Based on Tanh-Sinh Rules

**SPARSE_GRID_TANH_SINH**
is a dataset directory which
contains examples of sparse grids generated from
1D Tanh-Sinh quadrature rules.

A quadrature rule is a set of **n** points **x** and associated
weights **w** so that the integral of a function **f(x)** over
some particular region can be approximated by:

Integral f(x) dx = Sum ( 1 <= i <= n ) w(i) * f(x(i))

For this directory, a quadrature rule is stored as three files, containing the weights, the points, and a file containing two points defining the corners of the rectangular region. The dimension of the region is deduced implicitly from the dimension of the points.

Here are the three files associated with a sparse grid for two dimensions, based on a tanh-sinh rule, for level 2 which has 17 points.

Here is the text of an "W" file storing the weights of such a rule

```
-0.652737
0.127421
0.127421
0.127421
0.127421
0.018346
0.966102
0.966102
0.018346
0.051315
0.051315
0.051315
0.051315
0.018346
0.966102
0.966102
0.018346
```

Here is the text of a "X" file storing the abscissas of such a rule:

```
0.000000 0.000000
-0.951368 0.000000
0.951368 0.000000
0.000000 -0.951368
0.000000 0.951368
-0.997515 0.000000
-0.674271 0.000000
0.674271 0.000000
0.997515 0.000000
-0.951368 -0.951368
0.951368 -0.951368
-0.951368 0.951368
0.951368 0.951368
0.000000 -0.997515
0.000000 -0.674271
0.000000 0.674271
0.000000 0.997515
```

Here is the text of an "R" file storing the lower and upper limits of the region, needed to determine the integration region:

```
-1.000000 -1.000000
1.000000 1.000000
```

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

CLENSHAW_CURTIS_RULE, a FORTRAN90 library of routines which can compute Clenshaw Curtis grids.

NINT_EXACTNESS_MIXED, a C++ program which measures the polynomial exactness of a multidimensional quadrature rule based on a mixture of 1D quadrature rule factors.

SMOLPACK, a C library which estimates the integral of a function over a multidimensional hypercube using a sparse grid, by Knut Petras;

SPARSE_GRID_CC, a C++ library which creates sparse grids based on Clenshaw-Curtis rules.

SPARSE_GRID_CC_DATASET, a C++ program which creates a sparse grid dataset based on Clenshaw-Curtis rules.

SPARSE_GRID_DISPLAY, a MATLAB program which reads a file of points on a sparse grid, displays the grid and save the image in a PNG file;

SPARSE_GRID_MIXED, a C++ library which creates a sparse grid dataset based on a mixed set of 1D factor rules.

SPINTERP, a MATLAB library which carries out piecewise multilinear hierarchical sparse grid interpolation, by Andreas Klimke; ACM TOMS Algorithm 847.

TABLE, a data directory which contains examples of TABLE files, a simple format for N points in M dimensions;

**TS_D2_LEVEL0** is the level 0 rule for dimension 2, using 1 point.

- ts_d2_level0_x.txt, the X values.
- ts_d2_level0_w.txt, the W values.
- ts_d2_level0_r.txt, the R values.
- ts_d2_level0.png, a PNG image of the X values.

**TS_D2_LEVEL1** is the level 1 rule for dimension 2, using 5 points.

- ts_d2_level1_x.txt, the X values.
- ts_d2_level1_w.txt, the W values.
- ts_d2_level1_r.txt, the R values.
- ts_d2_level1.png, a PNG image of the X values.

**TS_D2_LEVEL0** is the level 2 rule for dimension 2, using 17 points.

- ts_d2_level2_x.txt, the X values.
- ts_d2_level2_w.txt, the W values.
- ts_d2_level2_r.txt, the R values.
- ts_d2_level2.png, a PNG image of the X values.

**TS_D2_LEVEL0** is the level 3 rule for dimension 2, using 49 points.

- ts_d2_level3_x.txt, the X values.
- ts_d2_level3_w.txt, the W values.
- ts_d2_level3_r.txt, the R values.
- ts_d2_level3.png, a PNG image of the X values.

**TS_D2_LEVEL0** is the level 4 rule for dimension 2, using 129 points.

- ts_d2_level4_x.txt, the X values.
- ts_d2_level4_w.txt, the W values.
- ts_d2_level4_r.txt, the R values.
- ts_d2_level4.png, a PNG image of the X values.

**TS_D2_LEVEL0** is the level 5 rule for dimension 2, using 321 points.

- ts_d2_level5_x.txt, the X values.
- ts_d2_level5_w.txt, the W values.
- ts_d2_level5_r.txt, the R values.
- ts_d2_level5.png, a PNG image of the X values.

**TS_D2_LEVEL0** is the level 6 rule for dimension 2, using 769 points.

- ts_d2_level6_x.txt, the X values.
- ts_d2_level6_w.txt, the W values.
- ts_d2_level6_r.txt, the R values.
- ts_d2_level6.png, a PNG image of the X values.

**TS_D2_LEVEL0** is the level 7 rule for dimension 2, using 1793 points.

- ts_d2_level7_x.txt, the X values.
- ts_d2_level7_w.txt, the W values.
- ts_d2_level7_r.txt, the R values.
- ts_d2_level7.png, a PNG image of the X values.

You can go up one level to the DATASETS page.