forked from johannesgerer/jburkardt-f
-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathvan_der_corput.html
433 lines (380 loc) · 12.2 KB
/
van_der_corput.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
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
<html>
<head>
<title>
VAN_DER_CORPUT - The van der Corput Quasirandom Sequence
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
VAN_DER_CORPUT <br> The van der Corput Quasirandom Sequence
</h1>
<hr>
<p>
<b>VAN_DER_CORPUT</b>
is a FORTRAN90 library which
computes the van der Corput
quasirandom sequence.
</p>
<p>
<b>VAN_DER_CORPUT</b> includes routines to make it easy to manipulate
this computation, to compute the next N entries, to compute a
particular entry, or to restart the sequence at a particular point.
</p>
<p>
The NDIM-dimensional Halton sequence is derived
from the 1-dimensional van der Corput sequence by using a set of
different (usually distinct prime) bases for each dimension,
and the Hammersley sequence is derived in almost the same way.
</p>
<p>
The van der Corput sequence is often used to generate a "subrandom"
sequence of points which have a better covering property
than pseudorandom points.
</p>
<p>
The van der Corput sequence generates a sequence of points in [0,1]
which (theoretically) never repeats. Except for SEED = 0, the
elements of the van der Corput sequence are strictly between 0 and 1.
</p>
<p>
The van der Corput sequence writes an integer in a given base B,
and then its digits are "reflected" about the decimal point.
This maps the numbers from 1 to N into a set of numbers in [0,1],
which are especially nicely distributed if N is one less
than a power of the base.
</p>
<p>
Hammersley suggested generating a set of N nicely distributed
points in two dimensions by setting the first component of the
Ith point to I/N, and the second to the van der Corput
value of I in base 2.
</p>
<p>
Halton suggested that in many cases, you might not know the number
of points you were generating, so Hammersley's formulation was
not ideal. Instead, he suggested that to generate a nicely
distributed sequence of points in M dimensions, you simply
choose the first M primes, P(1:M), and then for the J-th component of
the I-th point in the sequence, you compute the van der Corput
value of I in base P(J).
</p>
<p>
Thus, to generate a Halton sequence in a 2 dimensional space,
it is typical practice to generate a pair of van der Corput sequences,
the first with prime base 2, the second with prime base 3.
Similarly, by using the first K primes, a suitable sequence
in K-dimensional space can be generated.
</p>
<p>
The generation is quite simple. Given an integer SEED, the expansion
of SEED in base BASE is generated. Then, essentially, the result R
is generated by writing a decimal point followed by the digits of
the expansion of SEED, in reverse order. This decimal value is actually
still in base BASE, so it must be properly interpreted to generate
a usable value.
</p>
<p>
Here is an example in base 2:
<table border="1">
<tr>
<th>SEED (decimal)</th>
<th>SEED (binary)</th>
<th>VDC (binary)</th>
<th>VDC (decimal)</th>
</tr>
<tr>
<td>0</td><td>0</td><td>.0</td><td>0.0</td>
</tr>
<tr>
<td>1</td><td>1</td><td>.1</td><td>0.5</td>
</tr>
<tr>
<td>2</td><td>10</td><td>.01</td><td>0.25</td>
</tr>
<tr>
<td>3</td><td>11</td><td>.11</td><td>0.75</td>
</tr>
<tr>
<td>4</td><td>100</td><td>.001</td><td>0.125</td>
</tr>
<tr>
<td>5</td><td>101</td><td>.101</td><td>0.625</td>
</tr>
<tr>
<td>6</td><td>110</td><td>.011</td><td>0.375</td>
</tr>
<tr>
<td>7</td><td>111</td><td>.111</td><td>0.875</td>
</tr>
<tr>
<td>8</td><td>1000</td><td>.0001</td><td>0.0625</td>
</tr>
</table>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>VAN_DER_CORPUT</b> is available in
<a href = "../../cpp_src/van_der_corput/van_der_corput.html">a C++ version</a> and
<a href = "../../f_src/van_der_corput/van_der_corput.html">a FORTRAN90 version</a> and
<a href = "../../m_src/van_der_corput/van_der_corput.html">a MATLAB 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/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a FORTRAN90 library which
computes elements of a Niederreiter quasirandom sequence with base 2.
</p>
<p>
<a href = "../../f_src/normal/normal.html">
NORMAL</a>,
a FORTRAN90 library which
computes elements of a
sequence of pseudorandom normally distributed values.
</p>
<p>
<a href = "../../m_src/sequence_streak_display/sequence_streak_display.html">
SEQUENCE_STREAK_DISPLAY</a>,
a MATLAB program which
makes a "streak file" of a sequence.
</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 = "../../datasets/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a dataset directory which
contains examples of van der Corput sequences.
</p>
<p>
<a href = "../../f_src/van_der_corput_dataset/van_der_corput_dataset.html">
VAN_DER_CORPUT_DATASET</a>,
a FORTRAN90 program which
creates a van der Corput dataset and writes it to a file.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
John Halton,<br>
On the efficiency of certain quasi-random sequences of points
in evaluating multi-dimensional integrals,<br>
Numerische Mathematik,<br>
Volume 2, pages 84-90, 1960.
</li>
<li>
John Hammersley,<br>
Monte Carlo methods for solving multivariable problems,<br>
Proceedings of the New York Academy of Science,<br>
Volume 86, pages 844-874, 1960.
</li>
<li>
Johannes van der Corput,<br>
Verteilungsfunktionen I & II,<br>
Nederl. Akad. Wetensch. Proc.,<br>
Volume 38, 1935, pages 813-820, pages 1058-1066.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "van_der_corput.f90">van_der_corput.f90</a>, the source code.
</li>
<li>
<a href = "van_der_corput.sh">van_der_corput.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "van_der_corput_prb.f90">van_der_corput_prb.f90</a>,
a sample problem.
</li>
<li>
<a href = "van_der_corput_prb.sh">van_der_corput_prb.sh</a>,
commands to compile, link and run the sample problem.
</li>
<li>
<a href = "van_der_corput_prb_output.txt">van_der_corput_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CIRCLE_UNIT_VAN_DER_CORPUT</b> picks a van der Corput point on the unit circle.
</li>
<li>
<b>GET_SEED</b> returns a seed for the random number generator.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_LOG_2</b> returns the integer part of the logarithm base 2 of an I4.
</li>
<li>
<b>I4_TO_VAN_DER_CORPUT</b> computes an element of a van der Corput sequence.
</li>
<li>
<b>I4_TO_VAN_DER_CORPUT_SEQUENCE:</b> next N elements of a van der Corput sequence.
</li>
<li>
<b>I4VEC_INDICATOR</b> sets an I4VEC to the indicator vector A(I)=I.
</li>
<li>
<b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>VAN_DER_CORPUT</b> computes the next element in the van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_BASE_GET</b> gets the base for a van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_BASE_SET</b> sets the base for a van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_MEMORY</b> stores data for the van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_SEED_GET</b> gets the "seed" for the van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_SEED_SET</b> sets the "seed" for the van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_SEQUENCE:</b> next N elements in the van der Corput sequence.
</li>
<li>
<b>VDC_NUMERATOR_SEQUENCE:</b> van der Corput numerator sequence base 2.
</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 03 February 2011.
</i>
<!-- John Burkardt -->
</body>
</html>