# SPARSE_GRID_CCS Clenshaw Curtis Slow Growth Sparse Grids

SPARSE_GRID_CCS is a dataset directory which contains examples of multidimensional sparse grid quadrature rules based on the one-dimensional Clenshaw Curtis rule with Slow Exponential Growth (CCS)

The slow exponential growth rate refers to the relationship between the 1D level and order of the Clenshaw Curtis rules that are used. In the classical CC rule, the rules have orders of ( 1, 3, 5, 9, 17, 33, ... ) so that for 0 < L, the order O = 2^L+1.

At level L, the slow growth Clenshaw Curtis (CCS) rule chooses the classical Clenshaw of lowest order that equals or exceeds 2 * L + 1. Thus, the CCS orders begin with ( 1, 3, 5, 9, 9, 17, 17, 17, 17, 33, ... ).

Particularly for low dimensions, sparse grids made from the CCS family can have significantly fewer points than those made from the CC family, while maintaining a precision of 2*L+1.

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 the CCS rule, for level 2 which has 13 points.

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

``````
-0.3555555555555558
-0.0888888888888890
-0.0888888888888890
-0.0888888888888889
-0.0888888888888889
1.0666666666666667
1.0666666666666669
0.1111111111111111
0.1111111111111111
0.1111111111111111
0.1111111111111111
1.0666666666666667
1.0666666666666669
``````

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

``````
0.0000000000000001      0.0000000000000001
-1.0000000000000000      0.0000000000000001
1.0000000000000000      0.0000000000000001
0.0000000000000001     -1.0000000000000000
0.0000000000000001      1.0000000000000000
-0.7071067811865475      0.0000000000000001
0.7071067811865476      0.0000000000000001
-1.0000000000000000     -1.0000000000000000
-1.0000000000000000      1.0000000000000000
1.0000000000000000     -1.0000000000000000
1.0000000000000000      1.0000000000000000
0.0000000000000001     -0.7071067811865475
0.0000000000000001      0.7071067811865476
``````

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

``````
-1.0000000000000000        -1.0000000000000000
1.0000000000000000         1.0000000000000000
``````

### Related Data and Programs:

SPARSE_GRID_CCE, a dataset directory which contains multidimensional Smolyak sparse grids based on Clenshaw Curtis Exponential growth rule;

SPARSE_GRID_CCL, a dataset directory which contains M-dimensional Smolyak sparse grids based on Clenshaw Curtis Linear growth rule;

### Sample Files:

CCS rules in 2D:

CCS rules in 3D:

CCS rules in 6D:

You can go up one level to the DATASETS page.

Last revised on 20 February 2014.