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
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
|
*> \brief \b DGEQRT3
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition
* ==========
*
* RECURSIVE SUBROUTINE DGEQRT3( M, N, A, LDA, T, LDT, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N, LDT
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), T( LDT, * )
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> DGEQRT3 recursively computes a QR factorization of a real M-by-N
*> matrix A, using the compact WY representation of Q.
*>
*> Based on the algorithm of Elmroth and Gustavson,
*> IBM J. Res. Develop. Vol 44 No. 4 July 2000.
*>
*>\endverbatim
*
* Arguments
* =========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the real M-by-N matrix A. On exit, the elements on and
*> above the diagonal contain the N-by-N upper triangular matrix R; the
*> elements below the diagonal are the columns of V. See below for
*> further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,N)
*> The N-by-N upper triangular factor of the block reflector.
*> The elements on and above the diagonal contain the block
*> reflector T; the elements below the diagonal are not used.
*> See below for further details.
*> \endverbatim
*> \verbatim
*> LDT (intput) INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*>
*
* Authors
* =======
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleGEcomputational
*
*
* Further Details
* ===============
*>\details \b Further \b Details
*> \verbatim
*>
*> The matrix V stores the elementary reflectors H(i) in the i-th column
*> below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*> V = ( 1 )
*> ( v1 1 )
*> ( v1 v2 1 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> where the vi's represent the vectors which define H(i), which are returned
*> in the matrix A. The 1's along the diagonal of V are not stored in A. The
*> block reflector H is then given by
*>
*> H = I - V * T * V**T
*>
*> where V**T is the transpose of V.
*>
*> For details of the algorithm, see Elmroth and Gustavson (cited above).
*>
*> \endverbatim
*>
* =====================================================================
RECURSIVE SUBROUTINE DGEQRT3( M, N, A, LDA, T, LDT, INFO )
*
* -- LAPACK computational routine (version 3.?) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N, LDT
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), T( LDT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+00 )
* ..
* .. Local Scalars ..
INTEGER I, I1, J, J1, N1, N2, IINFO
* ..
* .. External Subroutines ..
EXTERNAL DLARFG, DTRMM, DGEMM, XERBLA
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N .LT. 0 ) THEN
INFO = -2
ELSE IF( M .LT. N ) THEN
INFO = -1
ELSE IF( LDA .LT. MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LDT .LT. MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEQRT3', -INFO )
RETURN
END IF
*
IF( N.EQ.1 ) THEN
*
* Compute Householder transform when N=1
*
CALL DLARFG( M, A, A( MIN( 2, M ), 1 ), 1, T )
*
ELSE
*
* Otherwise, split A into blocks...
*
N1 = N/2
N2 = N-N1
J1 = MIN( N1+1, N )
I1 = MIN( N+1, M )
*
* Compute A(1:M,1:N1) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
*
CALL DGEQRT3( M, N1, A, LDA, T, LDT, IINFO )
*
* Compute A(1:M,J1:N) = Q1^H A(1:M,J1:N) [workspace: T(1:N1,J1:N)]
*
DO J=1,N2
DO I=1,N1
T( I, J+N1 ) = A( I, J+N1 )
END DO
END DO
CALL DTRMM( 'L', 'L', 'T', 'U', N1, N2, ONE,
& A, LDA, T( 1, J1 ), LDT )
*
CALL DGEMM( 'T', 'N', N1, N2, M-N1, ONE, A( J1, 1 ), LDA,
& A( J1, J1 ), LDA, ONE, T( 1, J1 ), LDT)
*
CALL DTRMM( 'L', 'U', 'T', 'N', N1, N2, ONE,
& T, LDT, T( 1, J1 ), LDT )
*
CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( J1, 1 ), LDA,
& T( 1, J1 ), LDT, ONE, A( J1, J1 ), LDA )
*
CALL DTRMM( 'L', 'L', 'N', 'U', N1, N2, ONE,
& A, LDA, T( 1, J1 ), LDT )
*
DO J=1,N2
DO I=1,N1
A( I, J+N1 ) = A( I, J+N1 ) - T( I, J+N1 )
END DO
END DO
*
* Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
*
CALL DGEQRT3( M-N1, N2, A( J1, J1 ), LDA,
& T( J1, J1 ), LDT, IINFO )
*
* Compute T3 = T(1:N1,J1:N) = -T1 Y1^H Y2 T2
*
DO I=1,N1
DO J=1,N2
T( I, J+N1 ) = (A( J+N1, I ))
END DO
END DO
*
CALL DTRMM( 'R', 'L', 'N', 'U', N1, N2, ONE,
& A( J1, J1 ), LDA, T( 1, J1 ), LDT )
*
CALL DGEMM( 'T', 'N', N1, N2, M-N, ONE, A( I1, 1 ), LDA,
& A( I1, J1 ), LDA, ONE, T( 1, J1 ), LDT )
*
CALL DTRMM( 'L', 'U', 'N', 'N', N1, N2, -ONE, T, LDT,
& T( 1, J1 ), LDT )
*
CALL DTRMM( 'R', 'U', 'N', 'N', N1, N2, ONE,
& T( J1, J1 ), LDT, T( 1, J1 ), LDT )
*
* Y = (Y1,Y2); R = [ R1 A(1:N1,J1:N) ]; T = [T1 T3]
* [ 0 R2 ] [ 0 T2]
*
END IF
*
RETURN
*
* End of DGEQRT3
*
END
|
