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ctzrzf(3P)		    Sun Performance Library		    ctzrzf(3P)

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

SYNOPSIS
       SUBROUTINE CTZRZF(M, N, A, LDA, TAU, WORK, LWORK, INFO)

       COMPLEX A(LDA,*), TAU(*), WORK(*)
       INTEGER M, N, LDA, LWORK, INFO

       SUBROUTINE CTZRZF_64(M, N, A, LDA, TAU, WORK, LWORK, INFO)

       COMPLEX A(LDA,*), TAU(*), WORK(*)
       INTEGER*8 M, N, LDA, LWORK, INFO

   F95 INTERFACE
       SUBROUTINE TZRZF([M], [N], A, [LDA], TAU, [WORK], [LWORK], [INFO])

       COMPLEX, DIMENSION(:) :: TAU, WORK
       COMPLEX, DIMENSION(:,:) :: A
       INTEGER :: M, N, LDA, LWORK, INFO

       SUBROUTINE TZRZF_64([M], [N], A, [LDA], TAU, [WORK], [LWORK], [INFO])

       COMPLEX, DIMENSION(:) :: TAU, WORK
       COMPLEX, DIMENSION(:,:) :: A
       INTEGER(8) :: M, N, LDA, LWORK, INFO

   C INTERFACE
       #include <sunperf.h>

       void ctzrzf(int m, int n,  complex  *a,	int  lda,  complex  *tau,  int
		 *info);

       void ctzrzf_64(long m, long n, complex *a, long lda, complex *tau, long
		 *info);

PURPOSE
       ctzrzf 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) The number of rows of the matrix A.  M >= 0.

       N (input) The number of columns of the matrix A.	 N >= 0.

       A (input/output)
		 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)
		 The leading dimension of the array A.	LDA >= max(1,M).

       TAU (output)
		 The scalar factors of the elementary reflectors.

       WORK (workspace)
		 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK (input)
		 The  dimension	 of  the  array WORK.  LWORK >= max(1,M).  For
		 optimum performance LWORK >= M*NB, where NB  is  the  optimal
		 blocksize.

		 If LWORK = -1, then a workspace query is assumed; the routine
		 only calculates the optimal size of the WORK  array,  returns
		 this value as the first entry of the WORK array, and no error
		 message related to LWORK is issued by XERBLA.

       INFO (output)
		 = 0:  successful exit
		 < 0:  if INFO = -i, the i-th argument had an illegal value

FURTHER DETAILS
       Based on contributions by
	 A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

       The factorization is obtained by Householder's method.  The kth	transā€
       formation 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 ( 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 ).

				  6 Mar 2009			    ctzrzf(3P)
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