ZTZRQF  -  routine  is deprecated and has been replaced by
       routine ZTZRZF


SYNOPSIS

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

           INTEGER        INFO, LDA, M, N

           COMPLEX*16     A( LDA, * ), TAU( * )


PURPOSE

       This routine is deprecated and has been replaced  by  rou­
       tine  ZTZRZF.   ZTZRQF 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*16 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*16 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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