-
Notifications
You must be signed in to change notification settings - Fork 59
/
Copy pathr16_int_exactness_gen_hermite.html
740 lines (675 loc) · 24.6 KB
/
r16_int_exactness_gen_hermite.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
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
<html>
<head>
<title>
R16_INT_EXACTNESS_GEN_HERMITE -
Exactness of Generalized Gauss-Hermite Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
R16_INT_EXACTNESS_GEN_HERMITE <br>
Exactness of Generalized Gauss-Hermite Quadrature Rules
</h1>
<hr>
<p>
<b>R16_INT_EXACTNESS_GEN_HERMITE</b>
is a FORTRAN90 program which
investigates the polynomial exactness of a generalized Gauss-Hermite
quadrature rule for the infinite interval (-oo,+oo), using
"quadruple real precision" arithmetic.
</p>
<p>
The use of quadruple real precision arithmetic is motivated here by the extreme
nature of the calculations being performed. A "reasonable" calculation
might require checking a generalized Gauss-Hermite quadrature rule of order
16, with ALPHA = 1, against monomials up to degree 35. In effect,
this requires us to sum quantities involving some terms like
<b>|x| exp (-x^2) x^35</b> for values of <b>x</b> on the order of 4.
Even with double precision, a symmetric quadrature rule did not necessary
give a value of 0, or close to zero, when handling odd functions (when
the monomial exponent is odd). And results were also unsatisfactory for
monomials with a large even exponent. However, once we moved to
quadruple precision, results were vastly improved, even though we made
no numerical changes to the code. For instance, the Gamma function is
still computed using constants that were suitable for a good double precision
value; one might have thought an improved Gamma evaluator would have been needed,
but apparently it's primarily a simple arithmetic problem.
</p>
<p>
Standard generalized Gauss-Hermite quadrature assumes that the integrand we are
considering has a form like:
<pre>
Integral ( -oo < x < +oo ) |x|^alpha * exp(-x^2) * f(x) dx
</pre>
where the factor <b>|x|^alpha * exp(-x^2)</b> is regarded as a weight factor.
</p>
<p>
A <i>standard generalized Gauss-Hermite quadrature rule</i> is a set of <b>n</b>
positive weights <b>w</b> and abscissas <b>x</b> so that
<pre>
Integral ( -oo < x < +oo ) |x|^alpha * exp(-x^2) * f(x) dx
</pre>
may be approximated by
<pre>
Sum ( 1 <= I <= N ) w(i) * f(x(i))
</pre>
</p>
<p>
It is often convenient to consider approximating integrals in which
the weighting factor <b>|x|^alpha * exp(-x^2)</b> is implicit. In that case, we
are looking at approximating
<pre>
Integral ( -oo < x < +oo ) f(x) dx
</pre>
and it is easy to modify a standard generalized Gauss-Hermite quadrature rule
to handle this case directly.
</p>
<p>
A <i>modified generalized Gauss-Hermite quadrature rule</i> is a set of <b>n</b>
positive weights <b>w</b> and abscissas <b>x</b> so that
<pre>
Integral ( -oo < x < +oo ) f(x) dx
</pre>
may be approximated by
<pre>
Sum ( 1 <= I <= N ) w(i) * f(x(i))
</pre>
</p>
<p>
When using a generalized Gauss-Hermite quadrature rule, it's important to know whether
the rule has been developed for the standard or modified cases.
Basically, the only change is that the weights of the modified rule have
been divided by the weighting function evaluated at the corresponding abscissa.
</p>
<p>
For a standard generalized Gauss-Hermite 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 ( -oo < x < +oo ) |x|^alpha * exp(-x^2) * f(x) dx
</pre>
</p>
<p>
For a modified generalized Gauss-Hermite rule, polynomial exactness is defined in terms of
the function <b>f(x)</b> divided by the implicit weighting function. That is,
we say a modified generalized Gauss-Hermite rule is exact for polynomials up to degree
DEGREE_MAX if, for any integrand <b>f(x)</b> with the property that
<b>f(x)/(|x|^alpha*exp(-x^2))</b> is a polynomial of degree no more than DEGREE_MAX,
the quadrature rule will product the exact value of:
<pre>
Integral ( -oo < x < +oo ) f(x) 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>
If the program understands that the rule being considered is a modified rule,
then the monomials are multiplied by <b>|x|^alpha * exp(-x^2)</b> when performing the
exactness test.
</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>r16_int_exactness_gen_hermite</b> <i>prefix</i> <i>degree_max</i> <i>alpha</i>
<i>option</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 monomial degree to check. This would normally be
a relatively small nonnegative number, such as 5, 10 or 15.
</li>
<li>
<i>alpha</i> is the value of the parameter, which should be a real number greater than -1.0.
Setting <i>alpha</i> to 0.0 results in the basic (non-generalized) Gauss-Hermite rule.
</li>
<li>
<i>option</i>:<br>
0 indicates a standard rule for integrating |x|^alpha * exp(-x^2)*f(x).<br>
1 indicates a modified rule for integrating f(x).
</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">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/gen_hermite_rule/gen_hermite_rule.html">
GEN_HERMITE_RULE</a>,
a FORTRAN90 program which
can generate a generalized Gauss-Hermite quadrature
rule on request.
</p>
<p>
<a href = "../../f_src/int_exactness/int_exactness.html">
INT_EXACTNESS</a>,
a FORTRAN90 program which
tests the polynomial exactness of a quadrature rule for a finite interval.
</p>
<p>
<a href = "../../f_src/int_exactness_gen_hermite/int_exactness_gen_hermite.html">
INT_EXACTNESS_GEN_HERMITE</a>,
a FORTRAN90 program which
tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.
It is the double precision program upon which this quadruple precision program
is based.
</p>
<p>
<a href = "../../f_src/int_exactness_gen_laguerre/int_exactness_gen_laguerre.html">
INT_EXACTNESS_GEN_LAGUERRE</a>,
a FORTRAN90 program which
tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_hermite/int_exactness_hermite.html">
INT_EXACTNESS_HERMITE</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Hermite quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_jacobi/int_exactness_jacobi.html">
INT_EXACTNESS_JACOBI</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Jacobi quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_laguerre/int_exactness_laguerre.html">
INT_EXACTNESS_LAGUERRE</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Laguerre quadrature rules.
</p>
<p>
<a href = "../../f_src/int_exactness_legendre/int_exactness_legendre.html">
INT_EXACTNESS_LEGENDRE</a>,
a FORTRAN90 program which
tests the polynomial exactness of Gauss-Legendre quadrature rules.
</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/intlib/intlib.html">
INTLIB</a>,
a FORTRAN90 library which
numerically estimate integrals in one dimension.
</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_gen_hermite/quadrature_rules_gen_hermite.html">
QUADRATURE_RULES_GEN_HERMITE</a>,
a dataset directory which
contains sets of files that define generalized Gauss-Hermite quadrature rules.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
define quadrature rules on a
variety of intervals with different weight functions.
</p>
<p>
<a href = "../../f_src/r16_hermite_rule/r16_hermite_rule.html">
R16_HERMITE_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Hermite quadrature rule, using
"quadruple precision real" arithmetic.
</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_int_hermite/test_int_hermite.html">
TEST_INT_HERMITE</a>,
a FORTRAN90 library which
define integrand functions that can be approximately integrated by a Gauss-Hermite rule.
</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>
<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 = "r16_int_exactness_gen_hermite.f90">r16_int_exactness_gen_hermite.f90</a>, the source code.
</li>
<li>
<a href = "r16_int_exactness_gen_hermite.sh">r16_int_exactness_gen_hermite.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>GEN_HERM_O1_A1.0</b> is a standard generalized Gauss-Hermite order 1 rule with ALPHA = 1.0.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o1_a1.0_x.txt">
gen_herm_o1_a1.0_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o1_a1.0_w.txt">
gen_herm_o1_a1.0_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o1_a1.0_r.txt">
gen_herm_o1_a1.0_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_herm_o1_a1.0_exact.txt">gen_herm_o1_a1.0_exact.txt</a>,
the results of the command
<blockquote><b>
r16_int_exactness_gen_hermite gen_herm_o1_a1.0 5 1.0 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_HERM_O2_A1.0</b> is a standard generalized Gauss-Hermite order 2 rule with ALPHA = 1.0.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o2_a1.0_x.txt">
gen_herm_o2_a1.0_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o2_a1.0_w.txt">
gen_herm_o2_a1.0_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o2_a1.0_r.txt">
gen_herm_o2_a1.0_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_herm_o2_a1.0_exact.txt">gen_herm_o2_a1.0_exact.txt</a>,
the results of the command
<blockquote><b>
r16_int_exactness_gen_hermite gen_herm_o2_a1.0 5 1.0 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_HERM_O4_A1.0</b> is a standard generalized Gauss-Hermite order 4 rule with ALPHA = 1.0.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o4_a1.0_x.txt">
gen_herm_o4_a1.0_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o4_a1.0_w.txt">
gen_herm_o4_a1.0_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o4_a1.0_r.txt">
gen_herm_o4_a1.0_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_herm_o4_a1.0_exact.txt">gen_herm_o4_a1.0_exact.txt</a>,
the results of the command
<blockquote><b>
r16_int_exactness_gen_hermite gen_herm_o4_a1.0 10 1.0 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_HERM_O8_A1.0</b> is a standard generalized Gauss-Hermite order 8 rule with ALPHA = 1.0.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o8_a1.0_x.txt">
gen_herm_o8_a1.0_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o8_a1.0_w.txt">
gen_herm_o8_a1.0_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o8_a1.0_r.txt">
gen_herm_o8_a1.0_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_herm_o8_a1.0_exact.txt">gen_herm_o8_a1.0_exact.txt</a>,
the results of the command
<blockquote><b>
r16_int_exactness_gen_hermite gen_herm_o8_a1.0 18 1.0 0
</b></blockquote>
</ul>
</p>
<p>
<b>GEN_HERM_O16_A1.0</b> is a standard generalized Gauss-Hermite order 16 rule with ALPHA = 1.0.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o16_a1.0_x.txt">
gen_herm_o16_a1.0_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o16_a1.0_w.txt">
gen_herm_o16_a1.0_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o16_a1.0_r.txt">
gen_herm_o16_a1.0_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_herm_o16_a1.0_exact.txt">gen_herm_o16_a1.0_exact.txt</a>,
the results of the command
<blockquote><b>
r16_int_exactness_gen_hermite gen_herm_o16_a1.0 35 1.0 0
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_HERM_O1_A1.0_MODIFIED</b> is a modified generalized Gauss-Hermite order 1 rule with ALPHA = 1.0.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o1_a1.0_modified_x.txt">
gen_herm_o1_a1.0_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o1_a1.0_modified_w.txt">
gen_herm_o1_a1.0_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o1_a1.0_modified_r.txt">
gen_herm_o1_a1.0_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_herm_o1_a1.0_modified_exact.txt">gen_herm_o1_a1.0_modified_exact.txt</a>,
the results of the command
<blockquote><b>
r16_int_exactness_gen_hermite gen_herm_o1_a1.0_modified 5 1.0 1
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_HERM_O2_A1.0_MODIFIED</b> is a modified generalized Gauss-Hermite order 2 rule with ALPHA = 1.0.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o2_a1.0_modified_x.txt">
gen_herm_o2_a1.0_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o2_a1.0_modified_w.txt">
gen_herm_o2_a1.0_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o2_a1.0_modified_r.txt">
gen_herm_o2_a1.0_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_herm_o2_a1.0_modified_exact.txt">gen_herm_o2_a1.0_modified_exact.txt</a>,
the results of the command
<blockquote><b>
r16_int_exactness_gen_hermite gen_herm_o2_a1.0_modified 5 1.0 1
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_HERM_O4_A1.0_MODIFIED</b> is a modified generalized Gauss-Hermite order 4 rule with ALPHA = 1.0.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o4_a1.0_modified_x.txt">
gen_herm_o4_a1.0_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o4_a1.0_modified_w.txt">
gen_herm_o4_a1.0_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o4_a1.0_modified_r.txt">
gen_herm_o4_a1.0_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "herm_o4_a1.0_modified_exact.txt">gen_herm_o4_a1.0_modified_exact.txt</a>,
the results of the command
<blockquote><b>
int_exactness_gen_hermite gen_herm_o4_a1.0_modified 10 1.0 1
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>GEN_HERM_O8_A1.0_MODIFIED</b> is a modified generalized Gauss-Hermite order 8 rule with ALPHA = 1.0.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o8_a1.0_modified_x.txt">
gen_herm_o8_a1.0_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o8_a1.0_modified_w.txt">
gen_herm_o8_a1.0_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o8_a1.0_modified_r.txt">
gen_herm_o8_a1.0_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_herm_o8_a1.0_modified_exact.txt">gen_herm_o8_a1.0_modified_exact.txt</a>,
the results of the command
<blockquote><b>
r16_int_exactness_gen_hermite gen_herm_o8_a1.0_modified 18 1.0 1
</b></blockquote>
</ul>
</p>
<p>
<b>GEN_HERM_O16_A1.0_MODIFIED</b> is a modified generalized Gauss-Hermite order 16 rule with ALPHA = 1.0.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o16_a1.0_modified_x.txt">
gen_herm_o16_a1.0_modified_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o16_a1.0_modified_w.txt">
gen_herm_o16_a1.0_modified_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_gen_hermite/gen_herm_o16_a1.0_modified_r.txt">
gen_herm_o16_a1.0_modified_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gen_herm_o16_a1.0_modified_exact.txt">gen_herm_o16_a1.0_modified_exact.txt</a>,
the results of the command
<blockquote><b>
r16_int_exactness_gen_hermite gen_herm_o16_a1.0_modified 35 1.0 1
</b></blockquote>
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for R16_INT_EXACTNESS_GEN_HERMITE.
</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>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>HERMITE_INTEGRAL2</b> returns the value of a Hermite integral.
</li>
<li>
<b>MONOMIAL_QUADRATURE_GEN_HERMITE</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>R16MAT_DATA_READ</b> reads data from an R16MAT file.
</li>
<li>
<b>R16MAT_HEADER_READ</b> reads the header from an R16MAT file.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_R16</b> reads an R16 from a string.
</li>
<li>
<b>S_TO_R16VEC</b> reads an R16VEC 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>
</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 30 May 2010.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>