forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
sandia_sgmga.html
723 lines (670 loc) · 22.8 KB
/
sandia_sgmga.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
<html>
<head>
<title>
SANDIA_SGMGA - Sparse Grid Mixed Growth Anisotropic Rules.
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SANDIA_SGMGA <br> Sparse Grid Mixed Growth Anisotropic Rules.
</h1>
<hr>
<p>
<b>SANDIA_SGMGA</b>
is a C++ library which
implements a family of sparse grid rules. These rules are "mixed",
in that a different 1D quadrature rule can be specified for each dimension.
Moreover, each 1D quadrature rule comes in a family of increasing size
whose growth rate (typically linear or exponential) is chosen by the user.
Finally, the user may also specify different weights for each dimension,
resulting in anisotropic rules.
</p>
<p>
<b>SANDIA_SGMGA</b> is a variant of the
<b>SGMGA</b> library. It has the same capabilities as
that library, but it uses a different interface to the SANDIA_RULES routines
which compute points and weights of quadrater rules. Instead of passing
the parameters needed by some of those rules as function arguments, these
values are made available indirectly. This library implements that
indirect storage using a function called <b>parameter</b>, which must
therefore be supplied by the user as part of every program that calls
the library. Refer to the source code of the example programs to see
how <b>parameter</b> is defined and used.
</p>
<p>
<b>SANDIA_SGMGA</b> calls routines from the <b>SANDIA_RULES</b>
and <b>SANDIA_RULES2</b> libraries. Source code or compiled copies
of those libraries must be available when a program wishes to use
the <b>SANDIA_SGMGA</b> library.
</p>
<p>
<table border=1>
<tr>
<th>Name</th>
<th>Abbreviation</th>
<th>Interval</th>
<th>Weight function</th>
</tr>
<tr>
<td>Clenshaw-Curtis</td>
<td>CC</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Fejer Type 2</td>
<td>F2</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Gauss Patterson</td>
<td>GP</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Gauss-Legendre</td>
<td>GL</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>Gauss-Hermite</td>
<td>GH</td>
<td>(-oo,+oo)</td>
<td>e<sup>-x*x</sup></td>
</tr>
<tr>
<td>Generalized Gauss-Hermite</td>
<td>GGH</td>
<td>(-oo,+oo)</td>
<td>|x|<sup>alpha</sup> e<sup>-x*x</sup></td>
</tr>
<tr>
<td>Gauss-Laguerre</td>
<td>LG</td>
<td>[0,+oo)</td>
<td>e<sup>-x</sup></td>
</tr>
<tr>
<td>Generalized Gauss-Laguerre</td>
<td>GLG</td>
<td>[0,+oo)</td>
<td>x<sup>alpha</sup> e<sup>-x</sup></td>
</tr>
<tr>
<td>Gauss-Jacobi</td>
<td>GJ</td>
<td>[-1,+1]</td>
<td>(1-x)<sup>alpha</sup> (1+x)<sup>beta</sup></td>
</tr>
<tr>
<td>Hermite Genz-Keister</td>
<td>HGK</td>
<td>(-oo,+oo)</td>
<td>e<sup>-x*x</sup></td>
</tr>
</table>
</p>
<p>
In the sparse grid setting, for any 1D quadrature rule, it is necessary to
select a sequence of rules of increasing order (number of points), indexed
by a variable we will call the "level". Thus, although the Clenshaw Curtis
rule can be set up for any, a common procedure in sparse grids is to choose
select the rules of order 1, 3, 5, 9, 17, 33, ..., assigning these the
levels 0, 1, 2, 3, 4, 5 and so forth. The relationship between level (L)
and order (O) will be called the <i>growth rule</i>.
</p>
<p>
The details of growth rules vary somewhat, depending on whether there is
nesting to take advantage of, whether the user wants to economize as much
as possible in the number of points added, and so on. For each dimension,
the user must specify a growth rule appropriate for the chosen quadrature rule.
We provide a number of predefined growth rules that are suitable.
</p>
<p>
Here are the names of the growth rule functions, with a brief comment on
their behavior and use. These growth rule functions are available in
the <b>sandia_rules</b> library where their details may be examined.
<table border=1>
<tr>
<th>Growth Rule</th>
<th>Discussion</th>
</tr>
<tr>
<td>level_to_order_exp_cc()</td>
<td>Clenshaw Curtis rule. Fast growth is exponential</td>
</tr>
<tr>
<td>level_to_order_exp_f2()</td>
<td>Fejer Type 2 rule. Fast growth is exponential</td>
</tr>
<tr>
<td>level_to_order_exp_gauss()</td>
<td>Gaussian rules. Fast growth is exponential, O=2^(L+1)-1</td>
</tr>
<tr>
<td>level_to_order_exp_hgk()</td>
<td>Genz-Keister rules for Hermite weight;</td>
</tr>
<tr>
<td>level_to_order_linear_nn()</td>
<td>Linear growth for a non-nested rule;</td>
</tr>
<tr>
<td>level_to_order_linear_wn()</td>
<td>Linear growth for a weakly-nested rule (typically, an abscissas at 0 is common);</td>
</tr>
</table>
</p>
<p>
Each growth
rule has "slow", "moderate" and "fast" settings.
A scalar quantity GROWTH selects the rule order O for level L
from the three growth options for each 1D rule. In the case of exponentially
growing rules, the slow and moderate growth rules choose O indirectly,
by imposing a requirement on P, the degree of precision of the rule.
<table border=1>
<tr>
<th>Value</th>
<th>Name</th>
<th>Meaning</th>
</tr>
<tr>
<td>0</td>
<td>Slow</td>
<td>O=L+1 for linear rules, P=2*L+1 for exponential</td>
</tr>
<tr>
<td>1</td>
<td>Moderate</td>
<td>O=2*L+1 for linear rules, P=4*L+1 for exponential</td>
</tr>
<tr>
<td>2</td>
<td>Full</td>
<td>O=2*L+1 for linear rules, O = next rule in sequence for exponential</td>
</tr>
</table>
</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>SANDIA_SGMGA</b> is available in
<a href = "../../cpp_src/sandia_sgmga/sandia_sgmga.html">a C++ version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/nint_exactness_mixed/nint_exactness_mixed.html">
NINT_EXACTNESS_MIXED</a>,
a C++ program which
measures the polynomial exactness of a multidimensional quadrature rule
based on a mixture of 1D quadrature rule factors.
</p>
<p>
<a href = "../../cpp_src/quadrule/quadrule.html">
QUADRULE</a>,
a C++ library which
defines quadrature rules for various intervals and weight functions.
</p>
<p>
<a href = "../../cpp_src/sandia_rules/sandia_rules.html">
SANDIA_RULES</a>,
a C++ library which
produces 1D quadrature rules of
Chebyshev, Clenshaw Curtis, Fejer 2, Gegenbauer, generalized Hermite,
generalized Laguerre, Hermite, Jacobi, Laguerre, Legendre and Patterson types.
</p>
<p>
<a href = "../../cpp_src/sandia_rules2/sandia_rules2.html">
SANDIA_RULES2</a>,
a C++ library which
contains a very small selection of functions which serve as an interface
between SANDIA_SGMG or SANDIA_SGMGA and SANDIA_RULES.
</p>
<p>
<a href = "../../cpp_src/sandia_sgmg/sandia_sgmg.html">
SANDIA_SGMG</a>,
a C++ library which
creates a sparse grid dataset based on a mixed set of 1D factor rules,
and experiments with the use of a linear growth rate for the quadrature rules.
This is a version of SPARSE_GRID_MIXED_GROWTH that uses a different procedure
for supplying the parameters needed to evaluate certain quadrature rules.
</p>
<p>
<a href = "../../cpp_src/sandia_sparse/sandia_sparse.html">
SANDIA_SPARSE</a>,
a C++ library which
computes the points and weights of a Smolyak sparse
grid, based on a variety of 1-dimensional quadrature rules.
</p>
<p>
<a href = "../../cpp_src/sgmg/sgmg.html">
SGMG</a>,
a C++ library which
creates a sparse grid dataset based on a mixed set of 1D factor rules,
and experiments with the use of a linear growth rate for the quadrature rules.
</p>
<p>
<a href = "../../cpp_src/sgmga/sgmga.html">
SGMGA</a>,
a C++ library which
creates sparse grids based on a mixture of 1D quadrature rules,
allowing anisotropic weights for each dimension.
</p>
<p>
<a href = "../../c_src/smolpack/smolpack.html">
SMOLPACK</a>,
a C library which
implements Novak and Ritter's method for estimating the integral
of a function over a multidimensional hypercube using sparse grids,
by Knut Petras.
</p>
<p>
<a href = "../../m_src/sparse_grid_display/sparse_grid_display.html">
SPARSE_GRID_DISPLAY</a>,
a MATLAB program which
can display a 2D or 3D sparse grid.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a C++ library which
creates sparse grids based on a mix of 1D rules.
</p>
<p>
<a href = "../../m_src/toms847/toms847.html">
TOMS847</a>,
a MATLAB program which
uses sparse grids to carry out multilinear hierarchical interpolation.
It is commonly known as SPINTERP, and is by Andreas Klimke.
</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>
Charles Clenshaw, Alan Curtis,<br>
A Method for Numerical Integration on an Automatic Computer,<br>
Numerische Mathematik,<br>
Volume 2, Number 1, December 1960, pages 197-205.
</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>
Michael Eldred, John Burkardt,<br>
Comparison of Non-Intrusive Polynomial Chaos and Stochastic
Collocation Methods for Uncertainty Quantification,<br>
American Institute of Aeronautics and Astronautics,<br>
Paper 2009-0976,<br>
47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition,<br>
5 - 8 January 2009, Orlando, Florida.
</li>
<li>
Walter Gautschi,<br>
Numerical Quadrature in the Presence of a Singularity,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 4, Number 3, September 1967, pages 357-362.
</li>
<li>
Thomas Gerstner, Michael Griebel,<br>
Numerical Integration Using Sparse Grids,<br>
Numerical Algorithms,<br>
Volume 18, Number 3-4, 1998, pages 209-232.
</li>
<li>
Gene Golub, John Welsch,<br>
Calculation of Gaussian Quadrature Rules,<br>
Mathematics of Computation,<br>
Volume 23, Number 106, April 1969, pages 221-230.
</li>
<li>
Prem Kythe, Michael Schaeferkotter,<br>
Handbook of Computational Methods for Integration,<br>
Chapman and Hall, 2004,<br>
ISBN: 1-58488-428-2,<br>
LC: QA299.3.K98.
</li>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</li>
<li>
Fabio Nobile, Raul Tempone, Clayton Webster,<br>
A Sparse Grid Stochastic Collocation Method for Partial Differential
Equations with Random Input Data,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 46, Number 5, 2008, pages 2309-2345.
</li>
<li>
Fabio Nobile, Raul Tempone, Clayton Webster,<br>
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential
Equations with Random Input Data,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 46, Number 5, 2008, pages 2411-2442.
</li>
<li>
Thomas Patterson,<br>
The Optimal Addition of Points to Quadrature Formulae,<br>
Mathematics of Computation,<br>
Volume 22, Number 104, October 1968, pages 847-856.
</li>
<li>
Knut Petras,<br>
Smolyak Cubature of Given Polynomial Degree with Few Nodes
for Increasing Dimension,<br>
Numerische Mathematik,<br>
Volume 93, Number 4, February 2003, pages 729-753.
</li>
<li>
Sergey Smolyak,<br>
Quadrature and Interpolation Formulas for Tensor Products of
Certain Classes of Functions,<br>
Doklady Akademii Nauk SSSR,<br>
Volume 4, 1963, pages 240-243.
</li>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
<li>
Joerg Waldvogel,<br>
Fast Construction of the Fejer and Clenshaw-Curtis
Quadrature Rules,<br>
BIT Numerical Mathematics,<br>
Volume 43, Number 1, 2003, pages 1-18.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sandia_sgmga.cpp">sandia_sgmga.cpp</a>, the source code.
</li>
<li>
<a href = "sandia_sgmga.hpp">sandia_sgmga.hpp</a>, the include file.
</li>
<li>
<a href = "sandia_sgmga.sh">sandia_sgmga.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>SANDIA_SGMGA_ANISO_NORMALIZE_PRB</b> tests <b>SANDIA_SGMGA_ANISO_BALANCE</b>,
<b>SANDIA_SGMGA_ANISO_NORMALIZE</b> and <b>SANDIA_SGMGA_IMPORTANCE_TO_ANISO</b>.
<ul>
<li>
<a href = "sandia_sgmga_aniso_normalize_prb.cpp">sandia_sgmga_aniso_normalize_prb.cpp</a>,
the program.
</li>
<li>
<a href = "sandia_sgmga_aniso_normalize_prb.sh">sandia_sgmga_aniso_normalize_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sandia_sgmga_aniso_normalize_prb_output.txt">sandia_sgmga_aniso_normalize_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SANDIA_SGMGA_INDEX_PRB</b> tests <b>SANDIA_SGMGA_INDEX</b>.
<ul>
<li>
<a href = "sandia_sgmga_index_prb.cpp">sandia_sgmga_index_prb.cpp</a>,
the program.
</li>
<li>
<a href = "sandia_sgmga_index_prb.sh">sandia_sgmga_index_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sandia_sgmga_index_prb_output.txt">sandia_sgmga_index_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SANDIA_SGMGA_POINT_PRB</b> tests <b>SANDIA_SGMGA_POINT</b>.
<ul>
<li>
<a href = "sandia_sgmga_point_prb.cpp">sandia_sgmga_point_prb.cpp</a>,
the program.
</li>
<li>
<a href = "sandia_sgmga_point_prb.sh">sandia_sgmga_point_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sandia_sgmga_point_prb_output.txt">sandia_sgmga_point_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SANDIA_SGMGA_PRODUCT_WEIGHT_PRB</b> tests <b>SANDIA_SGMGA_PRODUCT_WEIGHT</b>.
<ul>
<li>
<a href = "sandia_sgmga_product_weight_prb.cpp">sandia_sgmga_product_weight_prb.cpp</a>,
the program.
</li>
<li>
<a href = "sandia_sgmga_product_weight_prb.sh">sandia_sgmga_product_weight_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sandia_sgmga_product_weight_prb_output.txt">sandia_sgmga_product_weight_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SANDIA_SGMGA_SIZE_PRB</b> tests <b>SANDIA_SGMGA_SIZE</b>
and <b>SANDIA_SGMGA_SIZE_TOTAL</b>.
<ul>
<li>
<a href = "sandia_sgmga_size_prb.cpp">sandia_sgmga_size_prb.cpp</a>,
the program.
</li>
<li>
<a href = "sandia_sgmga_size_prb.sh">sandia_sgmga_size_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sandia_sgmga_size_prb_output.txt">sandia_sgmga_size_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SANDIA_SGMGA_SIZE_TABLE</b> tabulates the point counts from <b>SANDIA_SGMGA_SIZE</b>
for an isotropic rule over a range of dimensions and levels.
<ul>
<li>
<a href = "sandia_sgmga_size_table.cpp">sandia_sgmga_size_table.cpp</a>,
the program.
</li>
<li>
<a href = "sandia_sgmga_size_table.sh">sandia_sgmga_size_table.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sandia_sgmga_size_table_output.txt">sandia_sgmga_size_table_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SANDIA_SGMGA_UNIQUE_INDEX_PRB</b> tests <b>SANDIA_SGMGA_UNIQUE_INDEX</b>.
<ul>
<li>
<a href = "sandia_sgmga_unique_index_prb.cpp">sandia_sgmga_unique_index_prb.cpp</a>,
the program.
</li>
<li>
<a href = "sandia_sgmga_unique_index_prb.sh">sandia_sgmga_unique_index_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sandia_sgmga_unique_index_prb_output.txt">sgmgav_index_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SANDIA_SGMGA_VCN_PRB</b> tests <b>SANDIA_SGMGA_VCN</b> and <b>SANDIA_SGMGA_VCN_ORDERED</b>.
<ul>
<li>
<a href = "sandia_sgmga_vcn_prb.cpp">sandia_sgmga_vcn_prb.cpp</a>,
the program.
</li>
<li>
<a href = "sandia_sgmga_vcn_prb.sh">sandia_sgmga_vcn_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sandia_sgmga_vcn_prb_output.txt">sandia_sgmga_vcn_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SANDIA_SGMGA_VCN_COEF_PRB</b> tests <b>SANDIA_SGMGA_VCN_COEF</b>.
<ul>
<li>
<a href = "sandia_sgmga_vcn_coef_prb.cpp">sandia_sgmga_vcn_coef_prb.cpp</a>,
the program.
</li>
<li>
<a href = "sandia_sgmga_vcn_coef_prb.sh">sandia_sgmga_vcn_coef_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sandia_sgmga_vcn_coef_prb_output.txt">sandia_sgmga_vcn_coef_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SANDIA_SGMGA_WEIGHT_PRB</b> tests <b>SANDIA_SGMGA_WEIGHT</b>.
<ul>
<li>
<a href = "sandia_sgmga_weight_prb.cpp">sandia_sgmga_weight_prb.cpp</a>,
the program.
</li>
<li>
<a href = "sandia_sgmga_weight_prb.sh">sandia_sgmga_weight_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sandia_sgmga_weight_prb_output.txt">sandia_sgmga_weight_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>SANDIA_SGMGA_ANISO_BALANCE</b> "balances" an anisotropic weight vector.
</li>
<li>
<b>SANDIA_SGMGA_ANISO_NORMALIZE</b> normalizes the SANDIA_SGMGA anisotropic weight vector.
</li>
<li>
<b>SANDIA_SGMGA_IMPORTANCE_TO_ANISO:</b> importance vector to anisotropic weight vector.
</li>
<li>
<b>SANDIA_SGMGA_INDEX</b> indexes an SANDIA_SGMGA grid.
</li>
<li>
<b>SANDIA_SGMGA_POINT</b> computes the points of an SANDIA_SGMGA rule.
</li>
<li>
<b>SANDIA_SGMGA_PRODUCT_WEIGHT</b> computes the weights of a mixed product rule.
</li>
<li>
<b>SANDIA_SGMGA_SIZE</b> sizes an SANDIA_SGMGA grid, discounting duplicates.
</li>
<li>
<b>SANDIA_SGMGA_SIZE_TOTAL</b> sizes an SANDIA_SGMGA grid, counting duplicates.
</li>
<li>
<b>SANDIA_SGMGA_UNIQUE_INDEX</b> maps nonunique to unique points.
</li>
<li>
<b>SANDIA_SGMGA_VCN</b> returns the next constrained vector.
</li>
<li>
<b>SANDIA_SGMGA_VCN_COEF</b> returns the "next" constrained vector's coefficient.
</li>
<li>
<b>SANDIA_SGMGA_VCN_COEF_NAIVE:</b> "next" constrained vector's coefficient.
</li>
<li>
<b>SANDIA_SGMGA_VCN_NAIVE</b> returns the next constrained vector.
</li>
<li>
<b>SANDIA_SGMGA_VCN_ORDERED</b> returns the "next" constrained vector, with ordering.
</li>
<li>
<b>SANDIA_SGMGA_VCN_ORDERED_NAIVE</b> returns the "next" constrained vector, with ordering.
</li>
<li>
<b>SANDIA_SGMGA_WEIGHT</b> computes weights for an SANDIA_SGMGA grid.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 23 January 2012.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>