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NAME
     ctzrqf - reduce the M-by-N  (  M<=N  )  complex  upper  tra-
     pezoidal  matrix A to upper triangular form by means of uni-
     tary transformations

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

     INTEGER INFO, LDA, M, N

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



     #include <sunperf.h>

     void ctzrqf(int m, int n, complex *ca, int lda,
                complex *tau, int *info) ;

PURPOSE
     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

FURTHER DETAILS
     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 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 ).

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