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CTZRZF(1)		 LAPACK routine (version 3.2)		     CTZRZF(1)

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



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

       CTZRZF 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

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

       WORK    (workspace/output) COMPLEX 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 opti‐
	       mum  performance	 LWORK >= M*NB, where NB is the optimal block‐
	       size.  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	trans‐
       formation 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 ) )
	  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 routine (version 3.2)	 November 2008			     CTZRZF(1)

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