```      SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
INTEGER            INFO, LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
INTEGER            JPVT( * )
DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGEQP3 computes a QR factorization with column pivoting of a
*  matrix A:  A*P = Q*R  using Level 3 BLAS.
*
*  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 trapezoidal 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(J).ne.0, the J-th column of A is permuted
*          to the front of A*P (a leading column); if JPVT(J)=0,
*          the J-th column of A is a free column.
*          On exit, if JPVT(J)=K, then the J-th column of A*P was the
*          the K-th column of A.
*
*  TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
*          The scalar factors of the elementary reflectors.
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*          On exit, if INFO=0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK. LWORK >= 3*N+1.
*          For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
*          is the optimal blocksize.
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  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(k), where k = min(m,n).
*
*  Each H(i) has the form
*
*     H(i) = I - tau * v * v'
*
*  where tau is a real/complex scalar, and v is a real/complex 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), and tau in TAU(i).
*
*  Based on contributions by
*    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*    X. Sun, Computer Science Dept., Duke University, USA
*
*  =====================================================================
*
*     .. Parameters ..
INTEGER            INB, INBMIN, IXOVER
PARAMETER          ( INB = 1, INBMIN = 2, IXOVER = 3 )
*     ..
*     .. Local Scalars ..
LOGICAL            LQUERY
INTEGER            FJB, IWS, J, JB, LWKOPT, MINMN, MINWS, NA, NB,
\$                   NBMIN, NFXD, NX, SM, SMINMN, SN, TOPBMN
*     ..
*     .. External Subroutines ..
EXTERNAL           DGEQRF, DLAQP2, DLAQPS, DORMQR, DSWAP, XERBLA
*     ..
*     .. External Functions ..
INTEGER            ILAENV
DOUBLE PRECISION   DNRM2
EXTERNAL           ILAENV, DNRM2
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          INT, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test input arguments
*     ====================
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
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.EQ.0 ) THEN
MINMN = MIN( M, N )
IF( MINMN.EQ.0 ) THEN
IWS = 1
LWKOPT = 1
ELSE
IWS = 3*N + 1
NB = ILAENV( INB, 'DGEQRF', ' ', M, N, -1, -1 )
LWKOPT = 2*N + ( N + 1 )*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( ( LWORK.LT.IWS ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQP3', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
*     Quick return if possible.
*
IF( MINMN.EQ.0 ) THEN
RETURN
END IF
*
*     Move initial columns up front.
*
NFXD = 1
DO 10 J = 1, N
IF( JPVT( J ).NE.0 ) THEN
IF( J.NE.NFXD ) THEN
CALL DSWAP( M, A( 1, J ), 1, A( 1, NFXD ), 1 )
JPVT( J ) = JPVT( NFXD )
JPVT( NFXD ) = J
ELSE
JPVT( J ) = J
END IF
NFXD = NFXD + 1
ELSE
JPVT( J ) = J
END IF
10 CONTINUE
NFXD = NFXD - 1
*
*     Factorize fixed columns
*     =======================
*
*     Compute the QR factorization of fixed columns and update
*     remaining columns.
*
IF( NFXD.GT.0 ) THEN
NA = MIN( M, NFXD )
*CC      CALL DGEQR2( M, NA, A, LDA, TAU, WORK, INFO )
CALL DGEQRF( M, NA, A, LDA, TAU, WORK, LWORK, INFO )
IWS = MAX( IWS, INT( WORK( 1 ) ) )
IF( NA.LT.N ) THEN
*CC         CALL DORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA,
*CC  \$                   TAU, A( 1, NA+1 ), LDA, WORK, INFO )
CALL DORMQR( 'Left', 'Transpose', M, N-NA, NA, A, LDA, TAU,
\$                   A( 1, NA+1 ), LDA, WORK, LWORK, INFO )
IWS = MAX( IWS, INT( WORK( 1 ) ) )
END IF
END IF
*
*     Factorize free columns
*     ======================
*
IF( NFXD.LT.MINMN ) THEN
*
SM = M - NFXD
SN = N - NFXD
SMINMN = MINMN - NFXD
*
*        Determine the block size.
*
NB = ILAENV( INB, 'DGEQRF', ' ', SM, SN, -1, -1 )
NBMIN = 2
NX = 0
*
IF( ( NB.GT.1 ) .AND. ( NB.LT.SMINMN ) ) THEN
*
*           Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( IXOVER, 'DGEQRF', ' ', SM, SN, -1,
\$           -1 ) )
*
*
IF( NX.LT.SMINMN ) THEN
*
*              Determine if workspace is large enough for blocked code.
*
MINWS = 2*SN + ( SN+1 )*NB
IWS = MAX( IWS, MINWS )
IF( LWORK.LT.MINWS ) THEN
*
*                 Not enough workspace to use optimal NB: Reduce NB and
*                 determine the minimum value of NB.
*
NB = ( LWORK-2*SN ) / ( SN+1 )
NBMIN = MAX( 2, ILAENV( INBMIN, 'DGEQRF', ' ', SM, SN,
\$                    -1, -1 ) )
*
*
END IF
END IF
END IF
*
*        Initialize partial column norms. The first N elements of work
*        store the exact column norms.
*
DO 20 J = NFXD + 1, N
WORK( J ) = DNRM2( SM, A( NFXD+1, J ), 1 )
WORK( N+J ) = WORK( J )
20    CONTINUE
*
IF( ( NB.GE.NBMIN ) .AND. ( NB.LT.SMINMN ) .AND.
\$       ( NX.LT.SMINMN ) ) THEN
*
*           Use blocked code initially.
*
J = NFXD + 1
*
*           Compute factorization: while loop.
*
*
TOPBMN = MINMN - NX
30       CONTINUE
IF( J.LE.TOPBMN ) THEN
JB = MIN( NB, TOPBMN-J+1 )
*
*              Factorize JB columns among columns J:N.
*
CALL DLAQPS( M, N-J+1, J-1, JB, FJB, A( 1, J ), LDA,
\$                      JPVT( J ), TAU( J ), WORK( J ), WORK( N+J ),
\$                      WORK( 2*N+1 ), WORK( 2*N+JB+1 ), N-J+1 )
*
J = J + FJB
GO TO 30
END IF
ELSE
J = NFXD + 1
END IF
*
*        Use unblocked code to factor the last or only block.
*
*
IF( J.LE.MINMN )
\$      CALL DLAQP2( M, N-J+1, J-1, A( 1, J ), LDA, JPVT( J ),
\$                   TAU( J ), WORK( J ), WORK( N+J ),
\$                   WORK( 2*N+1 ) )
*
END IF
*
WORK( 1 ) = IWS
RETURN
*
*     End of DGEQP3
*
END

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