/
jacobi_rule.html
447 lines (395 loc) · 12 KB
/
jacobi_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
435
436
437
438
439
440
441
442
443
444
445
446
447
<html>
<head>
<title>
JACOBI_RULE - Gauss-Jacobi Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
JACOBI_RULE <br> Gauss-Jacobi Quadrature Rules
</h1>
<hr>
<p>
<b>JACOBI_RULE</b>
is a FORTRAN90 program which
generates a specific Gauss-Jacobi 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 <i>Gauss-Jacobi quadrature rule </i> is used as follows:
<pre>
Integral ( A <= x <= B ) (B-x)^alpha (x-A)^beta f(x) dx
</pre>
is to be approximated by
<pre>
Sum ( 1 <= i <= order ) w(i) * f(x(i))
</pre>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>jacobi_rule</b> <i>order</i> <i>alpha</i> <i>beta</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.
</li>
<li>
<i>alpha</i> is the exponent of (1-x), which must be greater than -1.
</li>
<li>
<i>beta</i> is the exponent of (1+x), which must be greater than -1.
</li>
<li>
<i>a</i> is the left endpoint;
</li>
<li>
<i>b</i> is the right endpoint.
</li>
<li>
<i>filename</i> specifies how the rule is to be reported:
<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">
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>JACOBI_RULE</b> is available in
<a href = "../../cpp_src/jacobi_rule/jacobi_rule.html">a C++ version</a> and
<a href = "../../f_src/jacobi_rule/jacobi_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/jacobi_rule/jacobi_rule.html">a MATLAB version.</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/int_exactness.html">
INT_EXACTNESS</a>,
a FORTRAN90 program which
checks the polynomial exactness
of a 1-dimensional quadrature rule for a finite interval.
</p>
<p>
<a href = "../../f_src/int_exactness_jacobi/int_exactness_jacobi.html">
INT_EXACTNESS_JACOBI</a>,
a FORTRAN90 program which
checks the polynomial exactness
of a Gauss-Jacobi rule.
</p>
<p>
<a href = "../../f_src/intlib/intlib.html">
INTLIB</a>,
a FORTRAN90 library which
contains a variety
of routines for numerical estimation of integrals in 1D.
</p>
<p>
<a href = "../../f_src/jacobi_polynomial/jacobi_polynomial.html">
JACOBI_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Jacobi polynomial and associated functions.
</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
computes 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/patterson_rule/patterson_rule.html">
PATTERSON_RULE</a>,
a FORTRAN90 program which
computes a Gauss-Patterson quadrature rule.
</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_jacobi/quadrature_rules_jacobi.html">
QUADRATURE_RULES_JACOBI</a>,
a dataset directory which
contains triples of files defining Gauss-Jacobi quadrature rules.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
contains 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
defines functions that may be used as test integrands for
quadrature rules in 1D.
</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>
Sylvan Elhay, Jaroslav Kautsky,<br>
Algorithm 655:
IQPACK,
FORTRAN Subroutines for the Weights of Interpolatory Quadrature,<br>
ACM Transactions on Mathematical Software,<br>
Volume 13, Number 4, December 1987, pages 399-415.
</li>
<li>
Jaroslav Kautsky, Sylvan Elhay,<br>
Calculation of the Weights of Interpolatory Quadratures,<br>
Numerische Mathematik,<br>
Volume 40, 1982, pages 407-422.
</li>
<li>
Roger Martin, James Wilkinson,<br>
The Implicit QL Algorithm,<br>
Numerische Mathematik,<br>
Volume 12, Number 5, December 1968, pages 377-383.
</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 = "jacobi_rule.f90">jacobi_rule.f90</a>, the source code.
</li>
<li>
<a href = "jacobi_rule.sh">jacobi_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "jac_o4_a0.5_b1.5_r.txt">jac_o4_a0.5_b1.5_r.txt</a>,
the region file created by the command
<pre><b>
jacobi_rule 4 0.5 1.5 -1.0 +1.0 jac_o4_a0.5_b1.5
</b></pre>
</li>
<li>
<a href = "jac_o4_a0.5_b1.5_w.txt">jac_o4_a0.5_b1.5_w.txt</a>,
the weight file created by the command
<pre><b>
jacobi_rule 4 0.5 1.5 -1.0 +1.0 jac_o4_a0.5_b1.5
</b></pre>
</li>
<li>
<a href = "jac_o4_a0.5_b1.5_x.txt">jac_o4_a0.5_b1.5_x.txt</a>,
the abscissa file created by the command
<pre><b>
jacobi_rule 4 0.5 1.5 -1.0 +1.0 jac_o4_a0.5_b1.5
</b></pre>
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for JACOBI_RULE.
</li>
<li>
<b>CDGQF</b> computes a Gauss quadrature formula with default A, B and simple knots.
</li>
<li>
<b>CGQF</b> computes knots and weights of a Gauss quadrature formula.
</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>CLASS_MATRIX</b> computes the Jacobi matrix for a quadrature rule.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<b>PARCHK</b> checks parameters ALPHA and BETA for classical weight functions.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>RULE_WRITE</b> writes a quadrature rule to a 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>SCQF</b> scales a quadrature formula to a nonstandard interval.
</li>
<li>
<b>SGQF</b> computes knots and weights of a Gauss Quadrature formula.
</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 22 February 2010.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>