Sparse Grid Files
Newton Cotes Open

These are some examples of sparse grids, using the idea of a level to control the number of points, and assigning point locations using the Newton Cotes Open rule.

The file has a simple structure. There are a few initial comment lines, which begin with the "#" character. These can be removed if they are inconvenient. There then follow the coordinates of the points, with each point on a single line, and the coordinates of a point separated by spaces.

The points are ordered to respect the natural nesting of the sparse grids. That is, the file of level 3 points begins by listing the points in the level 2 grid.

To make full use of such a sparse grid, it is also necessary to compute the associated interpolation weights.

Example:

Here is the text of a file storing a Newton Cotes Open rule of level 2.

      0.500000      0.500000
      0.166667      0.500000
      0.833333      0.500000
      0.500000      0.166667
      0.500000      0.833333
      0.333333      0.500000
      0.666667      0.500000
      0.166667      0.166667
      0.166667      0.833333
      0.833333      0.166667
      0.833333      0.833333
      0.500000      0.333333
      0.500000      0.666667
      

Related Data and Programs:

• SMOLPACK is a C library of routines which implements Novak and Ritter's method for estimating the integral of a function over a multidimensional hypercube using sparse grids.
• SPARSE_GRID_DISPLAY is a library of MATLAB routines which can display a 2D or 3D sparse grid.
• SPINTERP is a MATLAB library of routines by Andreas Klimke which use a sparse grid to perform multilinear hierarchical interpolation.
• TABLE is the file format used to store this data;

• QUADRATURE_RULES is a dataset directory of files that define quadrature rules; a number of examples of sparse grid quadrature rules are included.
• SPARSE_GRID_CC some sample sparse grids from the Clenshaw Curtis rule.
• SPARSE_GRID_F1 some sample sparse grids from the Fejer Type 1 rule.
• SPARSE_GRID_F2 some sample sparse grids from the Fejer Type 2 rule.
• SPARSE_GRID_GP some sample sparse grids from the Gauss Patterson rule.
• SPARSE_GRID_NCC some sample sparse grids from the Newton Cotes Closed rule.
• SPARSE_GRID_NCOH some sample sparse grids from the Newton Cotes Open Half rule.

Sample Files:

Some grids in 2D:

• nco_d2_level0.txt, Newton Cotes Open rule in 2 dimensions, level 0, in [0,1], 1 point.
• nco_d2_level1.txt, Newton Cotes Open rule in 2 dimensions, level 1, in [0,1], 5 points.
• nco_d2_level2.txt, Newton Cotes Open rule in 2 dimensions, level 2, in [0,1], 17 points.
• nco_d2_level3.txt, Newton Cotes Open rule in 2 dimensions, level 3, in [0,1], 49 points.
• nco_d2_level4.txt, Newton Cotes Open rule in 2 dimensions, level 4, in [0,1], 129 points.
• nco_d2_level5.txt, Newton Cotes Open rule in 2 dimensions, level 5, in [0,1], 321 points.

SPARSE_GRID_DISPLAY can be used to display images of two dimensional grids, which can be saved in the PNG format:

• nco_d2_level0.png,
• nco_d2_level1.png,
• nco_d2_level2.png,
• nco_d2_level3.png,
• nco_d2_level4.png,
• nco_d2_level5.png,

You can go up one level to the DATASETS page.