# sgmga

sgmga, a C code 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.

The code 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 SGMGA().

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 Golub-Welsch GW Moderate Linear ? ?
11 Hermite Genz-Keister HGK Moderate Exponential (-oo,+oo) e-x*x

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.

A version of the sparse grid library is available in https://tasmanian.ornl.gov, the TASMANIAN library, available from Oak Ridge National Laboratory.

### Languages:

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

### Related Data and Programs:

QUADRULE, a C code which defines quadrature rules for various intervals and weight functions.

SANDIA_RULES, a C code which produces 1D quadrature rules of Chebyshev, Clenshaw Curtis, Fejer 2, Gegenbauer, generalized Hermite, generalized Laguerre, Hermite, Jacobi, Laguerre, Legendre and Patterson types.

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 code 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_HW, a C 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.

### Reference:

1. Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
2. Charles Clenshaw, Alan Curtis,
A Method for Numerical Integration on an Automatic Computer,
Numerische Mathematik,
Volume 2, Number 1, December 1960, pages 197-205.
3. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
4. Michael Eldred, John Burkardt,
Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification,
American Institute of Aeronautics and Astronautics,
Paper 2009-0976,
47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition,
5 - 8 January 2009, Orlando, Florida.
5. Walter Gautschi,
Numerical Quadrature in the Presence of a Singularity,
SIAM Journal on Numerical Analysis,
Volume 4, Number 3, September 1967, pages 357-362.
6. Thomas Gerstner, Michael Griebel,
Numerical Integration Using Sparse Grids,
Numerical Algorithms,
Volume 18, Number 3-4, 1998, pages 209-232.
7. Gene Golub, John Welsch,
Mathematics of Computation,
Volume 23, Number 106, April 1969, pages 221-230.
8. Prem Kythe, Michael Schaeferkotter,
Handbook of Computational Methods for Integration,
Chapman and Hall, 2004,
ISBN: 1-58488-428-2,
LC: QA299.3.K98.
9. Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
ISBN: 0-12-519260-6,
LC: QA164.N54.
10. 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.
11. Fabio Nobile, Raul Tempone, Clayton Webster,
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data,
SIAM Journal on Numerical Analysis,
Volume 46, Number 5, 2008, pages 2411-2442.
12. Thomas Patterson,
Mathematics of Computation,
Volume 22, Number 104, October 1968, pages 847-856.
13. Sergey Smolyak,
Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,
Volume 4, 1963, pages 240-243.
14. Arthur Stroud, Don Secrest,
Prentice Hall, 1966,
LC: QA299.4G3S7.
15. Joerg Waldvogel,
Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules,
BIT Numerical Mathematics,
Volume 43, Number 1, 2003, pages 1-18.

### Source Code:

Last revised on 04 August 2019.