SPARSE_GRID_LAGUERRE
Sparse Grids Based on Gauss-Laguerre Rules

SPARSE_GRID_LAGUERRE is a FORTRAN90 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:	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

Licensing:

The code described and made available on this web page is distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

QUADRATURE_RULES, a dataset directory which defines quadrature rules; a number of examples of sparse grid quadrature rules are included.

QUADRULE, a FORTRAN90 library which defines quadrature rules for various intervals and weight functions.

SGMGA, a FORTRAN90 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_CC, a dataset directory which contains files of the abscissas of sparse grids based on a Clenshaw Curtis rule.

SPARSE_GRID_F2, a dataset directory which contains files of the abscissas of sparse grids based on a Fejer Type 2 rule.

SPARSE_GRID_GL, a FORTRAN90 library which computes a sparse grid based on 1D Gauss-Legendre rules..

SPARSE_GRID_GP, a dataset directory which contains files of the abscissas of sparse grids based on a Gauss Patterson rule.

SPARSE_GRID_HERMITE, a FORTRAN90 library which creates sparse grids based on Gauss-Hermite rules.

SPARSE_GRID_HW, a FORTRAN90 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_LAGUERRE, a dataset directory which contains files of the abscissas of sparse grids based on a Gauss-Laguerre rule.

SPARSE_GRID_MIXED, a FORTRAN90 library which constructs a sparse grid using different rules in each spatial dimension.

SPARSE_GRID_NCC, a dataset directory which contains files of the abscissas of sparse grids based on a Newton Cotes closed rule.

SPARSE_GRID_NCO, a dataset directory which contains files of the abscissas of sparse grids based on a Newton Cotes open rule.

SPARSE_GRID_OPEN, a FORTRAN90 library which defines define sparse grids based on open nested quadrature rules.

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,
   Advances in Computational Mathematics,
   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,
   Academic Press, 1978,
   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,
   Doklady Akademii Nauk SSSR,
   Volume 4, 1963, pages 240-243.
6. Dennis Stanton, Dennis White,
   Constructive Combinatorics,
   Springer, 1986,
   ISBN: 0387963472,
   LC: QA164.S79.

Source Code:

• sparse_grid_laguerre.f90, the source code.

Examples and Tests:

• sparse_grid_laguerre_test.f90, a sample calling program.
• sparse_grid_laguerre_test.txt, the output file.
• 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.

List of Routines:

• CHOOSE computes the binomial coefficient C(N,K).
• COMP_NEXT computes the compositions of the integer N into K parts.
• GET_UNIT returns a free FORTRAN unit number.
• I4_HUGE returns a "huge" I4.
• I4_LOG_2 returns the integer part of the logarithm base 2 of an I4.
• LAGUERRE_ABSCISSA sets abscissas for multidimensional Gauss-Laguerre quadrature.
• LAGUERRE_INTEGRAL_ND integrates a Laguerre monomial.
• LAGUERRE_WEIGHTS returns weights for certain Gauss-Laguerre quadrature rules.
• LEVEL_TO_ORDER_OPEN converts a level to an order for open rules.
• MONOMIAL_QUADRATURE applies a quadrature rule to a monomial.
• MONOMIAL_VALUE evaluates a monomial.
• MULTIGRID_INDEX_ONE returns an indexed multidimensional grid.
• PRODUCT_WEIGHT_LAGUERRE: weights for a product Gauss-Laguerre rule.
• R8_FACTORIAL computes the factorial of N, also denoted "N!".
• R8_HUGE returns a very large R8.
• R8MAT_WRITE writes an R8MAT file.
• R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
• S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
• SPARSE_GRID_LAGUERRE computes a sparse grid of Gauss-Laguerre points.
• SPARSE_GRID_LAGUERRE_INDEX indexes points in a sparse Gauss-Laguerre grid.
• SPARSE_GRID_LAGUERRE_SIZE sizes a sparse grid of Gauss-Laguerre points.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• VEC_COLEX_NEXT2 generates vectors in colex order.

You can go up one level to the FORTRAN90 source codes.