-
Notifications
You must be signed in to change notification settings - Fork 59
/
Copy pathpatterson_rule.html
434 lines (382 loc) · 12 KB
/
patterson_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
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
<html>
<head>
<title>
PATTERSON_RULE - Gauss-Patterson Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
PATTERSON_RULE <br> Gauss-Patterson Quadrature Rules
</h1>
<hr>
<p>
<b>PATTERSON_RULE</b>
is a FORTRAN90 program which
generates a specific Gauss-Patterson quadrature rule,
based on user input.
</p>
<p>
The rule is written to three files for easy use as input
to other programs.
</p>
<p>
The Gauss-Patterson quadrature is a nested family which begins with
the Gauss-Legendre rules of orders 1 and 3, and then succesively inserts
one new abscissa in each subinterval. Thus, after the second rule, the
Gauss-Patterson rules do not have the super-high precision of the
Gauss-Legendre rules. They trade this precision in exchange for the
advantages of nestedness. This means that Gauss-Patterson rules are
only available for orders of 1, 3, 7, 15, 31, 63, 127 or 255.
</p>
<p>
The <i>standard Gauss-Patterson quadrature rule </i> is used as follows:
<pre>
Integral ( A <= x <= B ) f(x) dx
</pre>
is to be approximated by
<pre>
Sum ( 1 <= i <= order ) w(i) * f(x(i))
</pre>
</p>
<p>
The polynomial precision of a Gauss-Patterson rule can be checked
numerically by the <b>INT_EXACTNESS_LEGENDRE</b> program. We should expect
<table border=1>
<tr>
<th>Index</th><th>Order</th><th>Free+Fixed</th><th>Expected Precision</th><th>Actual Precision</th>
</tr>
<tr>
<td>0</td><td>1</td><td>1 + 0</td><td>2*1+0-1=1</td><td>1</td>
</tr>
<tr>
<td>1</td><td>3</td><td>3 + 0</td><td>2*3+0-1=5</td><td>5</td>
</tr>
<tr>
<td>2</td><td>7</td><td>4 + 3</td><td>2*4+3-1=10</td><td>10 + 1 = 11</td>
</tr>
<tr>
<td>3</td><td>15</td><td>8 + 7</td><td>2*8+7-1=22</td><td>22 + 1 = 23</td>
</tr>
<tr>
<td>4</td><td>31</td><td>16 + 15</td><td>2*16+15-1=46</td><td>46 + 1 = 47</td>
</tr>
<tr>
<td>5</td><td>63</td><td>32 + 31</td><td>2*32+31-1=94</td><td>94 + 1 = 95</td>
</tr>
<tr>
<td>6</td><td>127</td><td>64 + 63</td><td>2*64+63-1=190</td><td>190 + 1 = 191</td>
</tr>
<tr>
<td>7</td><td>255</td><td>128 + 127</td><td>2*128+127-1=382</td><td>382 + 1 = 383</td>
</tr>
</table>
where the extra 1 degree of precision comes about because the rules are symmetric,
and can integrate any odd monomial exactly. Thus, after the first rule, the
precision is 3*2^index - 1.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>patterson_rule</b> <i>order</i> <i>a</i> <i>b</i> <i>filename</i>
</blockquote>
where
<ul>
<li>
<i>order</i> is the number of points in the quadrature rule. Acceptable values are
1, 3, 7, 15, 31, 63, 127 or 255.
</li>
<li>
<i>a</i> is the left endpoint;
</li>
<li>
<i>b</i> is the right endpoint;
</li>
<li>
<i>filename</i> specifies the output filenames:
<i>filename</i><b>_w.txt</b>,
<i>filename</i><b>_x.txt</b>, and <i>filename</i><b>_r.txt</b>,
containing the weights, abscissas, and interval limits.
</li>
</ul>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>PATTERSON_RULE</b> is available in
<a href = "../../cpp_src/patterson_rule/patterson_rule.html">a C++ version</a> and
<a href = "../../f_src/patterson_rule/patterson_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/patterson_rule/patterson_rule.html">a MATLAB version</a>.
</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/ccn_rule/ccn_rule.html">
CCN_RULE</a>,
a FORTRAN90 program which
defines a nested Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../f_src/chebyshev1_rule/chebyshev1_rule.html">
CHEBYSHEV1_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Chebyshev type 1 quadrature rule.
</p>
<p>
<a href = "../../f_src/chebyshev2_rule/chebyshev2_rule.html">
CHEBYSHEV2_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Chebyshev type 2 quadrature rule.
</p>
<p>
<a href = "../../f_src/clenshaw_curtis_rule/clenshaw_curtis_rule.html">
CLENSHAW_CURTIS_RULE</a>,
a FORTRAN90 program which
defines a Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../f_src/gegenbauer_rule/gegenbauer_rule.html">
GEGENBAUER_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Gegenbauer quadrature rule.
</p>
<p>
<a href = "../../f_src/gen_hermite_rule/gen_hermite_rule.html">
GEN_HERMITE_RULE</a>,
a FORTRAN90 program which
can compute and print a generalized Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../f_src/gen_laguerre_rule/gen_laguerre_rule.html">
GEN_LAGUERRE_RULE</a>,
a FORTRAN90 program which
can compute and print a generalized Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../f_src/hermite_rule/hermite_rule.html">
HERMITE_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../f_src/int_exactness_legendre/int_exactness_legendre.html">
INT_EXACTNESS_LEGENDRE</a>,
a FORTRAN90 program which
checks the polynomial exactness of a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../f_src/jacobi_rule/jacobi_rule.html">
JACOBI_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Jacobi quadrature rule.
</p>
<p>
<a href = "../../f_src/laguerre_rule/laguerre_rule.html">
LAGUERRE_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../f_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../f_src/legendre_rule_fast/legendre_rule_fast.html">
LEGENDRE_RULE_FAST</a>,
a FORTRAN90 program which
uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.
</p>
<p>
<a href = "../../f_src/product_rule/product_rule.html">
PRODUCT_RULE</a>,
a FORTRAN90 program which
constructs a product rule
from 1D factor rules.
</p>
<p>
<a href = "../../f_src/quadpack/quadpack.html">
QUADPACK</a>,
a FORTRAN90 library which
contains routines for numerical estimation of integrals in 1D.
</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 = "../../datasets/quadrature_rules_patterson/quadrature_rules_patterson.html">
QUADRATURE_RULES_PATTERSON</a>,
a dataset directory which
contains triples of files defining standard Gauss-Patterson
quadrature rules.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines 1-dimensional quadrature rules.
</p>
<p>
<a href = "../../f_src/tanh_sinh_rule/tanh_sinh_rule.html">
TANH_SINH_RULE</a>,
a FORTRAN90 program which
computes and writes out a tanh-sinh quadrature rule of given order.
</p>
<p>
<a href = "../../f_src/test_int/test_int.html">
TEST_INT</a>,
a FORTRAN90 library which
contains functions that may be used as test integrands for
quadrature rules in 1D.
</p>
<p>
<a href = "../../f77_src/toms699/toms699.html">
TOMS699</a>,
a FORTRAN77 library which
implements a new representation of Patterson's quadrature formula;<br>
this is ACM TOMS algorithm 699.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<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>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "patterson_rule.f90">patterson_rule.f90</a>, the source code.
</li>
<li>
<a href = "patterson_rule.sh">patterson_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "gp_o15_r.txt">gp_o15_r.txt</a>,
the region file created by the command
<pre><b>
patterson_rule 15 gp_o15
</b></pre>
</li>
<li>
<a href = "gp_o15_w.txt">gp_o15_w.txt</a>,
the weight file created by the command
<pre><b>
patterson_rule 15 gp_o15
</b></pre>
</li>
<li>
<a href = "gp_o15_x.txt">gp_o15_x.txt</a>,
the abscissa file created by the command
<pre><b>
patterson_rule 15 gp_o15
</b></pre>
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for PATTERSON_RULE.
</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>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>ORDER_CHECK</b> checks the value of ORDER.
</li>
<li>
<b>PATTERSON_HANDLE</b> looks up the requested Gauss-Patterson rule and outputs it.
</li>
<li>
<b>PATTERSON_SET</b> sets abscissas and weights for Gauss-Patterson quadrature.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>RESCALE</b> rescales a Legendre quadrature rule from [-1,+1] to [A,B].
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 value from a string.
</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 16 February 2010.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>