forked from johannesgerer/jburkardt-f
-
Notifications
You must be signed in to change notification settings - Fork 1
/
triangle_exactness.html
453 lines (403 loc) · 12.8 KB
/
triangle_exactness.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
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
<html>
<head>
<title>
TRIANGLE_EXACTNESS - Exactness of Quadrature on Triangles
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGLE_EXACTNESS <br>
Exactness of Quadrature on Triangles
</h1>
<hr>
<p>
<b>TRIANGLE_EXACTNESS</b>
is a FORTRAN90 program which
investigates the polynomial exactness of a
quadrature rule for a triangle.
</p>
<p>
The polynomial exactness of a quadrature rule is defined as the
highest total degree <b>D</b> such that the quadrature rule is
guaranteed to integrate exactly all polynomials of total degree
<b>DEGREE_MAX</b> or less, ignoring roundoff.
The total degree of a polynomial is the maximum of the degrees
of all its monomial terms. For a triangle, the degree of a monomial
term is the sum of the exponents of <b>x</b> and <b>y</b>.
Thus, for instance, the <b>DEGREE</b> of
<blockquote><b>
x<sup>2</sup>y<sup>5</sup>
</b></blockquote>
is 2+5=7.
</p>
<p>
To be thorough, the program starts at <b>DEGREE</b> = 0, and then
proceeds to <b>DEGREE</b> = 1, 2, and so on up to a maximum degree
<b>DEGREE_MAX</b> specified by the user. At each value of <b>DEGREE</b>,
the program generates every possible monomial term, applies the
quadrature rule to it, and determines the quadrature error. The program
uses a scaling factor on each monomial so that the exact integral
should always be 1; therefore, each reported error can be compared
on a fixed scale.
</p>
<p>
The program is very flexible and interactive. The quadrature rule
is defined by three files, to be read at input, and the
maximum degree is specified by the user as well.
</p>
<p>
Note that the three files that define the quadrature rule
are assumed to have related names, of the form
<ul>
<li>
<i>prefix</i>_<b>x.txt</b>
</li>
<li>
<i>prefix</i>_<b>w.txt</b>
</li>
<li>
<i>prefix</i>_<b>r.txt</b>
</li>
</ul>
When running the program, the user only enters the common <i>prefix</i>
part of the file names, which is enough information for the program
to find all three files.
</p>
<p>
For information on the form of these files, see the
<b>QUADRATURE_RULES_TRI</b> directory listed below.
</p>
<p>
The exactness results are written to an output file with the
corresponding name:
<ul>
<li>
<i>prefix</i>_<b>exact.txt</b>
</li>
</ul>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>triangle_exactness</b> <i>prefix</i> <i>degree_max</i>
</blockquote>
where
<ul>
<li>
<i>prefix</i> is the common prefix for the files containing the abscissa, weight
and region information of the quadrature rule;
</li>
<li>
<i>degree_max</i> is the maximum total monomial degree to check. A value of 5 or 10
might be reasonable, but a value of 50 or 100 is probably never a
good input!
</li>
</ul>
</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>TRIANGLE_EXACTNESS</b> is available in
<a href = "../../cpp_src/triangle_exactness/triangle_exactness.html">a C++ version</a> and
<a href = "../../f_src/triangle_exactness/triangle_exactness.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangle_exactness/triangle_exactness.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/dunavant/dunavant.html">
DUNAVANT</a>,
a FORTRAN90 library which
sets up a Dunavant quadrature rule for a triangle.
</p>
<p>
<a href = "../../f_src/fekete/fekete.html">
FEKETE</a>,
a FORTRAN90 library which
sets up a Fekete quadrature rule for a triangle.
</p>
<p>
<a href = "../../f_src/gm_rule/gm_rule.html">
GM_RULE</a>,
a FORTRAN90 library which
defines a Grundmann-Moeller
rule for quadrature over a triangle, tetrahedron, or general
M-dimensional simplex.
</p>
<p>
<a href = "../../f_src/integral_test/integral_test.html">
INTEGRAL_TEST</a>,
a FORTRAN90 program which
uses test integrals to measure the effectiveness of
certain sets of quadrature rules.
</p>
<p>
<a href = "../../f_src/lyness_rule/lyness_rule.html">
LYNESS_RULE</a>,
a FORTRAN90 library which
returns Lyness-Jespersen quadrature rules for the triangle.
</p>
<p>
<a href = "../../f_src/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a FORTRAN90 program which
tests the exactness of integration rules over the interval,
rectangle, or multidimensional rectangle.
</p>
<p>
<a href = "../../f_src/nint_exactness_mixed/nint_exactness_mixed.html">
NINT_EXACTNESS_MIXED</a>,
a FORTRAN90 program which
measures the polynomial exactness of a multidimensional quadrature rule
based on a mixture of 1D quadrature rule factors.
</p>
<p>
<a href = "../../f_src/pyramid_exactness/pyramid_exactness.html">
PYRAMID_EXACTNESS</a>,
a FORTRAN90 program which
investigates the polynomial exactness of a quadrature rule for the pyramid.
</p>
<p>
<a href = "../../datasets/quadrature_rules_tri/quadrature_rules_tri.html">
QUADRATURE_RULES_TRI</a>,
a dataset directory which
contains sets of files that define quadrature
rules over the triangle.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules on a
variety of intervals with different weight functions.
</p>
<p>
<a href = "../../f_src/sphere_exactness/sphere_exactness.html">
SPHERE_EXACTNESS</a>,
a FORTRAN90 program which
tests the polynomial exactness of a quadrature rule for the unit sphere;
</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_tri_int/test_tri_int.html">
TEST_TRI_INT</a>,
a FORTRAN90 library which
defines integrand functions for testing
quadrature routines for the triangle.
</p>
<p>
<a href = "../../f_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a FORTRAN90 program which
investigates the polynomial exactness of a quadrature rule for the tetrahedron.
</p>
<p>
<a href = "../../f_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a FORTRAN90 program which
uses the Monte Carlo method to estimate integrals over a triangle.
</p>
<p>
<a href = "../../f_src/wandzura/wandzura.html">
WANDZURA</a>,
a FORTRAN90 library which
sets up a Wandzura quadrature rule for a triangle.
</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 = "triangle_exactness.f90">triangle_exactness.f90</a>, the source code.
</li>
<li>
<a href = "triangle_exactness.sh">triangle_exactness.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
The "W", "X" and "R" files associated with the point sets tested
here may be found in the <b>QUADRATURE_RULES_TRI</b> directory.
</p>
<p>
<ul>
<li>
<a href = "centroid_exact.txt">centroid_exact.txt</a>.
</li>
<li>
<a href = "gauss4x4_exact.txt">gauss4x4_exact.txt</a>
(the results here have significant error. The abscissas and
weights were not available to the full precision.)
</li>
<li>
<a href = "gauss8x8_exact.txt">gauss8x8_exact.txt</a>
(the results here have significant error. The abscissas and
weights were not available to the full precision.)
</li>
<li>
<a href = "seven_point_exact.txt">seven_point_exact.txt</a>.
</li>
<li>
<a href = "strang1_exact.txt">strang1_exact.txt</a>.
</li>
<li>
<a href = "strang2_exact.txt">strang2_exact.txt</a>.
</li>
<li>
<a href = "strang3_exact.txt">strang3_exact.txt</a>.
</li>
<li>
<a href = "strang4_exact.txt">strang4_exact.txt</a>.
</li>
<li>
<a href = "strang5_exact.txt">strang5_exact.txt</a>.
</li>
<li>
<a href = "strang6_exact.txt">strang6_exact.txt</a>.
</li>
<li>
<a href = "strang7_exact.txt">strang7_exact.txt</a>.
</li>
<li>
<a href = "strang8_exact.txt">strang8_exact.txt</a>.
</li>
<li>
<a href = "strang9_exact.txt">strang9_exact.txt</a>.
</li>
<li>
<a href = "strang10_exact.txt">strang10_exact.txt</a>.
</li>
<li>
<a href = "toms584_19_exact.txt">toms584_19_exact.txt</a>.
</li>
<li>
<a href = "toms612_19_exact.txt">toms612_19_exact.txt</a>
</li>
<li>
<a href = "toms612_28_exact.txt">toms612_28_exact.txt</a>.
</li>
<li>
<a href = "toms706_37_exact.txt">toms706_37_exact.txt</a>.
</li>
<li>
<a href = "vertex_exact.txt">vertex_exact.txt</a>.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TRIANGLE_EXACTNESS.
</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>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</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>MONOMIAL_VALUE</b> evaluates a monomial.
</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>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRIANGLE_AREA</b> computes the area of a triangle.
</li>
<li>
<b>TRIANGLE_ORDER3_PHYSICAL_TO_REFERENCE</b> maps physical points to reference points.
</li>
<li>
<b>TRIANGLE01_MONOMIAL_INTEGRAL:</b> integral of a monomial over the unit triangle.
</li>
<li>
<b>TRIANGLE01_MONOMIAL_QUADRATURE:</b> approximate monomial integral, unit triangle.
</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 September 2010.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>