          SUBROUTINE DTZRZF( M, N, A, LDA, 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 ..
          DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
    *  to upper triangular form by means of orthogonal transformations.
    *
    *  The upper trapezoidal matrix A is factored as
    *
    *     A = ( R  0 ) * Z,
    *
    *  where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
    *  triangular matrix.
    *
    *  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 >= M.
    *
    *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
    *          On entry, the leading M-by-N upper trapezoidal part of the
    *          array A must contain the matrix to be factorized.
    *          On exit, the leading M-by-M upper triangular part of A
    *          contains the upper triangular matrix R, and elements M+1 to
    *          N of the first M rows of A, with the array TAU, represent the
    *          orthogonal matrix Z as a product of M elementary reflectors.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,M).
    *
    *  TAU     (output) DOUBLE PRECISION array, dimension (M)
    *          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 >= max(1,M).
    *          For optimum performance LWORK >= M*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
    *  ===============
    *
    *  Based on contributions by
    *    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
    *
    *  The factorization is obtained by Householder's method.  The kth
    *  transformation matrix, Z( k ), which is used to introduce zeros into
    *  the ( m - k + 1 )th row of A, is given in the form
    *
    *     Z( k ) = ( I     0   ),
    *              ( 0  T( k ) )
    *
    *  where
    *
    *     T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
    *                                                 (   0    )
    *                                                 ( z( k ) )
    *
    *  tau is a scalar and z( k ) is an ( n - m ) element vector.
    *  tau and z( k ) are chosen to annihilate the elements of the kth row
    *  of X.
    *
    *  The scalar tau is returned in the kth element of TAU and the vector
    *  u( k ) in the kth row of A, such that the elements of z( k ) are
    *  in  a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
    *  the upper triangular part of A.
    *
    *  Z is given by
    *
    *     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO
          PARAMETER          ( ZERO = 0.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            LQUERY
          INTEGER            I, IB, IWS, KI, KK, LDWORK, LWKOPT, M1, MU, NB,
         $                   NBMIN, NX
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DLARZB, DLARZT, DLATRZ, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN
    *     ..
    *     .. External Functions ..
          INTEGER            ILAENV
          EXTERNAL           ILAENV
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input arguments
    *
          INFO = 0
          LQUERY = ( LWORK.EQ.-1 )
          IF( M.LT.0 ) THEN
             INFO = -1
          ELSE IF( N.LT.M ) THEN
             INFO = -2
          ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
             INFO = -4
          END IF
    *
          IF( INFO.EQ.0 ) THEN
             IF( M.EQ.0 .OR. M.EQ.N ) THEN
                LWKOPT = 1
             ELSE
    *
    *           Determine the block size.
    *
                NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
                LWKOPT = M*NB
             END IF
             WORK( 1 ) = LWKOPT
    *
             IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
                INFO = -7
             END IF
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DTZRZF', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( M.EQ.0 ) THEN
             RETURN
          ELSE IF( M.EQ.N ) THEN
             DO 10 I = 1, N
                TAU( I ) = ZERO
       10    CONTINUE
             RETURN
          END IF
    *
          NBMIN = 2
          NX = 1
          IWS = M
          IF( NB.GT.1 .AND. NB.LT.M ) THEN
    *
    *        Determine when to cross over from blocked to unblocked code.
    *
             NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) )
             IF( NX.LT.M ) THEN
    *
    *           Determine if workspace is large enough for blocked code.
    *
                LDWORK = M
                IWS = LDWORK*NB
                IF( LWORK.LT.IWS ) THEN
    *
    *              Not enough workspace to use optimal NB:  reduce NB and
    *              determine the minimum value of NB.
    *
                   NB = LWORK / LDWORK
                   NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1,
         $                 -1 ) )
                END IF
             END IF
          END IF
    *
          IF( NB.GE.NBMIN .AND. NB.LT.M .AND. NX.LT.M ) THEN
    *
    *        Use blocked code initially.
    *        The last kk rows are handled by the block method.
    *
             M1 = MIN( M+1, N )
             KI = ( ( M-NX-1 ) / NB )*NB
             KK = MIN( M, KI+NB )
    *
             DO 20 I = M - KK + KI + 1, M - KK + 1, -NB
                IB = MIN( M-I+1, NB )
    *
    *           Compute the TZ factorization of the current block
    *           A(i:i+ib-1,i:n)
    *
                CALL DLATRZ( IB, N-I+1, N-M, A( I, I ), LDA, TAU( I ),
         $                   WORK )
                IF( I.GT.1 ) THEN
    *
    *              Form the triangular factor of the block reflector
    *              H = H(i+ib-1) . . . H(i+1) H(i)
    *
                   CALL DLARZT( 'Backward', 'Rowwise', N-M, IB, A( I, M1 ),
         $                      LDA, TAU( I ), WORK, LDWORK )
    *
    *              Apply H to A(1:i-1,i:n) from the right
    *
                   CALL DLARZB( 'Right', 'No transpose', 'Backward',
         $                      'Rowwise', I-1, N-I+1, IB, N-M, A( I, M1 ),
         $                      LDA, WORK, LDWORK, A( 1, I ), LDA,
         $                      WORK( IB+1 ), LDWORK )
                END IF
       20    CONTINUE
             MU = I + NB - 1
          ELSE
             MU = M
          END IF
    *
    *     Use unblocked code to factor the last or only block
    *
          IF( MU.GT.0 )
         $   CALL DLATRZ( MU, N, N-M, A, LDA, TAU, WORK )
    *
          WORK( 1 ) = LWKOPT
    *
          RETURN
    *
    *     End of DTZRZF
    *
          END