forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
sparse_grid_open_dataset.html
579 lines (529 loc) · 16.8 KB
/
sparse_grid_open_dataset.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
<html>
<head>
<title>
SPARSE_GRID_OPEN_DATASET - Sparse Grid from Open 1D Quadrature Rule
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPARSE_GRID_OPEN_DATASET <br> Sparse Grid from Open 1D Quadrature Rule
</h1>
<hr>
<p>
<b>SPARSE_GRID_OPEN_DATASET</b>
is a C++ program which
computes a sparse quadrature rule for
an arbitrary spatial dimension, associated with a particular
"level" of the Smolyak construction,
and based on an open 1D quadrature rule.
</p>
<p>
The program offers a choice of open 1D quadrature rules to be used:
<ul>
<li>
<b>2: F2</b>, the Fejer type 2 rule;
</li>
<li>
<b>3: GP</b>, the Gauss-Patterson rule;
</li>
<li>
<b>4: NCO</b>, the Newton-Cotes Open rule;
</li>
<li>
<b>5: TS</b>, the Tanh-Sinh rule;
</li>
</ul>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>sparse_grid_open_dataset</b> <i>dim_num</i> <i>level_max</i> <i>rule</i>
</blockquote>
where
<ul>
<li>
<b>dim_num</b> is the spatial dimension;
</li>
<li>
<b>level_max</b> is the level of the Smolyak construction;
</li>
<li>
<b>rule</b> is the index (2/3/4/5) of the 1D quadrature rule to use.
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The code described and made available on this web page is distributed
under the
<a href = "gnu_lgpl.txt">GNU LGPL</a> license.
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SPARSE_GRID_OPEN_DATASET</b> is available in
<a href = "../../cpp_src/sparse_grid_open_dataset/sparse_grid_open_dataset.html">a C++ version</a> and
<a href = "../../f_src/sparse_grid_open_dataset/sparse_grid_open_dataset.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sparse_grid_open_dataset/sparse_grid_open_dataset.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/cc_display/cc_display.html">
CC_DISPLAY</a>,
a MATLAB library which
can compute and display Clenshaw Curtis grids in two dimensions,
as well as sparse grids formed from sums of Clenshaw Curtis grids.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
defines quadrature rules;
a number of examples of sparse grid quadrature rules are included.
</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/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.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_cc_dataset/sparse_grid_cc_dataset.html">
SPARSE_GRID_CC_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on Clenshaw-Curtis rules.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_closed_dataset/sparse_grid_closed_dataset.html">
SPARSE_GRID_CLOSED_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on
closed rules (Clenshaw-Curtis, Newton-Cotes-Closed).
</p>
<p>
<a href = "../../m_src/sparse_grid_display/sparse_grid_display.html">
SPARSE_GRID_DISPLAY</a>,
a MATLAB library which
can display a 2D or 3D sparse grid.
</p>
<p>
<a href = "../../datasets/sparse_grid_f2/sparse_grid_f2.html">
SPARSE_GRID_F2</a>,
a dataset directory which
contains sparse
grids based on a Fejer Type 2 rule.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_gl_dataset/sparse_grid_gl_dataset.html">
SPARSE_GRID_GL_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on Gauss-Legendre rules.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_hermite_dataset/sparse_grid_hermite_dataset.html">
SPARSE_GRID_HERMITE_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on Gauss-Hermite rules.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_laguerre_dataset/sparse_grid_laguerre_dataset.html">
SPARSE_GRID_LAGUERRE_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on Gauss-Laguerrre rules.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a C++ library which
constructs a sparse grid using different rules in each spatial dimension.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_mixed_dataset/sparse_grid_mixed_dataset.html">
SPARSE_GRID_MIXED_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on a mixture of 1D rules.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_mixed_growth/sparse_grid_mixed_growth.html">
SPARSE_GRID_MIXED_GROWTH</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 = "../../datasets/sparse_grid_ncc/sparse_grid_ncc.html">
SPARSE_GRID_NCC</a>,
a dataset directory which
contains sparse
grids based on a Newton Cotes closed rule.
</p>
<p>
<a href = "../../datasets/sparse_grid_nco/sparse_grid_nco.html">
SPARSE_GRID_NCO</a>,
a dataset directory which
contains sparse
grids based on a Newton Cotes open rule.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_open/sparse_grid_open.html">
SPARSE_GRID_OPEN</a>,
a C++ library which
defines define sparse grids based on open nested quadrature 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>
Volker Barthelmann, Erich Novak, Klaus Ritter,<br>
High Dimensional Polynomial Interpolation on Sparse Grids,<br>
Advances in Computational Mathematics,<br>
Volume 12, Number 4, 2000, pages 273-288.
</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>
Walter Gautschi,<br>
Numerical Quadrature in the Presence of a Singularity,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 4, Number 3, 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>
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>
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>
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>
Dennis Stanton, Dennis White,<br>
Constructive Combinatorics,<br>
Springer, 1986,<br>
ISBN: 0387963472,<br>
LC: QA164.S79.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sparse_grid_open_dataset.cpp">sparse_grid_open_dataset.cpp</a>, the source code.
</li>
<li>
<a href = "sparse_grid_open_dataset.sh">sparse_grid_open_dataset.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>F2_D2_LEVEL2</b> is an example computation based on a Fejer type 2 rule
in two dimensions and level 2.
<ul>
<li>
<a href = "f2_d2_level2_output.txt">
f2_d2_level2_output.txt</a>,
the printed output from a run of the program for DIM_NUM=2
and LEVEL_MAX=2 and RULE=2.
</li>
<li>
<a href = "f2_d2_level2_x.txt">
f2_d2_level2_x.txt</a>,
the abscissas of the computed quadrature rule.
</li>
<li>
<a href = "f2_d2_level2_w.txt">
f2_d2_level2_w.txt</a>,
the weights of the computed quadrature rule.
</li>
<li>
<a href = "f2_d2_level2_r.txt">
f2_d2_level2_r.txt</a>,
the integration region of the computed quadrature rule.
</li>
</ul>
</p>
<p>
<b>GP_D2_LEVEL2</b> is an example computation based on a Gauss-Patterson
rule in two dimensions and level 2.
<ul>
<li>
<a href = "gp_d2_level2_output.txt">
gp_d2_level2_output.txt</a>,
the printed output from a run of the program for DIM_NUM=2
and LEVEL_MAX=2 and RULE=3.
</li>
<li>
<a href = "gp_d2_level2_x.txt">
gp_d2_level2_x.txt</a>,
the abscissas of the computed quadrature rule.
</li>
<li>
<a href = "gp_d2_level2_w.txt">
gp_d2_level2_w.txt</a>,
the weights of the computed quadrature rule.
</li>
<li>
<a href = "gp_d2_level2_r.txt">
gp_d2_level2_r.txt</a>,
the integration region of the computed quadrature rule.
</li>
</ul>
</p>
<p>
<b>NCO_D2_LEVEL2</b> is an example computation based on a
Newton-Cotes Open rule in two dimensions and level 2.
<ul>
<li>
<a href = "nco_d2_level2_output.txt">
nco_d2_level2_output.txt</a>,
the printed output from a run of the program for DIM_NUM=2
and LEVEL_MAX=2 and RULE=4.
</li>
<li>
<a href = "nco_d2_level2_x.txt">
nco_d2_level2_x.txt</a>,
the abscissas of the computed quadrature rule.
</li>
<li>
<a href = "nco_d2_level2_w.txt">
nco_d2_level2_w.txt</a>,
the weights of the computed quadrature rule.
</li>
<li>
<a href = "nco_d2_level2_r.txt">
nco_d2_level2_r.txt</a>,
the integration region of the computed quadrature rule.
</li>
</ul>
</p>
<p>
<b>TS_D2_LEVEL4</b> is an example computation based on a
tanh-sinh rule in two dimensions and level 4.
<ul>
<li>
<a href = "ts_d2_level4_output.txt">
ts_d2_level4_output.txt</a>, the output file.
</li>
<li>
<a href = "ts_d2_level4_x.txt">
ts_d2_level4_x.txt</a>, the abscissas.
</li>
<li>
<a href = "ts_d2_level4_w.txt">
ts_d2_level4_w.txt</a>, the weights.
</li>
<li>
<a href = "ts_d2_level4_r.txt">
ts_d2_level4_r.txt</a>, the region.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for SPARSE_GRID_OPEN_DATASET.
</li>
<li>
<b>ABSCISSA_LEVEL_OPEN_ND:</b> first level at which given abscissa is generated.
</li>
<li>
<b>CHOOSE</b> computes the binomial coefficient C(N,K).
</li>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</li>
<li>
<b>F2_ABSCISSA</b> returns the I-th abscissa for the Fejer type 2 rule.
</li>
<li>
<b>F2_WEIGHTS</b> computes weights for a Fejer type 2 rule.
</li>
<li>
<b>GP_ABSCISSA</b> returns the I-th abscissa for a Gauss-Patterson rule.
</li>
<li>
<b>GP_WEIGHTS</b> sets weights for a Gauss-Patterson rule.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of an I4MAT, transposed.
</li>
<li>
<b>I4_TO_STRING</b> converts an I4 to a C++ string.
</li>
<li>
<b>I4VEC_PRODUCT</b> multiplies the entries of an I4VEC.
</li>
<li>
<b>INDEX_TO_LEVEL_OPEN</b> determines the level of a point given its index.
</li>
<li>
<b>LEVEL_TO_ORDER_OPEN</b> converts a level to an order for open rules.
</li>
<li>
<b>MULTIGRID_INDEX1</b> returns an indexed multidimensional grid.
</li>
<li>
<b>MULTIGRID_SCALE_OPEN</b> renumbers a grid as a subgrid on a higher level.
</li>
<li>
<b>NCO_ABSCISSA</b> returns the I-th abscissa for the Newton Cotes open rule.
</li>
<li>
<b>NCO_WEIGHTS</b> computes weights for a Newton-Cotes Open rule.
</li>
<li>
<b>PRODUCT_WEIGHTS_OPEN:</b> weights for an open product rule.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_COPY</b> copies an R8VEC.
</li>
<li>
<b>R8VEC_DIRECT_PRODUCT2</b> creates a direct product of R8VEC's.
</li>
<li>
<b>R8VEC_PRINT_SOME</b> prints "some" of an R8VEC.
</li>
<li>
<b>R8VEC_SUM</b> returns the sum of an R8VEC.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>SPARSE_GRID_OFN_SIZE</b> sizes a sparse grid using Open Fully Nested rules.
</li>
<li>
<b>LEVELS_OPEN_INDEX</b> computes open grids with 0 <= LEVEL <= LEVEL_MAX.
</li>
<li>
<b>SPGRID_OPEN_WEIGHTS</b> gathers the weights.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TS_ABSCISSA</b> returns the I-th abscissa for the tanh-sinh rule.
</li>
<li>
<b>TS_WEIGHTS</b> computes weights for a tanh-sinh rule.
</li>
<li>
<b>VEC_COLEX_NEXT2</b> generates vectors in colex order.
</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 December 2009.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>