1CTZRZF(1) LAPACK routine (version 3.2) CTZRZF(1)
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6 CTZRZF - reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A
7 to upper triangular form by means of unitary transformations
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10 SUBROUTINE CTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
11 INTEGER INFO, LDA, LWORK, M, N
13 COMPLEX A( LDA, * ), TAU( * ), WORK( * )
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17 CTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A
18 to upper triangular form by means of unitary transformations. The
19 upper trapezoidal matrix A is factored as
20 A = ( R 0 ) * Z,
21 where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular
22 matrix.
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25 M (input) INTEGER
26 The number of rows of the matrix A. M >= 0.
27 N (input) INTEGER
29 The number of columns of the matrix A. N >= M.
30 A (input/output) COMPLEX array, dimension (LDA,N)
32 On entry, the leading M-by-N upper trapezoidal part of the
33 array A must contain the matrix to be factorized. On exit, the
34 leading M-by-M upper triangular part of A contains the upper
35 triangular matrix R, and elements M+1 to N of the first M rows
36 of A, with the array TAU, represent the unitary matrix Z as a
37 product of M elementary reflectors.
38 LDA (input) INTEGER
40 The leading dimension of the array A. LDA >= max(1,M).
41 TAU (output) COMPLEX array, dimension (M)
43 The scalar factors of the elementary reflectors.
44 WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
46 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
47 LWORK (input) INTEGER
49 The dimension of the array WORK. LWORK >= max(1,M). For opti‐
50 mum performance LWORK >= M*NB, where NB is the optimal block‐
51 size. If LWORK = -1, then a workspace query is assumed; the
52 routine only calculates the optimal size of the WORK array,
53 returns this value as the first entry of the WORK array, and no
54 error message related to LWORK is issued by XERBLA.
55 INFO (output) INTEGER
57 = 0: successful exit
58 < 0: if INFO = -i, the i-th argument had an illegal value
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61 Based on contributions by
62 A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
63 The factorization is obtained by Householder's method. The kth trans‐
64 formation matrix, Z( k ), which is used to introduce zeros into the ( m
65 - k + 1 )th row of A, is given in the form
66 Z( k ) = ( I 0 ),
67 ( 0 T( k ) )
68 where
69 T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ),
70 ( 0 )
71 ( z( k ) ) tau is a
72 scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are
73 chosen to annihilate the elements of the kth row of X.
74 The scalar tau is returned in the kth element of TAU and the vector u(
75 k ) in the kth row of A, such that the elements of z( k ) are in a( k,
76 m + 1 ), ..., a( k, n ). The elements of R are returned in the upper
77 triangular part of A.
78 Z is given by
79 Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
80 LAPACK routine (version 3.2) November 2008 CTZRZF(1)