forked from johannesgerer/jburkardt-f
-
Notifications
You must be signed in to change notification settings - Fork 1
/
toms738.html
361 lines (319 loc) · 10 KB
/
toms738.html
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
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
<html>
<head>
<title>
TOMS738 - Niederreiter's Low Discrepancy Sequence
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TOMS738 <br> Niederreiter's Low Discrepancy Sequence
</h1>
<hr>
<p>
<b>TOMS738</b>
is a FORTRAN90 library which
implements ACM TOMS algorithm 738,
to compute Niederreiter's low discrepancy sequence.
</p>
<p>
A low discrepancy or quasirandom sequence, such as the Faure,
Halton, Hammersley, Niederreiter, or Sobol sequence,
is "less random" than a pseudorandom number sequence, but
more useful for such tasks as approximation of integrals in
higher dimensions, and in global optimization.
This is because low discrepancy sequences tend to sample
space "more uniformly" than random numbers. Algorithms
that use such sequences may have superior convergence.
</p>
<p>
The original, true, correct version of ACM TOMS Algorithm 738
is available through ACM:
<a href = "http://www.acm.org/pubs/calgo/">
http://www.acm.org/pubs/calgo</a>
or NETLIB:
<a href = "http://www.netlib.org/toms/index.html">
http://www.netlib.org/toms/index.html</a>
</p>
<p>
The version displayed here has been converted to FORTRAN90,
and other internal changes have been made to suit me.
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>TOMS738</b> is available in
<a href = "../../f77_src/toms738/toms738.html">a FORTRAN77 version</a> and
<a href = "../../f_src/toms738/toms738.html">a FORTRAN90 version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/cvt/cvt.html">
CVT</a>,
a FORTRAN90 library which
computes elements of a Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../f_src/faure/faure.html">
FAURE</a>,
a FORTRAN90 library which
computes elements of a Faure quasirandom sequence.
</p>
<p>
<a href = "../../f_src/grid/grid.html">
GRID</a>,
a FORTRAN90 library which
computes elements of a grid dataset.
</p>
<p>
<a href = "../../f_src/halton/halton.html">
HALTON</a>,
a FORTRAN90 library which
computes elements of a Halton quasirandom sequence.
</p>
<p>
<a href = "../../f_src/hammersley/hammersley.html">
HAMMERSLEY</a>,
a FORTRAN90 library which
computes elements of a Hammersley quasirandom sequence.
</p>
<p>
<a href = "../../f_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a FORTRAN90 library which
computes elements of a hexagonal grid dataset.
</p>
<p>
<a href = "../../f_src/hex_grid_angle/hex_grid_angle.html">
HEX_GRID_ANGLE</a>,
a FORTRAN90 library which
computes elements of an angled hexagonal grid dataset.
</p>
<p>
<a href = "../../f_src/ihs/ihs.html">
IHS</a>,
a FORTRAN90 library which
computes elements of an improved distributed Latin hypercube dataset.
</p>
<p>
<a href = "../../f_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing center points.
</p>
<p>
<a href = "../../f_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing edge points.
</p>
<p>
<a href = "../../f_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing
points at random.
</p>
<p>
<a href = "../../f_src/lcvt/lcvt.html">
LCVT</a>,
a FORTRAN90 library which
computes a latinized Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../f_src/niederreiter/niederreiter.html">
NIEDERREITER</a>,
a FORTRAN90 library which
is a modification of ACM TOMS algorithm 738,
which, among other things, allows the user to start the
sequence at any point. This is for the arbitrary base
calculations.
</p>
<p>
<a href = "../../f_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a FORTRAN90 library which
is a modification of ACM TOMS algorithm 738,
which, among other things, allows the user to start the
sequence at any point. This is for the base 2 calculations.
</p>
<p>
<a href = "../../f_src/sobol/sobol.html">
SOBOL</a>,
a FORTRAN90 library which
computes elements of a Sobol quasirandom sequence.
</p>
<p>
<a href = "../../f_src/uniform/uniform.html">
UNIFORM</a>,
a FORTRAN90 library which
computes elements of a uniform pseudorandom sequence.
</p>
<p>
<a href = "../../f_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a FORTRAN90 library which
computes elements of a van der Corput pseudorandom sequence.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Paul Bratley, Bennett Fox,<br>
Algorithm 659:
Implementing Sobol's Quasirandom Sequence Generator,<br>
ACM Transactions on Mathematical Software,<br>
Volume 14, Number 1, pages 88-100, 1988.
</li>
<li>
Paul Bratley, Bennett Fox, Harald Niederreiter,<br>
Algorithm 738:
Programs to Generate Niederreiter's Low-Discrepancy Sequences,<br>
ACM Transactions on Mathematical Software,<br>
Volume 20, Number 4, pages 494-495, 1994.
</li>
<li>
Paul Bratley, Bennett Fox, Harald Niederreiter,<br>
Implementation and Tests of Low Discrepancy Sequences,<br>
ACM Transactions on Modeling and Computer Simulation,<br>
Volume 2, Number 3, pages 195-213, 1992.
</li>
<li>
Bennett Fox,<br>
Algorithm 647:
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,<br>
ACM Transactions on Mathematical Software,<br>
Volume 12, Number 4, pages 362-376, 1986.
</li>
<li>
Rudolf Lidl, Harald Niederreiter, <br>
Finite Fields,<br>
Cambridge University Press, 1984, page 553.
</li>
<li>
Harald Niederreiter,<br>
Low-discrepancy and low-dispersion sequences,<br>
Journal of Number Theory,<br>
Volume 30, 1988, pages 51-70.
</li>
<li>
Harald Niederreiter,<br>
Random Number Generation and quasi-Monte Carlo Methods,<br>
SIAM, 1992.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<b>GFARIT</b> must be run first, to set up a tables
of addition and multiplication.
<ul>
<li>
<a href = "gfarit.f90">gfarit.f90</a>, the source code;
</li>
<li>
<a href = "gfarit.sh">gfarit.sh</a>,
commands to compile the source code;
</li>
<li>
<a href = "gfarit_output.txt">gfarit_output.txt</a>,
output from a run of GFARIT;
</li>
<li>
<a href = "gfarit.txt">gfarit.txt</a>,
the data file created by the run;
</li>
</ul>
</p>
<p>
<b>GFPLYS</b> must be run second, to set up a table
of irreducible polynomials.
<ul>
<li>
<a href = "gfplys.f90">gfplys.f90</a>, the source code;
</li>
<li>
<a href = "gfplys.sh">gfplys.sh</a>,
commands to compile the source code;
</li>
<li>
<a href = "gfplys_output.txt">gfplys_output.txt</a>,
output from a run of GFPLYS;
</li>
<li>
<a href = "gfplys.txt">gfplys.txt</a>,
the data file created by the run;
</li>
</ul>
</p>
<p>
<b>GENIN</b> can be run to generate a particular
Niederreiter sequence. The program is interactive, and requires
the user to specify a test integral, the spatial dimension, the
base, and the number of "warm up" values to skip.
<ul>
<li>
<a href = "genin.f90">genin.f90</a>, the source code;
</li>
<li>
<a href = "genin.sh">genin.sh</a>,
commands to compile the source code;
</li>
<li>
<a href = "genin_prb_output.txt">genin_prb_output.txt</a>,
output from a run of the program on all four test integrals;
</li>
</ul>
</p>
<p>
<b>GENIN_TWO</b> is a special, very efficient version of GENIN
for the special case where the base is 2:
<ul>
<li>
<a href = "genin_two.f90">genin_two.f90</a>, the source code;
</li>
<li>
<a href = "genin_two.sh">genin_two.sh</a>,
commands to compile the source code;
</li>
<li>
<a href = "genin_two_prb_output.txt">genin_two_prb_output.txt</a>,
output from a run of the program on all four test integrals;
</li>
</ul>
</p>
<p>
<b>SHOW_TWO</b> is a variation of GENIN_TWO which simply writes
the quasi-random numbers to a file.
<ul>
<li>
<a href = "show_two.f90">show_two.f90</a>, the source code;
</li>
<li>
<a href = "show_two.sh">show_two.sh</a>,
commands to compile the source code;
</li>
<li>
<a href = "show_two_prb_output.txt">show_two_prb_output.txt</a>,
output from a run of the program to compute the first 100 values;
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 15 September 2007.
</i>
<!-- John Burkardt -->
</body>
</html>