sparse_grid_laguerre, an Octave 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 computer code and data files described and made available on this web page are distributed under the MIT license
sparse_grid_laguerre is available in a C++ version and a Fortran90 version and a MATLAB version and an Octave version.
sparse_grid_composite, an Octave code which creates sparse grids based on 1d composite rules (currently only of order 1).
sparse_grid_gl, an Octave code which computes a sparse grid based on 1d Gauss-Legendre rules.
sparse_grid_hermite, an Octave code which creates sparse grids based on Gauss-Hermite rules.
sparse_grid_hw, an Octave code which creates sparse grids based on Gauss-Legendre, Gauss-Hermite, Gauss-Patterson, or a nested variation of Gauss-Hermite rules, by Florian Heiss and Viktor Winschel.
spquad, an Octave code which computes the points and weights of a sparse grid quadrature rule for a multidimensional integral, based on the Clenshaw-Curtis quadrature rule, by Greg von Winckel.
toms847, an Octave code which uses sparse grids to carry out multilinear hierarchical interpolation. It is commonly known as spinterp(), and is by Andreas Klimke.