) elementary reflectors.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A. LDA >= max(1,M).
    *
    *  JPVT    (input/output) INTEGER array, dimension (N)
    *          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
    *          to the front of A*P (a leading column); if JPVT(i) = 0,
    *          the i-th column of A is a free column.
    *          On exit, if JPVT(i) = k, then the i-th column of A*P
    *          was the k-th column of A.
    *
    *  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
    *          The scalar factors of the elementary reflectors.
    *
    *  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *
    *  Further Details
    *  ===============
    *
    *  The matrix Q is represented as a product of elementary reflectors
    *
    *     Q = H(1) H(2) . . . H(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(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).
    *
    *  The matrix P is represented in jpvt as follows: If
    *     jpvt(j) = i
    *  then the jth column of P is the ith canonical unit vector.
    *
    *  Partial column norm updating strategy modified by
    *    Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
    *    University of Zagreb, Croatia.
    *    June 2006.
    *  For more details see LAPACK Working Note 176.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, ITEMP, J, MA, MN, PVT
          DOUBLE PRECISION   AII, TEMP, TEMP2, TOL3Z
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DGEQR2, DLARF, DLARFG, DORM2R, DSWAP, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX, MIN, SQRT
    *     ..
    *     .. External Functions ..
          INTEGER            IDAMAX
          DOUBLE PRECISION   DLAMCH, DNRM2
          EXTERNAL           IDAMAX, DLAMCH, DNRM2
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input arguments
    *
          INFO = 0
          IF( M.LT.0 ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
             INFO = -4
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DGEQPF', -INFO )
             RETURN
          END IF
    *
          MN = MIN( M, N )
          TOL3Z = SQRT(DLAMCH('Epsilon'))
    *
    *     Move initial columns up front
    *
          ITEMP = 1
          DO 10 I = 1, N
             IF( JPVT( I ).NE.0 ) THEN
                IF( I.NE.ITEMP ) THEN
                   CALL DSWAP( M, A( 1, I ), 1, A( 1, ITEMP ), 1 )
                   JPVT( I ) = JPVT( ITEMP )
                   JPVT( ITEMP ) = I
                ELSE
                   JPVT( I ) = I
                END IF
                ITEMP = ITEMP + 1
             ELSE
                JPVT( I ) = I
             END IF
       10 CONTINUE
          ITEMP = ITEMP - 1
    *
    *     Compute the QR factorization and update remaining columns
    *
          IF( ITEMP.GT.0 ) THEN
             MA = MIN( ITEMP, M )
             CALL DGEQR2( M, MA, A, LDA, TAU, WORK, INFO )
             IF( MA.LT.N ) THEN
                CALL DORM2R( 'Left', 'Transpose', M, N-MA, MA, A, LDA, TAU,
         $                   A( 1, MA+1 ), LDA, WORK, INFO )
             END IF
          END IF
    *
          IF( ITEMP.LT.MN ) THEN
    *
    *        Initialize partial column norms. The first n elements of
    *        work store the exact column norms.
    *
             DO 20 I = ITEMP + 1, N
                WORK( I ) = DNRM2( M-ITEMP, A( ITEMP+1, I ), 1 )
                WORK( N+I ) = WORK( I )
       20    CONTINUE
    *
    *        Compute factorization
    *
             DO 40 I = ITEMP + 1, MN
    *
    *           Determine ith pivot column and swap if necessary
    *
                PVT = ( I-1 ) + IDAMAX( N-I+1, WORK( I ), 1 )
    *
                IF( PVT.NE.I ) THEN
                   CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
                   ITEMP = JPVT( PVT )
                   JPVT( PVT ) = JPVT( I )
                   JPVT( I ) = ITEMP
                   WORK( PVT ) = WORK( I )
                   WORK( N+PVT ) = WORK( N+I )
                END IF
    *
    *           Generate elementary reflector H(i)
    *
                IF( I.LT.M ) THEN
                   CALL DLARFG( M-I+1, A( I, I ), A( I+1, I ), 1, TAU( I ) )
                ELSE
                   CALL DLARFG( 1, A( M, M ), A( M, M ), 1, TAU( M ) )
                END IF
    *
                IF( I.LT.N ) THEN
    *
    *              Apply H(i) to A(i:m,i+1:n) from the left
    *
                   AII = A( I, I )
                   A( I, I ) = ONE
                   CALL DLARF( 'LEFT', M-I+1, N-I, A( I, I ), 1, TAU( I ),
         $                     A( I, I+1 ), LDA, WORK( 2*N+1 ) )
                   A( I, I ) = AII
                END IF
    *
    *           Update partial column norms
    *
                DO 30 J = I + 1, N
                   IF( WORK( J ).NE.ZERO ) THEN
    *
    *                 NOTE: The following 4 lines follow from the analysis in
    *                 Lapack Working Note 176.
    *
                      TEMP = ABS( A( I, J ) ) / WORK( J )
                      TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
                      TEMP2 = TEMP*( WORK( J ) / WORK( N+J ) )**2
                      IF( TEMP2 .LE. TOL3Z ) THEN
                         IF( M-I.GT.0 ) THEN
                            WORK( J ) = DNRM2( M-I, A( I+1, J ), 1 )
                            WORK( N+J ) = WORK( J )
                         ELSE
                            WORK( J ) = ZERO
                            WORK( N+J ) = ZERO
                         END IF
                      ELSE
                         WORK( J ) = WORK( J )*SQRT( TEMP )
                      END IF
                   END IF
       30       CONTINUE
    *
       40    CONTINUE
          END IF
          RETURN
    *
    *     End of DGEQPF
    *
          END