sparse_grid_laguerre, a C++ code which constructs sparse grids based on 1D Gauss-Laguerre rules.
Sparse grids are more naturally constructed from a nested family of quadrature rules. Gauss-Laguerre rules are not nested, but have higher accuracy. Thus, there can be a tradeoff. If we compare two sparse grids of the same "level", one using Gauss-Laguerre rules and the other a nested rule, then the Gauss-Laguerre sparse grid will have higher accuracy...but also a significantly greater number of points. When measuring efficiency, we really need to balance the cost in quadrature points against the accuracy, and so it is not immediately obvious which choice is best!
To slightly complicate matters, Gauss-Laguerre rules are not nested. A sparse grid constructed from Gauss-Laguerre rules will thus generally have more abscissas than a grid built of nested rules..
Here is a table showing the number of points in a sparse grid based on Gauss-Laguerre rules, indexed by the spatial dimension, and by the "level", which is simply an index for the family of sparse grids.
| DIM: | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| LEVEL_MAX | ||||||
| 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 3 | 7 | 10 | 13 | 16 | 19 |
| 2 | 7 | 29 | 58 | 95 | 141 | 196 |
| 3 | 15 | 95 | 255 | 515 | 906 | 1456 |
| 4 | 31 | 273 | 945 | 2309 | 4746 | 8722 |
| 5 | 63 | 723 | 3120 | 9065 | 21503 | 44758 |
| 6 | 127 | 1813 | 9484 | 32259 | 87358 | 204203 |
The code described and made available on this web page is distributed under the GNU LGPL license.
sparse_grid_laguerre is available in a C++ version and a Fortran90 version and a MATLAB version.
quad_rule, a C++ code which defines quadrature rules for various intervals and weight functions.
sparse_grid_gl, a C++ code which computes a sparse grid based on 1D Gauss-Legendre rules.
sparse_grid_hermite, a C++ code which creates sparse grids based on Gauss-Hermite rules.
sparse_grid_mixed, a C++ code which constructs a sparse grid using different rules in each spatial dimension.