forked from johannesgerer/jburkardt-f
-
Notifications
You must be signed in to change notification settings - Fork 2
/
Copy pathnint_exactness_mixed.html
756 lines (695 loc) · 24.1 KB
/
nint_exactness_mixed.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
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
<html>
<head>
<title>
NINT_EXACTNESS_MIXED - Exactness of Multidimensional Quadrature Using Mixed Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
NINT_EXACTNESS_MIXED <br> Exactness of Multidimensional Quadrature Using Mixed Rules
</h1>
<hr>
<p>
<b>NINT_EXACTNESS_MIXED</b>
is a FORTRAN90 program which
investigates the polynomial exactness of a multidimensional
quadrature rule based on a direct product of one-dimensional rules.
The one dimensional rules, in turn, may be any mixture of rules
for 6 standard quadrature problems.
</p>
<p>
The six standard 1d quadrature problems that may be used as factors
for the multidimensional problem are:
<ul>
<li>
<b>Legendre integration</b><br>
interval: [-1,+1],<br>
weight: w(x) = 1,<br>
rules: Clenshaw Curtis, Fejer Type 2, Gauss Legendre, Gauss Patterson.
</li>
<li>
<b>Jacobi integration</b>:<br>
interval: [-1,+1],<br>
weight: w(x) = (1-x)^alpha (1+x)^beta.<br>
rules: Gauss Jacobi.
</li>
<li>
<b>Laguerre integration</b> <br>
interval: [0,+oo),<br>
weight: w(x) = exp(-x).<br>
rules: Gauss Laguerre.
</li>
<li>
<b>Generalized Laguerre integration</b><br>
interval: [0,+oo),<br>
weight: w(x) = x^alpha exp(-x).<br>
rules: Gauss Laguerre.
</li>
<li>
<b>Hermite integration</b><br>
interval: (-oo,+oo),<br>
weight: w(x) = exp(-x*x).<br>
rules: Gauss Hermite.
</li>
<li>
<b>Generalized Hermite integration</b> <br>
interval: (-oo,+oo),<br>
weight: w(x) = |x|^alpha exp(-x*x).<br>
rules: generalized Gauss Hermite.
</li>
</ul>
</p>
<p>
The M-dimensional quadrature region R based on mixed factors is formed by the
direct product
<blockquote><b>
R = R1 x R2 x ... x Rm
</b></blockquote>
where each factor region Ri is the region associated with one of the six
rules. Thus, R is a sort of generalized hyperrectangle, with the understanding
that in some coordinate directions the region may be semi-infinite or
infinite.
</p>
<p>
The M-dimensional weight function W based on mixed factors is formed by the
dproduct
<blockquote><b>
w(x1,x2,...xm) = w1(x1) x w2(x2) x ... x wm(xm)
</b></blockquote>
where each factor weight wi(xi) is the weight function associated with one of the six
rules. Some weight functions include parameters alpha and beta, and these
parameters may be specified independently in each dimension.
</p>
<p>
For a quadrature region R based on mixed factors, the corresponding
monomial integrand has the form
<blockquote><b>
Mono(X,E) = X1^E1 x X2^E2 x ... x Xm^Em
</b></blockquote>
where each exponent Ei is a nonnegative integer.
</p>
<p>
The total degree of a monomial Mono(X,E) is:
<blockquote><b>
TotalDegree(Mono(X,E)) = Sum ( 1 <= I <= M ) E(I)
</b></blockquote>
</p>
<p>
Thus, for instance, the total degree of
<blockquote><b>
x1<sup>2</sup> * x2 * x3<sup>5</sup>
</b></blockquote>
is 2+1+5=8.
</p>
<p>
The corresponding monomial integral is:
<blockquote><b>
Integral ( X in R ) Mono(X,E) W(X) dX
</b></blockquote>
where each exponent Ei is a nonnegative integer.
</p>
<p>
The <b>monomial exactness</b> of a quadrature rule is the maximum number D
such that, for every monomial of total degree D or less, the quadrature
rule produces the exact value of the monomial integral.
</p>
<p>
The <b>polynomial exactness</b> of a quadrature rule is the maximum number D
such that, for every polynomial of total degree D or less, the quadrature
rule produces the exact value of the polynomial integral. The total degree
of a polynomial is simply the maximum of the total degrees of the monomials
that form the polynomial.
</p>
<p>
This program is given a quadrature rule based on mixed factors, and seeks
to determine the polynomial exactness of the rule. It does this simply
by applying the quadrature rule to all the monomials of a total degree
0 up to some limit specified by the user.
</p>
<p>
The program is very flexible and interactive. The quadrature rule
is defined by five files, to be read at input, and the
maximum degree is specified by the user as well.
</p>
<p>
The files that define the quadrature rule
are assumed to have related names, of the form
<ul>
<li>
<i>prefix</i>_<b>a.txt</b>, the "ALPHA" file;
</li>
<li>
<i>prefix</i>_<b>b.txt</b>, the "BETA" file;
</li>
<li>
<i>prefix</i>_<b>r.txt</b>, the "REGION" file;
</li>
<li>
<i>prefix</i>_<b>w.txt</b>, the "WEIGHT" file;
</li>
<li>
<i>prefix</i>_<b>x.txt</b>, the "ABSCISSA" file.
</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 the files.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>nint_exactness_mixed</b> <i>prefix</i> <i>degree_max</i>
</blockquote>
where
<ul>
<li>
<i>prefix</i> is the common prefix for the files containing the alpha, beta, region, weight
and abscissa information of the quadrature rule;
</li>
<li>
<i>degree_max</i> is the maximum total monomial degree to check. This should be
a relatively small nonnegative number, particularly if the
spatial dimension is high. A value of 5 or 10 might be
reasonable, but a value of 50 or 100 is probably never a
good input!
</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>NINT_EXACTNESS_MIXED</b> is available in
<a href = "../../cpp_src/nint_exactness_mixed/nint_exactness_mixed.html">a C++ version</a> and
<a href = "../../f_src/nint_exactness_mixed/nint_exactness_mixed.html">a FORTRAN90 version</a> and
<a href = "../../m_src/nint_exactness_mixed/nint_exactness_mixed.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/int_exactness/int_exactness.html">
INT_EXACTNESS</a>,
a FORTRAN90 program which
tests the polynomial exactness of one dimensional 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/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a FORTRAN90 program which
tests the polynomial exactness of integration rules for the unit hypercube.
</p>
<p>
<a href = "../../f_src/nintlib/nintlib.html">
NINTLIB</a>,
a FORTRAN90 library which
numerically estimates integrals in multiple dimensions.
</p>
<p>
<a href = "../../f_src/pyramid_exactness/pyramid_exactness.html">
PYRAMID_EXACTNESS</a>,
a FORTRAN90 program which
investigates the polynomial exactness of a quadrature rule for the pyramid.
</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 = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules on a
variety of intervals with different weight functions.
</p>
<p>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on a mixed set of 1D factor rules.
</p>
<p>
<a href = "../../f_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a FORTRAN90 library which
defines multidimensional quadrature rules using sparse grid techniques,
based on a mixed set of 1D quadrature rules.
</p>
<p>
<a href = "../../f_src/sphere_exactness/sphere_exactness.html">
SPHERE_EXACTNESS</a>,
a FORTRAN90 program which
tests the polynomial exactness of a quadrature rule for the unit sphere;
</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_nint/test_nint.html">
TEST_NINT</a>,
a FORTRAN90 library which
defines integrand functions for testing
multidimensional quadrature routines.
</p>
<p>
<a href = "../../f_src/testpack/testpack.html">
TESTPACK</a>,
a FORTRAN90 library which
defines a set of integrands used to test multidimensional quadrature.
</p>
<p>
<a href = "../../f_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a FORTRAN90 program which
investigates the polynomial exactness of a quadrature rule for the tetrahedron.
</p>
<p>
<a href = "../../f_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a FORTRAN90 program which
investigates the polynomial exactness of a quadrature rule for the triangle.
</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 = "nint_exactness_mixed.f90">nint_exactness_mixed.f90</a>, the source code.
</li>
<li>
<a href = "nint_exactness_mixed.sh">nint_exactness_mixed.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>SPARSE_GRID_MIXED_D2_L2_CCXCC</b> is a level 2 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Clenshaw Curtis, Clenshaw Curtis].
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxcc_a.txt">
sparse_grid_mixed_d2_l2_ccxcc_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxcc_b.txt">
sparse_grid_mixed_d2_l2_ccxcc_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxcc_r.txt">
sparse_grid_mixed_d2_l2_ccxcc_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxcc_w.txt">
sparse_grid_mixed_d2_l2_ccxcc_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxcc_x.txt">
sparse_grid_mixed_d2_l2_ccxcc_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l2_ccxcc_exactness.txt">
sparse_grid_mixed_d2_l2_ccxcc_exactness.txt</a>,
the output file for DEGREE_MAX = 7.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D2_L2_CCXGL</b> is a level 2 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Legendre].
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxgl_a.txt">
sparse_grid_mixed_d2_l2_ccxgl_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxgl_b.txt">
sparse_grid_mixed_d2_l2_ccxgl_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxgl_r.txt">
sparse_grid_mixed_d2_l2_ccxgl_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxgl_w.txt">
sparse_grid_mixed_d2_l2_ccxgl_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxgl_x.txt">
sparse_grid_mixed_d2_l2_ccxgl_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l2_ccxgl_exactness.txt">
sparse_grid_mixed_d2_l2_ccxgl_exactness.txt</a>,
the output file for DEGREE_MAX = 7.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D2_L2_CCXGLG</b> is a level 2 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Clenshaw Curtis, Generalized Gauss Laguerre].
The Generalized Gauss Laguerre rule uses ALPHA = 1.5.
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxglg_a.txt">
sparse_grid_mixed_d2_l2_ccxglg_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxglg_b.txt">
sparse_grid_mixed_d2_l2_ccxglg_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxglg_r.txt">
sparse_grid_mixed_d2_l2_ccxglg_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxglg_w.txt">
sparse_grid_mixed_d2_l2_ccxglg_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxglg_x.txt">
sparse_grid_mixed_d2_l2_ccxglg_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l2_ccxglg_exactness.txt">
sparse_grid_mixed_d2_l2_ccxglg_exactness.txt</a>,
the output file for DEGREE_MAX = 7.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D2_L2_CCXLG</b> is a level 2 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Laguerre].
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxlg_a.txt">
sparse_grid_mixed_d2_l2_ccxlg_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxlg_b.txt">
sparse_grid_mixed_d2_l2_ccxlg_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxlg_r.txt">
sparse_grid_mixed_d2_l2_ccxlg_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxlg_w.txt">
sparse_grid_mixed_d2_l2_ccxlg_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxlg_x.txt">
sparse_grid_mixed_d2_l2_ccxlg_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l2_ccxlg_exactness.txt">
sparse_grid_mixed_d2_l2_ccxlg_exactness.txt</a>,
the output file for DEGREE_MAX = 7.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D2_L2_F2XGJ</b> is a level 2 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Fejer Type 2, Gauss Jacobi]. The
Gauss Jacobi rule uses ALPHA = 0.5, BETA = 1.5.
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_f2xgj_a.txt">
sparse_grid_mixed_d2_l2_f2xgj_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_f2xgj_b.txt">
sparse_grid_mixed_d2_l2_f2xgj_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_f2xgj_r.txt">
sparse_grid_mixed_d2_l2_f2xgj_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_f2xgj_w.txt">
sparse_grid_mixed_d2_l2_f2xgj_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_f2xgj_x.txt">
sparse_grid_mixed_d2_l2_f2xgj_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l2_f2xgj_exactness.txt">
sparse_grid_mixed_d2_l2_f2xgj_exactness.txt</a>,
the output file for DEGREE_MAX = 7.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D2_L3_CCXGP</b> is a level 3 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Patterson].
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l3_ccxgp_a.txt">
sparse_grid_mixed_d2_l3_ccxgp_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l3_ccxgp_b.txt">
sparse_grid_mixed_d2_l3_ccxgp_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l3_ccxgp_r.txt">
sparse_grid_mixed_d2_l3_ccxgp_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l3_ccxgp_w.txt">
sparse_grid_mixed_d2_l3_ccxgp_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l3_ccxgp_x.txt">
sparse_grid_mixed_d2_l3_ccxgp_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l3_ccxgp_exactness.txt">
sparse_grid_mixed_d2_l3_ccxgp_exactness.txt</a>,
the output file for DEGREE_MAX = 9.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D3_L2_CCXF2XGH</b> is a level 2 sparse grid quadrature rule for
dimension 3, based on 1D factors of [Clenshaw Curtis, Fejer Type 2, Gauss Hermite].
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d3_l2_ccxf2xgh_a.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d3_l2_ccxf2xgh_b.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d3_l2_ccxf2xgh_r.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d3_l2_ccxf2xgh_w.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d3_l2_ccxf2xgh_x.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d3_l2_ccxf2xgh_exactness.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_exactness.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for NINT_EXACTNESS_MIXED.
</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>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</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>MONOMIAL_INTEGRAL_GENERALIZED_HERMITE</b> evaluates a 1D monomial generalized Hermite integral.
</li>
<li>
<b>MONOMIAL_INTEGRAL_GENERALIZED_LAGUERRE</b> evaluates a 1D monomial generalized Laguerre integral.
</li>
<li>
<b>MONOMIAL_INTEGRAL_HERMITE</b> evaluates a 1D monomial Hermite integral.
</li>
<li>
<b>MONOMIAL_INTEGRAL_JACOBI</b> evaluates the integral of a monomial with Jacobi weight.
</li>
<li>
<b>MONOMIAL_INTEGRAL_LAGUERRE</b> evaluates a 1D monomial Laguerre integral.
</li>
<li>
<b>MONOMIAL_INTEGRAL_LEGENDRE</b> evaluates a 1D monomial Legendre integral.
</li>
<li>
<b>MONOMIAL_INTEGRAL_MIXED</b> evaluates a multi-D monomial mixed integral.
</li>
<li>
<b>MONOMIAL_QUADRATURE</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>MONOMIAL_VALUE</b> evaluates a monomial.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function N!!
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>R8_HUGE</b> returns a very large R8.
</li>
<li>
<b>R8_HYPER_2F1</b> evaluates the hypergeometric function 2F1(A,B,C,X).
</li>
<li>
<b>R8_PSI</b> evaluates the function Psi(X).
</li>
<li>
<b>R8MAT_DATA_READ</b> reads data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT 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>S_TO_R8VEC</b> reads an R8VEC 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 14 October 2009.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>