# SPARSE_GRID_LAGUERRE Sparse Grids Based on Gauss-Laguerre Rules

SPARSE_GRID_LAGUERRE is a MATLAB library 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:123456
LEVEL_MAX
0111111
13710131619
27295895141196
315952555159061456
431273945230947468722
563723312090652150344758
6127181394843225987358204203

### Languages:

SPARSE_GRID_LAGUERRE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

SGMGA, a MATLAB library which creates sparse grids based on a mixture of 1D quadrature rules, allowing anisotropic weights for each dimension.

SMOLPACK, a C library which implements Novak and Ritter's method for estimating the integral of a function over a multidimensional hypercube using sparse grids.

SPARSE_GRID_COMPOSITE, a MATLAB library which creates sparse grids based on 1D composite rules (currently only of order 1).

SPARSE_GRID_GL, a MATLAB library which computes a sparse grid based on 1D Gauss-Legendre rules.

SPARSE_GRID_HERMITE, a MATLAB library which creates sparse grids based on Gauss-Hermite rules.

SPARSE_GRID_HW, a MATLAB library 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.

SPARSE_GRID_MIXED, a MATLAB library which constructs a sparse grid using different rules in each spatial dimension.

SPARSE_GRID_OPEN, a MATLAB library which define define sparse grids based on open nested quadrature rules.

SPQUAD, a MATLAB library 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, a MATLAB program which uses sparse grids to carry out multilinear hierarchical interpolation. It is commonly known as SPINTERP, and is by Andreas Klimke.

### Reference:

1. Volker Barthelmann, Erich Novak, Klaus Ritter,
High Dimensional Polynomial Interpolation on Sparse Grids,
Volume 12, Number 4, 2000, pages 273-288.
2. Thomas Gerstner, Michael Griebel,
Numerical Integration Using Sparse Grids,
Numerical Algorithms,
Volume 18, Number 3-4, 1998, pages 209-232.
3. Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
ISBN: 0-12-519260-6,
LC: QA164.N54.
4. Fabio Nobile, Raul Tempone, Clayton Webster,
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data,
SIAM Journal on Numerical Analysis,
Volume 46, Number 5, 2008, pages 2309-2345.
5. Sergey Smolyak,
Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,
Volume 4, 1963, pages 240-243.
6. Dennis Stanton, Dennis White,
Constructive Combinatorics,
Springer, 1986,
ISBN: 0387963472,
LC: QA164.S79.

### Examples and Tests:

The test program creates some data files:

• lg_d2_level3_r.txt, the "R" file for a sparse grid quadrature rule for spatial dimension 2 and level 3.
• lg_d2_level3_w.txt, the "W" file for a sparse grid quadrature rule for spatial dimension 2 and level 3.
• lg_d2_level3_x.txt, the "X" file for a sparse grid quadrature rule for spatial dimension 2 and level 3.

You can go up one level to the MATLAB source codes.

Last revised on 11 October 2007.