DTZRQF  -  routine  is deprecated and has been replaced by
       routine DTZRZF


SYNOPSIS

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

           INTEGER        INFO, LDA, M, N

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


PURPOSE

       This routine is deprecated and has been replaced  by  rou­
       tine  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.


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,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) DOUBLE PRECISION 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 ille­
               gal 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 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 ele­
       ments 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 ).


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