-
Notifications
You must be signed in to change notification settings - Fork 57
/
cc_project.html
306 lines (281 loc) · 8.92 KB
/
cc_project.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
<html>
<head>
<title>
CC_PROJECT - Clenshaw-Curtis-Like Quadrature for Semi-Infinite and Infinite Intervals
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
CC_PROJECT <br> Clenshaw-Curtis-Like Quadrature for Semi-Infinite and Infinite Intervals
</h1>
<hr>
<p>
<b>CC_PROJECT</b>
is a MATLAB library which
investigates the extension of a Clenshaw-Curtis-like quadrature
scheme to semi-infinite and infinite intervals, and to integrands
with a specified density function.
</p>
<p>
The Legendre integral for f(x) is:
<pre>
I(f) = integral ( -1 <= x <= +1 ) f(x) dx
</pre>
Quadrature rules for the Legendre integral include:
<ul>
<li>
Clenshaw-Curtis quadrature, a sequence of nested quadrature rules,
which include the endpoints; the rule of order N has exactness
N-1 (if N is even) or N (if N is odd);
</li>
<li>
Fejer Type 2 quadrature, a sequence of nested quadrature rules,
which include the endpoints; the rule of order N has exactness
N-1 (if N is even) or N (if N is odd);
</li>
</ul>
</p>
<p>
The Laguerre integral for f(x) is:
<pre>
I(f) = integral ( 0 <= x <= +oo ) f(x) rho(x) dx
</pre>
Depending on the value of the density function rho(x) we have:
<ul>
<li>
rho(x) = exp(-x): Laguerre 0 integral;
</li>
<li>
rho(x) = 1: Laguerre 1 integral.
</li>
</ul>
Quadrature rules include:
<ul>
<li>
CCFI_0 rules for integral ( 0 <= x <= +oo ) f(x) exp(-x) dx;
</li>
<li>
CCFI_1 rules for integral ( 0 <= x <= +oo ) f(x) dx.
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files 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>CC_PROJECT</b> is available in
<a href = "../../m_src/cc_project/cc_project.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
John Boyd,<br>
Exponentially convergent Fourier-Chebyshev quadrature schemes on
bounded and infinite intervals,<br>
Journal of Scientific Computing,<br>
Volume 2, Number 2, 1987, pages 99-109.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cardinal_cos.m">cardinal_cos.m</a>,
evaluates a cardinal cosine interpolation basis function.
</li>
<li>
<a href = "cardinal_sin.m">cardinal_sin.m</a>,
evaluates a cardinal sine interpolation basis function.
</li>
<li>
<a href = "ccff.m">ccff.m</a>,
defines points and weights for Boyd's quadrature rule
for [-1,1] with density 1.
</li>
<li>
<a href = "ccff_asymptotic.m">ccff_asymptotic.m</a>,
examines asymptotic error for a given integrand,
for Boyd's quadrature rule for [-1,+1]
with density 1.
</li>
<li>
<a href = "ccfi_0.m">ccfi_0.m</a>,
defines points and weights for Boyd's quadrature rule
for [0,+oo) with density exp(-x).
</li>
<li>
<a href = "ccfi_1.m">ccfi_1.m</a>,
defines points and weights for Boyd's quadrature rule
for [0,+oo) with density 1.
</li>
<li>
<a href = "ccii_0.m">ccii_0.m</a>,
defines points and weights for Boyd's quadrature rule
for (-oo,+oo) with density exp(-x^2).
</li>
<li>
<a href = "ccii_1.m">ccii_1.m</a>,
defines points and weights for Boyd's quadrature rule
for (-oo,+oo) with density 1.
</li>
<li>
<a href = "chebyshev1_compute.m">chebyshev1_compute.m</a>
computes a Gauss-Chebyshev type 1 quadrature rule.
</li>
<li>
<a href = "chebyshev2_compute.m">chebyshev2_compute.m</a>
computes a Gauss-Chebyshev type 2 quadrature rule.
</li>
<li>
<a href = "chebyshev3_compute.m">chebyshev3_compute.m</a>,
computes a Gauss-Chebyshev type 3 quadrature rule.
</li>
<li>
<a href = "fejer1_compute.m">fejer1_compute.m</a>,
computes a Fejer type 1 quadrature rule.
</li>
<li>
<a href = "fejer2_compute.m">fejer2_compute.m</a>,
computes a Fejer type 2 quadrature rule.
</li>
<li>
<a href = "r8_factorial.m">r8_factorial.m</a>,
evaluates the factorial function.
</li>
<li>
<a href = "r8_mop.m">r8_mop.m</a>,
evaluates an integer power of -1 as an R8.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "cc_project_test.m">cc_project_test.m</a>,
calls all the tests;
</li>
<li>
<a href = "cc_project_test_output.txt">
cc_project_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "cardinal_cos_test.m">cardinal_cos_test.m</a>,
plots a cardinal cosine interpolant basis function.
</li>
<li>
<a href = "cardinal_cos.png">cardinal_cos.png</a>,
a plot of a cardinal cosine interpolant basis function.
</li>
<li>
<a href = "cardinal_sin_test.m">cardinal_sin_test.m</a>,
plots a cardinal sine interpolant basis function.
</li>
<li>
<a href = "cardinal_sin.png">cardinal_sin.png</a>,
a plot of a cardinal sine interpolant basis function.
</li>
<li>
<a href = "cardinal_test.m">cardinal_test.m</a>,
checks the Lagrange property for the cardinal cosine and sine
families.
</li>
<li>
<a href = "ccff_asymptotic_test.m">ccff_asymptotic_test.m</a>,
tests ccff_asymptotic() for a specific integrand.
</li>
<li>
<a href = "ccff_tabulate.m">ccff_tabulate.m</a>,
tabulates CCFF quadrature rules for the Legendre integral.
</li>
<li>
<a href = "ccfi_0_tabulate.m">ccfi_0_tabulate.m</a>,
tabulates CCFI_0 quadrature rules for the Laguerre integral
with density exp(-x).
</li>
<li>
<a href = "ccfi_1_tabulate.m">ccfi_1_tabulate.m</a>,
tabulates CCFI_1 quadrature rules for the Laguerre integral
with density 1.
</li>
<li>
<a href = "ccii_0_tabulate.m">ccii_0_tabulate.m</a>,
tabulates CCII_0 quadrature rules for the Hermite integral
with density exp(-x^2).
</li>
<li>
<a href = "ccii_1_tabulate.m">ccii_1_tabulate.m</a>,
tabulates CCII_1 quadrature rules for the Hermite integral
with density 1.
</li>
<li>
<a href = "compare_ff_test.m">
compare_ff_test.m</a>,
prints out the order 5 version of Boyd's Clenshaw-Curtis type
rule for the Legendre integral, comparing it with several other
rules.
</li>
<li>
<a href = "legendre_exactness.m">
legendre_exactness.m</a>,
tests a quadrature rule for exactness on the Legendre integral.
</li>
<li>
<a href = "legendre_integral.m">
legendre_integral.m</a>,
returns the value of the Legendre integral of a monomial.
</li>
<li>
<a href = "legendre_monomial_quadrature.m">
legendre_monomial_quadrature.m</a>,
determines the error when a quadrature rule is applied to the
Legendre integral of a monomial.
</li>
<li>
<a href = "legendre_test_integral.m">
legendre_test_integral.m</a>,
returns the exact value of the Legendre integral of the
test integrand.
</li>
<li>
<a href = "legendre_test_integrand.m">
legendre_test_integrand.m</a>,
evaluates a test integrand for the Legendre integral.
</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 modified on 25 May 2014.
</i>
<!-- John Burkardt -->
</body>
</html>