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CTZRQF(3F)							    CTZRQF(3F)

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

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

	 INTEGER	INFO, LDA, M, N

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

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 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 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

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CTZRQF(3F)							    CTZRQF(3F)

	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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