CTZRQF  -  routine  is deprecated and has been replaced by
       routine CTZRZF


SYNOPSIS

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

           INTEGER        INFO, LDA, M, N

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


PURPOSE

       This routine is deprecated and has been replaced  by  rou­
       tine  CTZRZF.   CTZRQF 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 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 uni­
               tary matrix Z as a product of M elementary reflec­
               tors.

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

       TAU     (output) COMPLEX 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  ),  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  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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