forked from xiaolongma/jburkardt-f
-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathball_volume_quad.html
266 lines (230 loc) · 7.09 KB
/
ball_volume_quad.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
<html>
<head>
<title>
BALL_VOLUME_QUAD - Multidimensional Ball Volume by Quadrature
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
BALL_VOLUME_QUAD <br> Multidimensional Ball Volume by Quadrature
</h1>
<hr>
<p>
<b>BALL_VOLUME_QUAD</b>
is a FORTRAN90 program which
investigates the behavior of a quadrature rule when it is applied
to the characteristic function of the unit M-dimensional
hyperball, which is a discontinuous function.
</p>
<p>
The program integrates the function over the unit hypercube [-1,+1]^M,
where the spatial dimension M is arbitrary. The value of M is inferred
from the user input quadrature rule information.
</p>
<p>
The problem used as example input to the program works in M = 6 dimensions.
In that case, the volume of the hypercube is 64; the volume of the
hyperball is pi^3/6, or about 5.16771.
</p>
<p>
Because the integrand is discontinuous, any quadrature rule based on
the idea of interpolation will probably be unable to do a good job.
A family of quadrature rules, which rely on increasing the order of
interpolation to improve accuracy, will probably get increasingly
bad answers.
</p>
<p>
By contrast, a basic Monte Carlo rule, which assumes nothing about
the function, integrates this function just as well as it integrates
most any other square-integrable function. (That's both the strength
and weakness of the blunt instrument we call Monte Carlo integration.)
</p>
<p>
The program assumes that the quadrature rule is defined by three text files.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>ball_volume_quad</b> <i>prefix</i>
</blockquote>
where
<ul>
<li>
<i>prefix</i>_r.txt contains the coordinates of two points that
define the quadrature region.
</li>
<li>
<i>prefix</i>_w.txt contains the weights;
</li>
<li>
<i>prefix</i>_x.txt contains the points;
</li>
</ul>
For information on the form of these files, see the
<b>QUADRATURE_RULES</b> directory listed below.
</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">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/ball_volume_monte_carlo/ball_volume_monte_carlo.html">
BALL_VOLUME_MONTE_CARLO</a>,
a FORTRAN90 program which
applies a Monte Carlo method to estimate the volume of the unit 6D sphere;
</p>
<p>
<a href = "../../f_src/nintlib/nintlib.html">
NINTLIB</a>,
a FORTRAN90 library which
numerically estimates integrals in multiple dimensions.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
contains sets of files that define quadrature
rules over various 1D intervals or multidimensional hypercubes.
</p>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
defines quadrature rules for a variety of unusual areas, surfaces
and volumes in 2D, 3D and multiple dimensions.
</p>
<p>
<a href = "../../f_src/test_nint/test_nint.html">
TEST_NINT</a>,
a FORTRAN90 library which
defines integrand functions for testing
multidimensional quadrature routines.
</p>
<p>
<a href = "../../f_src/testpack/testpack.html">
TESTPACK</a>,
a FORTRAN90 library which
defines a set of integrands used to test multidimensional quadrature.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "ball_volume_quad.f90">ball_volume_quad.f90</a>, the source code.
</li>
<li>
<a href = "ball_volume_quad.sh">ball_volume_quad.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "ball_volume_quad_test.sh">ball_volume_quad_test.sh</a>,
runs BALL_VOLUME_QUAD on 6 sparse grid Clenshaw Curtis rules, and then
on six Monte Carlo "rules" that match the number of points used by the
Clenshaw Curtis rules.
</li>
<li>
<a href = "ball_volume_quad_output.txt">ball_volume_quad_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for BALL_VOLUME_QUAD.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>SPHERE_INDICATOR</b> evaluates the unit sphere indicator function.
</li>
<li>
<b>SPHERE_UNIT_VOLUME_ND</b> computes the volume of a unit sphere in multi-dimensions.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</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 27 August 2010.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>