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

6       ZTZRQF - i deprecated and has been replaced by routine ZTZRZF
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SYNOPSIS

9       SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO )
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11           INTEGER        INFO, LDA, M, N
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13           COMPLEX*16     A( LDA, * ), TAU( * )
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PURPOSE

16       This routine is deprecated and has been replaced by routine ZTZRZF.
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18       ZTZRQF  reduces  the M-by-N ( M<=N ) complex upper trapezoidal matrix A
19       to upper triangular form by means of unitary transformations.
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21       The upper trapezoidal matrix A is factored as
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23          A = ( R  0 ) * Z,
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25       where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular
26       matrix.
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ARGUMENTS

30       M       (input) INTEGER
31               The number of rows of the matrix A.  M >= 0.
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33       N       (input) INTEGER
34               The number of columns of the matrix A.  N >= M.
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36       A       (input/output) COMPLEX*16 array, dimension (LDA,N)
37               On  entry,  the  leading  M-by-N  upper trapezoidal part of the
38               array A must contain the matrix to be factorized.  On exit, the
39               leading  M-by-M  upper  triangular part of A contains the upper
40               triangular matrix R, and elements M+1 to N of the first M  rows
41               of  A,  with the array TAU, represent the unitary matrix Z as a
42               product of M elementary reflectors.
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44       LDA     (input) INTEGER
45               The leading dimension of the array A.  LDA >= max(1,M).
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47       TAU     (output) COMPLEX*16 array, dimension (M)
48               The scalar factors of the elementary reflectors.
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50       INFO    (output) INTEGER
51               = 0: successful exit
52               < 0: if INFO = -i, the i-th argument had an illegal value
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FURTHER DETAILS

55       The  factorization is obtained by Householder's method.  The kth trans‐
56       formation  matrix,  Z( k ), whose conjugate transpose is used to intro‐
57       duce zeros into the (m - k + 1)th row of A, is given in the form
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59          Z( k ) = ( I     0   ),
60                   ( 0  T( k ) )
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62       where
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64          T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
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66                                                      ( z( k ) )
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68       tau is a scalar and z( k ) is an ( n - m ) element vector.  tau and  z(
69       k ) are chosen to annihilate the elements of the kth row of X.
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71       The  scalar tau is returned in the kth element of TAU and the vector u(
72       k ) in the kth row of A, such that the elements of z( k ) are in  a( k,
73       m  +  1  ), ..., a( k, n ). The elements of R are returned in the upper
74       triangular part of A.
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76       Z is given by
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78          Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
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83 LAPACK routine (version 3.1)    November 2006                       ZTZRQF(1)