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dtzrqf.f(3)				      LAPACK				      dtzrqf.f(3)

       dtzrqf.f -

       subroutine dtzrqf (M, N, A, LDA, TAU, INFO)

Function/Subroutine Documentation
   subroutine dtzrqf (integerM, integerN, double precision, dimension( lda, * )A, integerLDA,
       double precision, dimension( * )TAU, integerINFO)


	    This routine is deprecated and has been replaced by routine DTZRZF.

	    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.


		     M is INTEGER
		     The number of rows of the matrix A.  M >= 0.


		     N is INTEGER
		     The number of columns of the matrix A.  N >= M.


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


		     TAU is DOUBLE PRECISION array, dimension (M)
		     The scalar factors of the elementary reflectors.


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

	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

	   November 2011

       Further Details:

	     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 form

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


		T( k ) = I - tau*u( k )*u( k )**T,   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 ).

       Definition at line 139 of file dtzrqf.f.

       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dtzrqf.f(3)
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