ztzrqf(3F)

NAME
     ZTZRQF - reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix A to
     upper triangular form by means of unitary transformations

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

         INTEGER        INFO, LDA, M, N

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

PURPOSE
     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 unitary 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) 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 illegal 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 ).