1ZTZRZF(1)                LAPACK routine (version 3.1)                ZTZRZF(1)
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NAME

6       ZTZRZF  -  the  M-by-N  (  M<=N ) complex upper trapezoidal matrix A to
7       upper triangular form by means of unitary transformations
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SYNOPSIS

10       SUBROUTINE ZTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
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12           INTEGER        INFO, LDA, LWORK, M, N
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14           COMPLEX*16     A( LDA, * ), TAU( * ), WORK( * )
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PURPOSE

17       ZTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal  matrix  A
18       to upper triangular form by means of unitary transformations.
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20       The upper trapezoidal matrix A is factored as
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22          A = ( R  0 ) * Z,
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24       where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular
25       matrix.
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ARGUMENTS

29       M       (input) INTEGER
30               The number of rows of the matrix A.  M >= 0.
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32       N       (input) INTEGER
33               The number of columns of the matrix A.  N >= M.
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35       A       (input/output) COMPLEX*16 array, dimension (LDA,N)
36               On entry, the leading M-by-N  upper  trapezoidal  part  of  the
37               array A must contain the matrix to be factorized.  On exit, the
38               leading M-by-M upper triangular part of A  contains  the  upper
39               triangular  matrix R, and elements M+1 to N of the first M rows
40               of A, with the array TAU, represent the unitary matrix Z  as  a
41               product of M elementary reflectors.
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43       LDA     (input) INTEGER
44               The leading dimension of the array A.  LDA >= max(1,M).
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46       TAU     (output) COMPLEX*16 array, dimension (M)
47               The scalar factors of the elementary reflectors.
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49       WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
50               On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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52       LWORK   (input) INTEGER
53               The dimension of the array WORK.  LWORK >= max(1,M).  For opti‐
54               mum performance LWORK >= M*NB, where NB is the  optimal  block‐
55               size.
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57               If  LWORK  = -1, then a workspace query is assumed; the routine
58               only calculates the optimal size of  the  WORK  array,  returns
59               this  value  as the first entry of the WORK array, and no error
60               message related to LWORK is issued by XERBLA.
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62       INFO    (output) INTEGER
63               = 0:  successful exit
64               < 0:  if INFO = -i, the i-th argument had an illegal value
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FURTHER DETAILS

67       Based on contributions by
68         A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
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70       The factorization is obtained by Householder's method.  The kth  trans‐
71       formation matrix, Z( k ), which is used to introduce zeros into the ( m
72       - k + 1 )th row of A, is given in the form
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74          Z( k ) = ( I     0   ),
75                   ( 0  T( k ) )
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77       where
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79          T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
80                                                      (   0    )
81                                                      ( z( k ) )
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83       tau is a scalar and z( k ) is an ( n - m ) element vector.  tau and  z(
84       k ) are chosen to annihilate the elements of the kth row of X.
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86       The  scalar tau is returned in the kth element of TAU and the vector u(
87       k ) in the kth row of A, such that the elements of z( k ) are in  a( k,
88       m  +  1  ), ..., a( k, n ). The elements of R are returned in the upper
89       triangular part of A.
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91       Z is given by
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93          Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
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98 LAPACK routine (version 3.1)    November 2006                       ZTZRZF(1)