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
|
#include "Tasmanian.hpp"
using namespace std;
/*!
* \internal
* \file example_sparse_grids_02.cpp
* \brief Examples for the Tasmanian Sparse Grid module.
* \author Miroslav Stoyanov
* \ingroup TasmanianSGExamples
*
* Tasmanian Sparse Grids Example 2.
* \endinternal
*/
/*!
* \ingroup TasmanianSGExamples
* \addtogroup TasmanianSGExamples2 Tasmanian Sparse Grids module, example 2
*
* \par Example 2
* Compute \f$ \int_{-5}^{+5} \int_{-2}^{+3} \exp(-x^2) \cos(y) dy dx \f$
* using a sparse grid with Gauss-Patterson nodes and weights.
* The grid is constructed over a non-canononical domain and the points
* are selected based on a total degree polynomial space.
*/
/*!
* \ingroup TasmanianSGExamples2
* \brief Sparse Grids Example 2: integrate a simple function over a canonical domain.
*
* \snippet SparseGrids/Examples/example_sparse_grids_02.cpp SG_Example_02 example
*/
void sparse_grids_example_02(){
#ifndef __TASMANIAN_DOXYGEN_SKIP
//! [SG_Example_02 example]
#endif
cout << "\n---------------------------------------------------------------------------------------------------\n";
cout << std::scientific; cout.precision(17);
cout << "Example 2: integrate f(x,y) = exp(-x^2) * cos(y) over [-5,5] x [-2,3]\n"
<< " using Gauss-Patterson nodes and total degree polynomial space\n";
int dimension = 2;
int exactness = 20;
// the type_qptotal will guarantee exact integral for all polynomials with degree 20 or less
auto grid = TasGrid::makeGlobalGrid(dimension, 0, exactness,
TasGrid::type_qptotal, TasGrid::rule_gausspatterson);
grid.setDomainTransform({-5.0, -2.0}, {5.0, 3.0}); // set the non-canonical domain
auto points = grid.getPoints();
auto weights = grid.getQuadratureWeights();
double I = 0.0;
for(int i=0; i<grid.getNumPoints(); i++){
double x = points[i*dimension];
double y = points[i*dimension+1];
I += weights[i] * std::exp(-x*x) * std::cos(y);
}
double exact = 1.861816427518323e+00;
double E = std::abs(exact - I);
cout << " at polynomial exactness: " << exactness
<< "\n the grid has: " << weights.size()
<< "\n integral: " << I
<< "\n error: " << E << "\n\n";
exactness = 40;
grid = TasGrid::makeGlobalGrid(dimension, 0, exactness,
TasGrid::type_qptotal, TasGrid::rule_gausspatterson);
grid.setDomainTransform({-5.0, -2.0}, {5.0, 3.0}); // set the non-canonical domain
points = grid.getPoints();
weights = grid.getQuadratureWeights();
I = 0.0;
for(int i=0; i<grid.getNumPoints(); i++){
double x = points[i*dimension];
double y = points[i*dimension+1];
I += weights[i] * std::exp(-x*x) * std::cos(y);
}
E = std::abs(exact - I);
cout << " at polynomial exactness: " << exactness
<< "\n the grid has: " << weights.size()
<< "\n integral: " << I
<< "\n error: " << E << endl;
#ifndef __TASMANIAN_DOXYGEN_SKIP
//! [SG_Example_02 example]
#endif
}
|
