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ZTZRZF(l)					)					ZTZRZF(l)

       ZTZRZF - reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix A to upper triangular
       form by means of unitary transformations



	   COMPLEX*16	  A( LDA, * ), TAU( * ), WORK( * )

       ZTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A to upper  triangular
       form by means of unitary transformations.  The upper trapezoidal matrix A is factored as

	  A = ( R  0 ) * Z,

       where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular matrix.

       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	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 unitary 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) COMPLEX*16 array, dimension (M)
	       The scalar factors of the elementary reflectors.

       WORK    (workspace/output) COMPLEX*16 array, dimension (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

       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

	  Z( k ) = ( I	   0   ),
		   ( 0	T( k ) )


	  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 ).

LAPACK version 3.0			   15 June 2000 				ZTZRZF(l)
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