-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathQR3D.SIF
230 lines (166 loc) · 4.68 KB
/
QR3D.SIF
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
***************************
* SET UP THE INITIAL DATA *
***************************
NAME QR3D
* Problem :
* *********
* Find the QR factorization of a tridiagonal matrix A.
* The problem is formulated as a system of quadratic equations
* whose unknowns are the elements of the orthogonal matrix Q and of
* the upper triangular matrix R. In this version of the problem,
* the banded structure of R is not imposed as a constraint. See problem
* QR3DBD for the case where this structure is explicitly used.
* The problem is non-convex.
* Source:
* Ph. Toint, private communication.
* SIF input: Ph. Toint, Nov 1993
* classification NQR2-AN-V-V
* Define the matrix order M ( M >= 3 ).
* There are M * ( 3M + 1) / 2 variables and equations.
*IE M 5 $-PARAMETER n = 40
*IE M 10 $-PARAMETER n = 155 original value
IE M 20 $-PARAMETER n = 610
* Define useful parameters
IE 1 1
IE 2 2
IA M-1 M -1
RI RM M
* Define the matrix A to factorize
* a(i,i-1)=a(i,i+1)= (1-i)/m, a(i,i) = 2i/m
RD 2/M RM 2.0
A= A(1,1) 2/M
AE A(1,2) 0.0
DO I 2 M-1
IA I+1 I 1
IA I-1 I -1
IM 1-I I-1 -1
RI R1-I 1-I
R/ 1-I/M R1-I RM
IM 2I I 2
RI R2I 2I
R/ 2I/M R2I RM
A= A(I,I-1) 1-I/M
A= A(I,I) 2I/M
A= A(I,I+1) 1-I/M
OD I
RI RM-1 M-1
RM 1-M RM-1 -1.0
R/ 1-M/M 1-M RM
RM 2M RM 2.0
A= A(M,M-1) 1-M/M
A= A(M,M) 2M
VARIABLES
* The orthogonal matrix Q
DO I 1 M
DO J 1 M
X Q(I,J)
OD J
OD I
* The upper triangular matrix R
DO I 1 M
DO J I M
X R(I,J)
OD J
OD I
GROUPS
* Orthogonality of the Q matrix
DO I 1 M
DO J I M
XE O(I,J)
OD J
OD I
* Factorization equations A= QR
DO I 1 M
DO J 1 M
XE F(I,J)
OD J
OD I
CONSTANTS
* Orthogonality constraints
DO I 1 M
X QR3D O(I,I) 1.0
OD J
* Factorization constraints
Z QR3D F(1,1) A(1,1)
Z QR3D F(1,2) A(1,2)
DO I 2 M-1
IA I-1 I -1
IA I+1 I 1
Z QR3D F(I,I-1) A(I,I-1)
Z QR3D F(I,I) A(I,I)
Z QR3D F(I,I+1) A(I,I+1)
OD I
Z QR3D F(M,M-1) A(M,M-1)
Z QR3D F(M,M) A(M,M)
BOUNDS
* All variables are free, except the diagonals of R
* which are nonnegative
FR QR3D 'DEFAULT'
DO I 1 M
XL QR3D R(I,I) 0.0
OD I
START POINT
* Initialize Q to the identity
DO I 1 M
XV QR3D Q(I,I) 1.0
OD I
* Initialize R to the upper triangular part of A
DO I 1 M-1
IA I+1 I 1
ZV QR3D R(I,I) A(I,I)
ZV QR3D R(I,I+1) A(I,I+1)
OD I
ZV QR3D R(M,M) A(M,M)
ELEMENT TYPE
EV 2PR V1 V2
ELEMENT USES
XT 'DEFAULT' 2PR
DO I 1 M
DO J I M
DO K 1 M
ZV C(I,J,K) V1 Q(I,K)
ZV C(I,J,K) V2 Q(J,K)
OD K
OD J
OD I
DO I 1 M
DO J 1 M
DO K 1 J
ZV B(I,J,K) V1 Q(I,K)
ZV B(I,J,K) V2 R(K,J)
OD K
OD J
OD I
GROUP USES
* Orthogonality constraints
DO I 1 M
DO J I M
DO K 1 M
XE O(I,J) C(I,J,K)
OD K
OD J
OD I
* Factorization constraints
DO I 1 M
DO J 1 M
DO K 1 J
XE F(I,J) B(I,J,K)
OD K
OD J
OD I
OBJECT BOUND
* Solution
*LO SOLTN 0.0
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS QR3D
INDIVIDUALS
T 2PR
F V1 * V2
G V1 V2
G V2 V1
H V1 V2 1.0
ENDATA