```      SUBROUTINE ZGERQF( 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 ..
COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  ZGERQF computes an RQ factorization of a complex M-by-N matrix A:
*  A = R * Q.
*
*  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) COMPLEX*16 array, dimension (LDA,N)
*          On entry, the M-by-N matrix A.
*          On exit,
*          if m <= n, the upper triangle of the subarray
*          A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
*          if m >= n, the elements on and above the (m-n)-th subdiagonal
*          contain the M-by-N upper trapezoidal matrix R;
*          the remaining elements, with the array TAU, represent the
*          unitary matrix Q as a product of min(m,n) elementary
*          reflectors (see Further Details).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,M).
*
*  TAU     (output) COMPLEX*16 array, dimension (min(M,N))
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  WORK    (workspace/output) COMPLEX*16 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
*  ===============
*
*  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 complex scalar, and v is a complex vector with
*  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on
*  exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i).
*
*  =====================================================================
*
*     .. Local Scalars ..
LOGICAL            LQUERY
INTEGER            I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
\$                   MU, NB, NBMIN, NU, NX
*     ..
*     .. External Subroutines ..
EXTERNAL           XERBLA, ZGERQ2, ZLARFB, ZLARFT
*     ..
*     .. 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.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
*
IF( INFO.EQ.0 ) THEN
K = MIN( M, N )
IF( K.EQ.0 ) THEN
LWKOPT = 1
ELSE
NB = ILAENV( 1, 'ZGERQF', ' ', 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( 'ZGERQF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
*     Quick return if possible
*
IF( K.EQ.0 ) THEN
RETURN
END IF
*
NBMIN = 2
NX = 1
IWS = M
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
*        Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'ZGERQF', ' ', M, N, -1, -1 ) )
IF( NX.LT.K ) 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, 'ZGERQF', ' ', M, N, -1,
\$                 -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
*        Use blocked code initially.
*        The last kk rows are handled by the block method.
*
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
*
DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
IB = MIN( K-I+1, NB )
*
*           Compute the RQ factorization of the current block
*           A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1)
*
CALL ZGERQ2( IB, N-K+I+IB-1, A( M-K+I, 1 ), LDA, TAU( I ),
\$                   WORK, IINFO )
IF( M-K+I.GT.1 ) THEN
*
*              Form the triangular factor of the block reflector
*              H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL ZLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB,
\$                      A( M-K+I, 1 ), LDA, TAU( I ), WORK, LDWORK )
*
*              Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
*
CALL ZLARFB( 'Right', 'No transpose', 'Backward',
\$                      'Rowwise', M-K+I-1, N-K+I+IB-1, IB,
\$                      A( M-K+I, 1 ), LDA, WORK, LDWORK, A, LDA,
\$                      WORK( IB+1 ), LDWORK )
END IF
10    CONTINUE
MU = M - K + I + NB - 1
NU = N - K + I + NB - 1
ELSE
MU = M
NU = N
END IF
*
*     Use unblocked code to factor the last or only block
*
IF( MU.GT.0 .AND. NU.GT.0 )
\$   CALL ZGERQ2( MU, NU, A, LDA, TAU, WORK, IINFO )
*
WORK( 1 ) = IWS
RETURN
*
*     End of ZGERQF
*
END

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