SPARSE_GRID_TANH_SINH - Sparse Grids
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.

Example:

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  
      

Licensing:

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

Related Data and Programs:

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;

Sample Files:

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.