Sparse Grid Mixed Growth Anisotropic Rules.

SGMGA is a FORTRAN90 library which implements a family of sparse grid rules. These rules are "mixed", in that a different 1D quadrature rule can be specified for each dimension. Moreover, each 1D quadrature rule comes in a family of increasing size whose growth rate (typically linear or exponential) is chosen by the user. Finally, the user may also specify different weights for each dimension, resulting in anisotropic rules.

SGMGA calls many routines from the SANDIA_RULES library. Source code or compiled copies of both libraries must be available when a program wishes to use the SGMGA library.

Thanks to Drew Kouri, who pointed out a discrepancy in the computation of the variable level_1d_max which meant that certain sparse grids requested the generation of a 1D rule of a level that was higher than necessary by 1. In particular, if the Gauss-Patterson rule was involved, sparse grids that actually only needed rules of level 7 would ask also for level 8, resulting in the computation being terminated. This problem was corrected on 25 April 2011.

Index Name Abbreviation Default Growth Rule Interval Weight function
1 Clenshaw-Curtis CC Moderate Exponential [-1,+1] 1
2 Fejer Type 2 F2 Moderate Exponential [-1,+1] 1
3 Gauss Patterson GP Moderate Exponential [-1,+1] 1
4 Gauss-Legendre GL Moderate Linear [-1,+1] 1
5 Gauss-Hermite GH Moderate Linear (-oo,+oo) e-x*x
6 Generalized Gauss-Hermite GGH Moderate Linear (-oo,+oo) |x|alpha e-x*x
7 Gauss-Laguerre LG Moderate Linear [0,+oo) e-x
8 Generalized Gauss-Laguerre GLG Moderate Linear [0,+oo) xalpha e-x
9 Gauss-Jacobi GJ Moderate Linear [-1,+1] (1-x)alpha (1+x)beta
10 Hermite Genz-Keister HGK Moderate Exponential (-oo,+oo) e-x*x
11 User Supplied Open UO Moderate Linear ? ?
12 User Supplied Closed UC Moderate Linear ? ?

For a given family, a growth rule can be prescribed, which determines the orders O of the sequence of rules selected from the family. The selected rules are indexed by L, which starts at 0. The polynomial precision P of the rule is sometimes used to determine the appropriate order O.
Index Name Order Formula
0 Default "DF", moderate exponential or moderate linear
1 "SL", Slow linear O=L+1
2 "SO", Slow Linear Odd O=1+2*((L+1)/2)
3 "ML", Moderate Linear O=2L+1
4 "SE", Slow Exponential select smallest exponential order O so that 2L+1 <= P
5 "ME", Moderate Exponential select smallest exponential order O so that 4L+1 <= P
6 "FE", Full Exponential O=2^L+1 for Clenshaw Curtis, O=2^(L+1)-1 otherwise.

Web Link:

A version of the sparse grid library is available in, the TASMANIAN library, available from Oak Ridge National Laboratory.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


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

Related Data and Programs:

NINT_EXACTNESS_MIXED, a FORTRAN90 program which measures the polynomial exactness of a multidimensional quadrature rule based on a mixture of 1D quadrature rule factors.

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

SANDIA_RULES, a FORTRAN90 library which generates Gauss quadrature rules of various orders and types.

SANDIA_SPARSE, a FORTRAN90 library which computes the points and weights of a Smolyak sparse grid, based on a variety of 1-dimensional quadrature rules.

SGMGA, a dataset directory which contains SGMGA files (Sparse Grid Mixed Growth Anisotropic), that is, multidimensional Smolyak sparse grids based on a mixture of 1D rules, and with a choice of exponential and linear growth rates for the 1D rules and anisotropic weights for the dimensions.

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, by Knut Petras.

SPARSE_GRID_MIXED, a FORTRAN90 library which creates a sparse grid dataset based on a mixed set of 1D factor 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.


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Source Code:

Examples and Tests:






SGMGA_SIZE_TABLE tabulates the point counts from SGMGA_SIZE for an isotropic rule over a range of dimensions and levels.






List of Routines:

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

Last revised on 29 July 2010.