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

6       DTZRZF  -  the M-by-N ( M<=N ) real upper trapezoidal matrix A to upper
7       triangular form by means of orthogonal transformations
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SYNOPSIS

10       SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
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12           INTEGER        INFO, LDA, LWORK, M, N
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14           DOUBLE         PRECISION A( LDA, * ), TAU( * ), WORK( * )
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PURPOSE

17       DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix  A  to
18       upper triangular form by means of orthogonal 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 orthogonal matrix and R is an M-by-M upper trian‐
25       gular 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) DOUBLE PRECISION 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 orthogonal matrix Z  as
41               a 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) DOUBLE PRECISION array, dimension (M)
47               The scalar factors of the elementary reflectors.
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49       WORK       (workspace/output)   DOUBLE   PRECISION   array,   dimension
50       (MAX(1,LWORK))
51               On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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53       LWORK   (input) INTEGER
54               The dimension of the array WORK.  LWORK >= max(1,M).  For opti‐
55               mum  performance  LWORK >= M*NB, where NB is the optimal block‐
56               size.
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58               If LWORK = -1, then a workspace query is assumed;  the  routine
59               only  calculates  the  optimal  size of the WORK array, returns
60               this value as the first entry of the WORK array, and  no  error
61               message related to LWORK is issued by XERBLA.
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63       INFO    (output) INTEGER
64               = 0:  successful exit
65               < 0:  if INFO = -i, the i-th argument had an illegal value
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FURTHER DETAILS

68       Based on contributions by
69         A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
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71       The  factorization is obtained by Householder's method.  The kth trans‐
72       formation matrix, Z( k ), which is used to introduce zeros into the ( m
73       - k + 1 )th row of A, is given in the form
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75          Z( k ) = ( I     0   ),
76                   ( 0  T( k ) )
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78       where
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80          T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
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82                                                      ( z( k ) )
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84       tau  is a scalar and z( k ) is an ( n - m ) element vector.  tau and z(
85       k ) are chosen to annihilate the elements of the kth row of X.
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87       The scalar tau is returned in the kth element of TAU and the vector  u(
88       k ) in the kth row of A, such that the elements of z( k ) are in  a( k,
89       m + 1 ), ..., a( k, n ). The elements of R are returned  in  the  upper
90       triangular part of A.
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92       Z is given by
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94          Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
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99 LAPACK routine (version 3.1)    November 2006                       DTZRZF(1)