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STZRQF(l)					)					STZRQF(l)

       STZRQF - routine is deprecated and has been replaced by routine STZRZF



	   REAL 	  A( LDA, * ), TAU( * )

       This routine is deprecated and has been replaced by routine STZRZF.  STZRQF reduces the M-
       by-N ( M<=N ) real upper trapezoidal matrix A to upper triangular form by means of orthog-
       onal 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.

       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) REAL 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 orthogonal 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) REAL array, dimension (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

       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 				STZRQF(l)
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