1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
|
*> \brief \b DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATRZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrz.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
*
* .. Scalar Arguments ..
* INTEGER L, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
*> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means
*> of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal
*> matrix and, R and A1 are M-by-M upper triangular matrices.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] L
*> \verbatim
*> L is INTEGER
*> The number of columns of the matrix A containing the
*> meaningful part of the Householder vectors. N-M >= L >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> 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 N-L+1 to
*> N of the first M rows of A, with the array TAU, represent the
*> orthogonal matrix Z as a product of M elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (M)
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup doubleOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The factorization is obtained by Householder's method. The kth
*> transformation 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 )**T, u( k ) = ( 1 ),
*> ( 0 )
*> ( z( k ) )
*>
*> tau is a scalar and z( k ) is an l element vector. tau and z( k )
*> are chosen to annihilate the elements of the kth row of A2.
*>
*> The scalar tau is returned in the kth element of TAU and the vector
*> u( k ) in the kth row of A2, such that the elements of z( k ) are
*> in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
*> the upper triangular part of A1.
*>
*> Z is given by
*>
*> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER L, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. External Subroutines ..
EXTERNAL DLARFG, DLARZ
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
* Quick return if possible
*
IF( M.EQ.0 ) THEN
RETURN
ELSE IF( M.EQ.N ) THEN
DO 10 I = 1, N
TAU( I ) = ZERO
10 CONTINUE
RETURN
END IF
*
DO 20 I = M, 1, -1
*
* Generate elementary reflector H(i) to annihilate
* [ A(i,i) A(i,n-l+1:n) ]
*
CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) )
*
* Apply H(i) to A(1:i-1,i:n) from the right
*
CALL DLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA,
$ TAU( I ), A( 1, I ), LDA, WORK )
*
20 CONTINUE
*
RETURN
*
* End of DLATRZ
*
END
|
