ZTZRQF(l)		LAPACK routine (version	1.1)		    ZTZRQF(l)

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

SYNOPSIS

  SUBROUTINE ZTZRQF( M,	N, A, LDA, TAU,	INFO )

      INTEGER	     INFO, LDA,	M, N

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

PURPOSE
  ZTZRQF 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.

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) COMPLEX*16 array, dimension (LDA,max(1,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 elemen-
	  tary reflectors.

  LDA	  (input) INTEGER
	  The leading dimension	of the array A.	 LDA >=	max(1,M).

  TAU	  (output) COMPLEX*16 array, dimension (max(1,M))
	  The scalar factors of	the elementary reflectors.

  INFO	  (output) INTEGER
	  = 0: successful exit
	  < 0: if INFO = -i, the i-th argument had an illegal value

FURTHER	DETAILS
  The  factorization is	obtained by Householder's method.  The kth transfor-
  mation matrix, Z( k ), whose conjugate transpose 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 ).