forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
tetrahedron_nco_rule.html
352 lines (306 loc) · 10.5 KB
/
tetrahedron_nco_rule.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
<html>
<head>
<title>
TETRAHEDRON_NCO_RULE - Newton-Cotes Open Quadrature for the Tetrahedron
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TETRAHEDRON_NCO_RULE <br> Newton-Cotes Open Quadrature for the Tetrahedron
</h1>
<hr>
<p>
<b>TETRAHEDRON_NCO_RULE</b>
is a C++ library which
defines the weights and abscisass for a sequence of
7 Newton-Cotes open quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
Newton_Cotes rules have the characteristic that the abscissas
are equally spaced. For a tetrahedron, this refers to spacing
in the unit reference tetrahedron, or in the barycentric coordinate
system. These rules may be mapped to an arbitrary tetrahedron,
and will still be valid.
</p>
<p>
The rules are said to be "open" when they do not include points on
the boundary of the tetrahedron.
</p>
<p>
The use of equally spaced abscissas may be important for your
application. That may how your data was collected, for instance.
On the other hand, the use of equally spaced abscissas carries
a few costs. In particular, for a given degree of polynomial
accuracy, there will be rules that achieve this accuracy, but
use fewer abscissas than Newton-Cotes. Moreover, the Newton-Cotes
approach almost always results in negative weights for some
abscissas. This is generally an undesirable feature, particularly
when higher order quadrature rules are being used.
</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>TETRAHEDRON_NCO_RULE</b> is available in
<a href = "../../c_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">a C version</a> and
<a href = "../../cpp_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">a C++ version</a> and
<a href = "../../f_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">a MATLAB version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/cube_felippa_rule/cube_felippa_rule.html">
CUBE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../cpp_src/line_nco_rule/line_nco_rule.html">
LINE_NCO_RULE</a>,
a C++ library which
computes a Newton Cotes Open (NCO) quadrature rule,
using equally spaced points,
over the interior of a line segment in 1D.
</p>
<p>
<a href = "../../cpp_src/pyramid_felippa_rule/pyramid_felippa_rule.html">
PYRAMID_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../cpp_src/simplex_gm_rule/simplex_gm_rule.html">
SIMPLEX_GM_RULE</a>,
a C++ library which
defines Grundmann-Moeller quadrature rules
over the interior of a simplex in M dimensions.
</p>
<p>
<a href = "../../cpp_src/square_felippa_rule/square_felippa_rule.html">
SQUARE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a square in 2D.
</p>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ library which
defines quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_arbq_rule/tetrahedron_arbq_rule.html">
TETRAHEDRON_ARBQ_RULE</a>,
a C++ library which
returns quadrature rules,
with exactness up to total degree 15,
over the interior of a tetrahedron in 3D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a C++ program which
investigates the monomial exactness of a quadrature rule
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_integrals/tetrahedron_integrals.html">
TETRAHEDRON_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_keast_rule/tetrahedron_keast_rule.html">
TETRAHEDRON_KEAST_RULE</a>,
a C++ library which
defines ten quadrature rules, with exactness degrees 0 through 8,
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">
TETRAHEDRON_MONTE_CARLO</a>,
a C++ library which
uses the Monte Carlo method to estimate integrals
over the interior of the unit tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">
TETRAHEDRON_NCC_RULE</a>,
a C++ library which
defines Newton-Cotes Closed (NCC) quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a C++ library which
defines Fekete rules for interpolation or quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_felippa_rule/triangle_felippa_rule.html">
TRIANGLE_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_nco_rule/triangle_nco_rule.html">
TRIANGLE_NCO_RULE</a>,
a C++ library which
defines Newton-Cotes open quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_felippa_rule/wedge_felippa_rule.html">
WEDGE_FELIPPA_RULE</a>,
a C++ library which
returns quadratures rules for approximating integrals
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Peter Silvester,<br>
Symmetric Quadrature Formulae for Simplexes,<br>
Mathematics of Computation,<br>
Volume 24, Number 109, January 1970, pages 95-100.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "tetrahedron_nco_rule.cpp">tetrahedron_nco_rule.cpp</a>, the source code.
</li>
<li>
<a href = "tetrahedron_nco_rule.hpp">tetrahedron_nco_rule.hpp</a>, the include file.
</li>
<li>
<a href = "tetrahedron_nco_rule.sh">tetrahedron_nco_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "tetrahedron_nco_rule_prb.cpp">tetrahedron_nco_rule_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "tetrahedron_nco_rule_prb.sh">tetrahedron_nco_rule_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "tetrahedron_nco_rule_prb_output.txt">tetrahedron_nco_rule_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>R8MAT_DET_4D</b> computes the determinant of a 4 by 4 R8MAT.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_T4</b> maps T4 reference points to physical points.
</li>
<li>
<b>TETRAHEDRON_NCO_DEGREE:</b> degree of an NCO rule for the tetrahedron.
</li>
<li>
<b>TETRAHEDRON_NCO_ORDER_NUM:</b> order of an NCO rule for the tetrahedron.
</li>
<li>
<b>TETRAHEDRON_NCO_RULE</b> returns the points and weights of an NCO rule.
</li>
<li>
<b>TETRAHEDRON_NCO_RULE_NUM</b> returns the number of NCO rules available.
</li>
<li>
<b>TETRAHEDRON_NCO_SUBORDER</b> returns the suborders for an NCO rule.
</li>
<li>
<b>TETRAHEDRON_NCO_SUBORDER_NUM</b> returns the number of suborders for an NCO rule.
</li>
<li>
<b>TETRAHEDRON_NCO_SUBRULE</b> returns a compressed NCO rule.
</li>
<li>
<b>TETRAHEDRON_NCO_SUBRULE_01</b> returns a compressed NCO rule 1.
</li>
<li>
<b>TETRAHEDRON_NCO_SUBRULE_02</b> returns a compressed NCO rule 2.
</li>
<li>
<b>TETRAHEDRON_NCO_SUBRULE_03</b> returns a compressed NCO rule 3.
</li>
<li>
<b>TETRAHEDRON_NCO_SUBRULE_04</b> returns a compressed NCO rule 4.
</li>
<li>
<b>TETRAHEDRON_NCO_SUBRULE_05</b> returns a compressed NCO rule 5.
</li>
<li>
<b>TETRAHEDRON_NCO_SUBRULE_06</b> returns a compressed NCO rule 6.
</li>
<li>
<b>TETRAHEDRON_NCO_SUBRULE_07</b> returns a compressed NCO rule 7.
</li>
<li>
<b>TETRAHEDRON_VOLUME</b> computes the volume of a tetrahedron.
</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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 15 June 2014.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>