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      SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )

*

*  -- LAPACK routine (version 3.2) --

*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

*  -- Univ. of California Berkeley, Univ. of Colorado Denver 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