DTZRQF(l)		LAPACK routine (version	1.1)		    DTZRQF(l)

NAME
  DTZRQF - reduce the M-by-N ( M<=N ) real upper trapezoidal matrix A to
  upper	triangular form	by means of orthogonal transformations

SYNOPSIS

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

      INTEGER	     INFO, LDA,	M, N

      DOUBLE	     PRECISION A( LDA, * ), TAU( * )

PURPOSE
  DTZRQF 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,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 orthogonal matrix Z as a	pro-
	  duct of M elementary reflectors.

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

  TAU	  (output) DOUBLE PRECISION 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 transforma-
  tion 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 ).