ZLATRZ  -  factor the M-by-(M+L) complex upper trapezoidal
       matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0
       )  *  Z by means of unitary transformations, where Z is an
       (M+L)-by-(M+L) unitary matrix and, R  and  A1  are  M-by-M
       upper triangular matrices


SYNOPSIS

       SUBROUTINE ZLATRZ( M, N, L, A, LDA, TAU, WORK )

           INTEGER        L, LDA, M, N

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


PURPOSE

       ZLATRZ  factors  the  M-by-(M+L) complex upper trapezoidal
       matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0
       )  *  Z by means of unitary transformations, where Z is an
       (M+L)-by-(M+L) unitary matrix and, R  and  A1  are  M-by-M
       upper triangular matrices.


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 >= 0.

       L       (input) INTEGER
               The  number  of columns of the matrix A containing
               the meaningful part of the Householder vectors. N-
               M >= L >= 0.

       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 N-L+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.

       WORK    (workspace) COMPLEX*16 array, dimension (M)


FURTHER DETAILS

       Based on contributions by

       Knoxville, USA

       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 l element vector. tau and
       z( k ) are chosen to annihilate the elements  of  the  kth
       row of A2.

       The  scalar  tau is returned in the kth element of TAU and
       the vector u( k ) in the kth row of A2, such that the ele­
       ments of z( k ) are in  a( k, l + 1 ), ..., a( k, n ). The
       elements of R are returned in the upper triangular part of
       A1.

       Z is given by

          Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).


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