sparse_grid_hw
sparse_grid_hw,
a C++ code which
computes 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,
by Florian Heiss and Viktor Winschel.
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 function 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.
Licensing:
The information on this web page is distributed under the MIT license.
Languages:
sparse_grid_hw is available in
a C version and
a C++ version and
a Fortran90 version and
a MATLAB version and
an Octave version.
Related Data and Programs:
sparse_grid_hw_test
quad_rule,
a C++ code which
defines quadrature rules for various intervals and weight functions.
sparse_grid_cc,
a C++ code which
creates sparse grids based on Clenshaw-Curtis rules.
sparse_grid_gl,
a C++ code which
creates sparse grids based on Gauss-legendre rules.
sparse_grid_hermite,
a C++ code which
creates sparse grids based on Gauss-Hermite rules.
sparse_grid_laguerre,
a C++ code which
creates sparse grids based on Gauss-Laguerre rules.
Author:
Original MATLAB code by Florian Heiss and Viktor Winschel.
This version by John Burkardt.
Reference:
-
Alan Genz, Bradley Keister,
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,
The optimal addition of points to quadrature formulae,
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 15 April 2020.