sparse_grid_hw
sparse_grid_hw,
a MATLAB 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 version of the 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 MATLAB 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:
-
ccu, Clenshaw-Curtis 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.
-
ccs, slow Clenshaw-Curtis 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.
-
glo, Gauss-Legendre odd 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/2)+1.
Licensing:
The computer code and data files described and made available on this web page
are 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
Related Data and Programs:
grid_display,
a MATLAB code which
can display a 2d or 3d grid or sparse grid.
nint_exactness_mixed,
a MATLAB code which
measures the polynomial exactness of a multidimensional quadrature rule
based on a mixture of 1d quadrature rule factors.
product_rule,
a MATLAB code which
constructs a product quadrature rule from identical 1d factor rules.
quadrule,
a MATLAB code which
defines quadrature rules for various intervals and weight functions.
sparse_grid_cc,
a MATLAB code which
can define a multidimensional sparse grid based on a
1d clenshaw curtis rule.
sparse_grid_gl,
a MATLAB code which
creates sparse grids based on gauss-legendre rules.
sparse_grid_hermite,
a MATLAB code which
creates sparse grids based on gauss-hermite rules.
sparse_grid_hw_test
sparse_grid_laguerre,
a MATLAB code which
creates sparse grids based on gauss-laguerre rules.
spinterp,
a MATLAB code which
carries out piecewise multilinear hierarchical sparse grid interpolation;
an earlier version of this software is acm toms algorithm 847,
by andreas klimke;
spquad,
a MATLAB code which
computes the points and weights of a sparse grid quadrature rule
for a multidimensional integral, based on the clenshaw-curtis quadrature rule,
by greg von winckel.
Author:
Original MATLAB code by Florian Heiss and Viktor Winschel.
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:
-
cc.m
returns one rule from the CCU family defined on [-1,+1]
with unit weight, indexed by order.
-
cce.m
returns one rule from the CCU (Clenshaw-Curtis Exponential growth)
family defined on [-1,+1] with unit weight,
indexed by level.
-
cce_order.m
computes the order of a Clenshaw-Curtis Exponential (CCE)
rule from the level.
-
ccl.m
returns one rule from the CCL (Clenshaw-Curtis Linear growth) family
defined on [-1,+1] with unit weight, indexed by level.
-
ccl_order.m
computes the order of a CCL (Clenshaw-Curtis Linear growth)
rule from the level.
-
ccs.m
returns one rule from the CCS (Clenshaw-Curtis Slow growth) family
defined on [-1,+1] with unit weight, indexed by level.
-
ccs_order.m
computes the order of a CCS (Clenshaw Curtis Slow growth)
rule from the level.
-
get_seq.m
generates integer vectors that describe component tensor products.
-
gqn.m
returns one rule from the GQN family.
-
gqn_order.m
computes the order of a GQN rule from the level, N = L.
-
gqn2_order.m
computes the order of a GQN rule from the level, N = 2*L-1.
-
gqu.m
returns one rule from the GQU family.
-
gqu_order.m
computes the order of a GQU rule from the level.
-
i4_factorial2.m
computes the double factorial function.
-
i4vec_print.m
prints an I4VEC.
-
kpn.m
returns one rule from the KPN family.
-
kpn_order.m
returns the order of a member of the KPN family given the level.
-
kpu.m
returns one rule from the KPU family.
-
kpu_order.m
returns the order of a member of the KPU family given the level.
-
nwspgr.m
the main routine, which the user calls in order to compute
a particular sparse grid.
-
nwspgr_size.m
returns the size of a particular sparse grid.
-
quad_rule_print.m
prints a multidimensional quadrature rule.
-
rule_adjust.m
adjusts a 1D quadrature rule from [A,B] to [C,D].
-
symmetric_sparse_size.m
sizes a symmetric sparse rule.
-
tensor_product.m
computes a tensor product quadrature rule.
Last revised on 20 March 2019.