-
Notifications
You must be signed in to change notification settings - Fork 57
/
Copy pathint_exactness_chebyshev2.html
427 lines (378 loc) · 12.6 KB
/
int_exactness_chebyshev2.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
<html>
<head>
<title>
INT_EXACTNESS_CHEBYSHEV2 - Exactness of Gauss-Chebyshev Type 2 Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
INT_EXACTNESS_CHEBYSHEV2 <br> Exactness of Gauss-Chebyshev Type 2 Quadrature Rules
</h1>
<hr>
<p>
<b>INT_EXACTNESS_CHEBYSHEV2</b>
is a MATLAB program which
investigates the polynomial exactness of a Gauss-Chebyshev type 2
quadrature rule for the interval [-1,+1].
</p>
<p>
Standard Gauss-Chebyshev type 2 quadrature assumes that the integrand we are
considering has a form like:
<pre>
Integral ( -1 <= x <= +1 ) f(x) * sqrt ( 1 - x^2 ) dx
</pre>
</p>
<p>
A <i>standard Gauss-Chebyshev type 2 quadrature rule</i> is a set of <b>n</b>
positive weights <b>w</b> and abscissas <b>x</b> so that
<pre>
Integral ( -1 <= x <= +1 ) f(x) * ( sqrt ( 1 - x^2 ) dx
</pre>
may be approximated by
<pre>
Sum ( 1 <= I <= N ) w(i) * f(x(i))
</pre>
</p>
<p>
For a standard Gauss-Chebyshev type 2 rule, polynomial exactness is defined in terms of
the function <b>f(x)</b>. That is, we say the rule is exact for polynomials
up to degree DEGREE_MAX if, for any polynomial <b>f(x)</b> of that degree or
less, the quadrature rule will produce the exact value of
<pre>
Integral ( -1 <= x <= +1 ) f(x) * sqrt ( 1 - x^2 ) dx
</pre>
</p>
<p>
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 the corresponding 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 top be checked 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</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>int_exactness_chebyshev2</b> ( <i>'prefix'</i>, <i>degree_max</i> )
</blockquote>
where
<ul>
<li>
<i>'prefix'</i> is a quoted string, 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 monomial degree to check. This would normally be
a relatively small nonnegative number, such as 5, 10 or 15.
</li>
</ul>
</p>
<p>
If the arguments are not supplied on the command line, the
program will prompt for them.
</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>INT_EXACTNESS_CHEBYSHEV2</b> is available in
<a href = "../../cpp_src/int_exactness_chebyshev2/int_exactness_chebyshev2.html">a C++ version</a> and
<a href = "../../f_src/int_exactness_chebyshev2/int_exactness_chebyshev2.html">a FORTRAN90 version</a> and
<a href = "../../m_src/int_exactness_chebyshev2/int_exactness_chebyshev2.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a MATLAB library which
evaluates the Chebyshev polynomial and associated functions.
</p>
<p>
<a href = "../../m_src/chebyshev2_rule/chebyshev2_rule.html">
CHEBYSHEV2_RULE</a>,
a MATLAB program which
generates a Gauss-Chebyshev type 2 quadrature rule.
</p>
<p>
<a href = "../../m_src/hermite_exactness/hermite_exactness.html">
HERMITE_EXACTNESS</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Hermite quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness/int_exactness.html">
INT_EXACTNESS</a>,
a MATLAB program which
tests the polynomial exactness of a quadrature rule for a finite interval.
</p>
<p>
<a href = "../../m_src/int_exactness_chebyshev1/int_exactness_chebyshev1.html">
INT_EXACTNESS_CHEBYSHEV1</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness_gegenbauer/int_exactness_gegenbauer.html">
INT_EXACTNESS_GEGENBAUER</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness_gen_hermite/int_exactness_gen_hermite.html">
INT_EXACTNESS_GEN_HERMITE</a>,
a MATLAB program which
tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness_gen_laguerre/int_exactness_gen_laguerre.html">
INT_EXACTNESS_GEN_LAGUERRE</a>,
a MATLAB program which
tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness_jacobi/int_exactness_jacobi.html">
INT_EXACTNESS_JACOBI</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Jacobi quadrature rules.
</p>
<p>
<a href = "../../m_src/laguerre_exactness/laguerre_exactness.html">
LAGUERRE_EXACTNESS</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Laguerre quadrature rules
for integration over [0,+oo) with density function exp(-x).
</p>
<p>
<a href = "../../m_src/legendre_exactness/legendre_exactness.html">
LEGENDRE_EXACTNESS</a>,
a MATLAB program which
tests the monomial exactness of quadrature rules for the Legendre problem
of integrating a function with density 1 over the interval [-1,+1].
</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 = "int_exactness_chebyshev2.m">int_exactness_chebyshev2.m</a>,
is the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>CHEBY2_O1</b> is a standard Gauss-Chebyshev type 2 order 1 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o1_x.txt">
cheby2_o1_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o1_w.txt">
cheby2_o1_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o1_r.txt">
cheby2_o1_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cheby2_o1_exact.txt">cheby2_o1_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_chebyshev2 ( 'cheby2_o1', 5 )
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>CHEBY2_O2</b> is a standard Gauss-Chebyshev type 2 order 2 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o2_x.txt">
cheby2_o2_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o2_w.txt">
cheby2_o2_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o2_r.txt">
cheby2_o2_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cheby2_o2_exact.txt">cheby2_o2_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_chebyshev2 ( 'cheby2_o2', 5 )
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>CHEBY2_O4</b> is a standard Gauss-Chebyshev type 2 order 4 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o4_x.txt">
cheby2_o4_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o4_w.txt">
cheby2_o4_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o4_r.txt">
cheby2_o4_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cheby2_o4_exact.txt">cheby2_o4_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_chebyshev2 ( 'cheby2_o4', 10 )
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>CHEBY2_O8</b> is a standard Gauss-Chebyshev type 2 order 8 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o8_x.txt">
cheby2_o8_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o8_w.txt">
cheby2_o8_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o8_r.txt">
cheby2_o8_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cheby2_o8_exact.txt">cheby2_o8_exact.txt</a>,
the results of the exactness test.
</li>
the results of the command
<blockquote><b>
int_exactness_chebyshev2 ( 'cheby2_o8', 18 )
</b></blockquote>
</ul>
</p>
<p>
<b>CHEBY2_O16</b> is a standard Gauss-Chebyshev type 2 order 16 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o16_x.txt">
cheby2_o16_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o16_w.txt">
cheby2_o16_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_chebyshev2/cheby2_o16_r.txt">
cheby2_o16_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cheby2_o16_exact.txt">cheby2_o16_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_chebyshev2 ( 'cheby2_o16', 35 )
</b></blockquote>
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 04 March 2008.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>