          SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
         $                   VN2, AUXV, F, LDF )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            KB, LDA, LDF, M, N, NB, OFFSET
    *     ..
    *     .. Array Arguments ..
          INTEGER            JPVT( * )
          DOUBLE PRECISION   A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
         $                   VN1( * ), VN2( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLAQPS computes a step of QR factorization with column pivoting
    *  of a real M-by-N matrix A by using Blas-3.  It tries to factorize
    *  NB columns from A starting from the row OFFSET+1, and updates all
    *  of the matrix with Blas-3 xGEMM.
    *
    *  In some cases, due to catastrophic cancellations, it cannot
    *  factorize NB columns.  Hence, the actual number of factorized
    *  columns is returned in KB.
    *
    *  Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
    *
    *  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
    *
    *  OFFSET  (input) INTEGER
    *          The number of rows of A that have been factorized in
    *          previous steps.
    *
    *  NB      (input) INTEGER
    *          The number of columns to factorize.
    *
    *  KB      (output) INTEGER
    *          The number of columns actually factorized.
    *
    *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
    *          On entry, the M-by-N matrix A.
    *          On exit, block A(OFFSET+1:M,1:KB) is the triangular
    *          factor obtained and block A(1:OFFSET,1:N) has been
    *          accordingly pivoted, but no factorized.
    *          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
    *          been updated.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A. LDA >= max(1,M).
    *
    *  JPVT    (input/output) INTEGER array, dimension (N)
    *          JPVT(I) = K <==> Column K of the full matrix A has been
    *          permuted into position I in AP.
    *
    *  TAU     (output) DOUBLE PRECISION array, dimension (KB)
    *          The scalar factors of the elementary reflectors.
    *
    *  VN1     (input/output) DOUBLE PRECISION array, dimension (N)
    *          The vector with the partial column norms.
    *
    *  VN2     (input/output) DOUBLE PRECISION array, dimension (N)
    *          The vector with the exact column norms.
    *
    *  AUXV    (input/output) DOUBLE PRECISION array, dimension (NB)
    *          Auxiliar vector.
    *
    *  F       (input/output) DOUBLE PRECISION array, dimension (LDF,NB)
    *          Matrix F' = L*Y'*A.
    *
    *  LDF     (input) INTEGER
    *          The leading dimension of the array F. LDF >= max(1,N).
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
    *    X. Sun, Computer Science Dept., Duke University, USA
    *
    *  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            ITEMP, J, K, LASTRK, LSTICC, PVT, RK
          DOUBLE PRECISION   AKK, TEMP, TEMP2, TOL3Z
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DGEMM, DGEMV, DLARFG, DSWAP
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, DBLE, MAX, MIN, NINT, SQRT
    *     ..
    *     .. External Functions ..
          INTEGER            IDAMAX
          DOUBLE PRECISION   DLAMCH, DNRM2
          EXTERNAL           IDAMAX, DLAMCH, DNRM2
    *     ..
    *     .. Executable Statements ..
    *
          LASTRK = MIN( M, N+OFFSET )
          LSTICC = 0
          K = 0
          TOL3Z = SQRT(DLAMCH('Epsilon'))
    *
    *     Beginning of while loop.
    *
       10 CONTINUE
          IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN
             K = K + 1
             RK = OFFSET + K
    *
    *        Determine ith pivot column and swap if necessary
    *
             PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 )
             IF( PVT.NE.K ) THEN
                CALL DSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 )
                CALL DSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF )
                ITEMP = JPVT( PVT )
                JPVT( PVT ) = JPVT( K )
                JPVT( K ) = ITEMP
                VN1( PVT ) = VN1( K )
                VN2( PVT ) = VN2( K )
             END IF
    *
    *        Apply previous Householder reflectors to column K:
    *        A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)'.
    *
             IF( K.GT.1 ) THEN
                CALL DGEMV( 'No transpose', M-RK+1, K-1, -ONE, A( RK, 1 ),
         $                  LDA, F( K, 1 ), LDF, ONE, A( RK, K ), 1 )
             END IF
    *
    *        Generate elementary reflector H(k).
    *
             IF( RK.LT.M ) THEN
                CALL DLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) )
             ELSE
                CALL DLARFG( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) )
             END IF
    *
             AKK = A( RK, K )
             A( RK, K ) = ONE
    *
    *        Compute Kth column of F:
    *
    *        Compute  F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)'*A(RK:M,K).
    *
             IF( K.LT.N ) THEN
                CALL DGEMV( 'Transpose', M-RK+1, N-K, TAU( K ),
         $                  A( RK, K+1 ), LDA, A( RK, K ), 1, ZERO,
         $                  F( K+1, K ), 1 )
             END IF
    *
    *        Padding F(1:K,K) with zeros.
    *
             DO 20 J = 1, K
                F( J, K ) = ZERO
       20    CONTINUE
    *
    *        Incremental updating of F:
    *        F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)'
    *                    *A(RK:M,K).
    *
             IF( K.GT.1 ) THEN
                CALL DGEMV( 'Transpose', M-RK+1, K-1, -TAU( K ), A( RK, 1 ),
         $                  LDA, A( RK, K ), 1, ZERO, AUXV( 1 ), 1 )
    *
                CALL DGEMV( 'No transpose', N, K-1, ONE, F( 1, 1 ), LDF,
         $                  AUXV( 1 ), 1, ONE, F( 1, K ), 1 )
             END IF
    *
    *        Update the current row of A:
    *        A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)'.
    *
             IF( K.LT.N ) THEN
                CALL DGEMV( 'No transpose', N-K, K, -ONE, F( K+1, 1 ), LDF,
         $                  A( RK, 1 ), LDA, ONE, A( RK, K+1 ), LDA )
             END IF
    *
    *        Update partial column norms.
    *
             IF( RK.LT.LASTRK ) THEN
                DO 30 J = K + 1, N
                   IF( VN1( J ).NE.ZERO ) THEN
    *
    *                 NOTE: The following 4 lines follow from the analysis in
    *                 Lapack Working Note 176.
    *
                      TEMP = ABS( A( RK, J ) ) / VN1( J )
                      TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
                      TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
                      IF( TEMP2 .LE. TOL3Z ) THEN
                         VN2( J ) = DBLE( LSTICC )
                         LSTICC = J
                      ELSE
                         VN1( J ) = VN1( J )*SQRT( TEMP )
                      END IF
                   END IF
       30       CONTINUE
             END IF
    *
             A( RK, K ) = AKK
    *
    *        End of while loop.
    *
             GO TO 10
          END IF
          KB = K
          RK = OFFSET + KB
    *
    *     Apply the block reflector to the rest of the matrix:
    *     A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) -
    *                         A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)'.
    *
          IF( KB.LT.MIN( N, M-OFFSET ) ) THEN
             CALL DGEMM( 'No transpose', 'Transpose', M-RK, N-KB, KB, -ONE,
         $               A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF, ONE,
         $               A( RK+1, KB+1 ), LDA )
          END IF
    *
    *     Recomputation of difficult columns.
    *
       40 CONTINUE
          IF( LSTICC.GT.0 ) THEN
             ITEMP = NINT( VN2( LSTICC ) )
             VN1( LSTICC ) = DNRM2( M-RK, A( RK+1, LSTICC ), 1 )
    *
    *        NOTE: The computation of VN1( LSTICC ) relies on the fact that
    *        SNRM2 does not fail on vectors with norm below the value of
    *        SQRT(DLAMCH('S'))
    *
             VN2( LSTICC ) = VN1( LSTICC )
             LSTICC = ITEMP
             GO TO 40
          END IF
    *
          RETURN
    *
    *     End of DLAQPS
    *
          END