# sparse_grid_hw

sparse_grid_hw, a FORTRAN90 code which can compute sparse grids for multidimensional integration, based on 1D rules for the unit interval with unit weight function, or for the real line with the Gauss-Hermite weight function. The original MATLAB code is by Florian Heiss and Viktor Winschel.

The original version of this software, and other information, is available at https://sparse-grids.de .

Four built-in 1D families of quadrature rules are supplied, and the user can extend the package by supplying any family of 1D quadrature rules.

The built-in families are identified by a 3-letter key which is also the name of the routine that returns members of the family:

• gqu, standard Gauss-Legendre quadrature rules, for the unit interval [0,1], with weight function w(x) = 1.
• gqn, standard Gauss-Hermite quadrature rules, for the infinite interval (-oo,+oo), with weight function w(x) = exp(-x*x/2)/sqrt(2*pi).
• kpu, Kronrod-Patterson quadrature rules, for the unit interval [0,1], with weight function w(x) = 1. These sacrifice some of the precision of gqu in order to provide a family of nested rules.
• kpn, Kronrod-Patterson quadrature rules, for the infinite interval (-oo,+oo), with weight function w(x) = exp(-x*x/2)/sqrt(2*pi). These sacrifice some of the precision of gqn in order to provide a family of nested rules.

The user can build new sparse grids by supplying a 1D quadrature family. Examples provided include:

• cce_order, Clenshaw-Curtis Exponential quadrature rules, for the unit interval [0,1], with weight function w(x) = 1. The K-th call returns the rule of order 1 if K is 1, and 2*(K-1)+1 otherwise.
• ccl_order, Clenshaw-Curtis Linear quadrature rules, for the unit interval [0,1], with weight function w(x) = 1. The K-th call returns the rule of order 2*K-1.
• ccs_order, slow Clenshaw-Curtis Slow quadrature rules, for the unit interval [0,1], with weight function w(x) = 1. The K-th call returns the rule of order 1 if K is 1, and otherwise a rule whose order N has the form 2^E+1 and is the lowest such order with precision at least 2*K-1.

### Languages:

sparse_grid_hw is available in a C version and a C++ version and a Fortran90 version and a MATLAB version

### Related Data and Programs:

product_rule, a FORTRAN90 code which constructs a product quadrature rule from identical 1D factor rules.

quad_rule, a FORTRAN90 code which defines quadrature rules for various intervals and weight functions.

### Author:

Original MATLAB code by Florian Heiss and Viktor Winschel. FORTRAN90 version by John Burkardt.

### Reference:

Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight,
Journal of Computational and Applied Mathematics,
Volume 71, 1996, pages 299-309.
• Florian Heiss, Viktor Winschel,
Likelihood approximation by numerical integration on sparse grids,
Journal of Econometrics,
Volume 144, Number 1, May 2008, pages 62-80.
• Thomas Patterson,
Mathematics of Computation,
Volume 22, Number 104, October 1968, pages 847-856.
• Knut Petras,
Smolyak Cubature of Given Polynomial Degree with Few Nodes for Increasing Dimension,
Numerische Mathematik,
Volume 93, Number 4, February 2003, pages 729-753.

### Source Code:

Last revised on 28 August 2020.