SPARSE_GRID_HW
Sparse Grids for Uniform and Normal Weights
Heiss and Winschel
SPARSE_GRID_HW
is a C++ library 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 GaussHermite 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
http://sparsegrids.de .
Four builtin 1D families of quadrature rules are supplied, and the
user can extend the package by supplying any family of 1D quadrature
rules.
The builtin families are identified by a 3letter key which is also
the name of the function that returns members of the family:

gqu, standard GaussLegendre quadrature rules, for
the unit interval [0,1], with weight function w(x) = 1.

gqn, standard GaussHermite quadrature rules, for
the infinite interval (oo,+oo), with weight function
w(x) = exp(x*x/2)/sqrt(2*pi).

kpu, KronrodPatterson 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, KronrodPatterson 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, ClenshawCurtis Exponential quadrature rules, for
the unit interval [0,1], with weight function w(x) = 1.
The Kth call returns the rule of order 1
if K is 1, and 2*(K1)+1 otherwise.

ccl_order, ClenshawCurtis Linear quadrature rules, for
the unit interval [0,1], with weight function w(x) = 1.
The Kth call returns the rule of order 2*K1.

ccs_order, slow ClenshawCurtis Slow quadrature rules, for
the unit interval [0,1], with weight function w(x) = 1.
The Kth 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*K1.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL 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:
QUADRULE,
a C++ library which
defines quadrature rules for various intervals and weight functions.
SANDIA_RULES,
a C++ library which
generates Gauss quadrature rules of various orders and types.
SGMGA,
a C++ library which
creates sparse grids based on a mixture of 1D quadrature rules,
allowing anisotropic weights for each dimension.
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.
Author:
Original MATLAB code by Florian Heiss and Viktor Winschel.
C++ 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 299309.

Florian Heiss, Viktor Winschel,
Likelihood approximation by numerical integration on sparse grids,
Journal of Econometrics,
Volume 144, Number 1, May 2008, pages 6280.

Thomas Patterson,
The optimal addition of points to quadrature formulae,
Mathematics of Computation,
Volume 22, Number 104, October 1968, pages 847856.

Knut Petras,
Smolyak Cubature of Given Polynomial Degree with Few Nodes
for Increasing Dimension,
Numerische Mathematik,
Volume 93, Number 4, February 2003, pages 729753.
Source Code:
Examples and Tests:
List of Routines:

CCE_ORDER: order of a ClenshawCurtis Exponential rule from the level.

CCL_ORDER computes the order of a CCL rule from the level.

CCS_ORDER: order of a "slow growth" Clenshaw Curtis quadrature rule.

CC computes a Clenshaw Curtis quadrature rule based on order.

CPU_TIME reports the elapsed CPU time.

FN_INTEGRAL is the integral of the Hermite test function.

FN_VALUE is a Hermite test function.

FU_INTEGRAL is the integral of the test function for the [0,1]^D interval.

FU_VALUE is a sample function for the [0,1]^D interval.

GET_SEQ generates all positive integer Dvectors that sum to NORM.

GQN provides data for Gauss quadrature with a normal weight.

GQN_ORDER computes the order of a GQN rule from the level.

GQN2_ORDER computes the order of a GQN rule from the level.

GQU provides data for Gauss quadrature with a uniform weight.

GQU_ORDER computes the order of a GQU rule from the level.

I4_CHOOSE computes the binomial coefficient C(N,K).

I4_FACTORIAL2 computes the double factorial function.

I4_MAX returns the maximum of two I4's.

I4_MIN returns the minimum of two I4's.

I4_MOP returns the Ith power of 1 as an I4 value.

I4_POWER returns the value of I^J.

I4MAT_PRINT prints an I4MAT.

I4MAT_PRINT_SOME prints some of an I4MAT.

I4VEC_CUM0_NEW computes the cumulutive sum of the entries of an I4VEC.

I4VEC_PRINT prints an I4VEC.

I4VEC_PRODUCT multiplies the entries of an I4VEC.

I4VEC_SUM sums the entries of an I4VEC.

I4VEC_TRANSPOSE_PRINT prints an I4VEC "transposed".

KPN provides data for KronrodPatterson quadrature with a normal weight.

KPN_ORDER computes the order of a KPN rule from the level.

KPU provides data for KronrodPatterson quadrature with a uniform weight.

KPU_ORDER computes the order of a KPU rule from the level.

NUM_SEQ returns the number of compositions of the integer N into K parts.

NWSPGR generates nodes and weights for sparse grid integration.

NWSPGR_SIZE determines the size of a sparse grid rule.

QUAD_RULE_PRINT prints a multidimensional quadrature rule.

R8_ABS returns the absolute value of an R8.

R8_UNIFORM_01 returns a unit pseudorandom R8.

R8CVV_OFFSET determines the row offsets of an R8CVV.

R8CVV_PRINT prints an R8CVV.

R8CVV_RGET_NEW gets row I from an R8CVV.

R8CVV_RSET sets row I from an R8CVV.

R8MAT_NORMAL_01_NEW returns a unit pseudonormal R8MAT.

R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.

R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.

R8MAT_UNIFORM_01_NEW returns a unit pseudorandom R8MAT.

R8VEC_COPY copies an R8VEC.

R8VEC_DIRECT_PRODUCT creates a direct product of R8VEC's.

R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.

R8VEC_DOT_PRODUCT computes the dot product of a pair of R8VEC's.

R8VEC_NORMAL_01_NEW returns a unit pseudonormal R8VEC.

R8VEC_PRINT prints an R8VEC.

R8VEC_SUM returns the sum of an R8VEC.

R8VEC_TRANSPOSE_PRINT prints an R8VEC "transposed".

R8VEC_UNIFORM_01_NEW returns a new unit pseudorandom R8VEC.

RULE_SORT sorts a multidimensional quadrature rule.

SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.

SYMMETRIC_SPARSE_SIZE sizes a symmetric sparse rule.

TENSOR_PRODUCT generates a tensor product quadrature rule.

TENSOR_PRODUCT_CELL generates a tensor product quadrature rule.

TIMESTAMP prints the current YMDHMS date as a time stamp.
You can go up one level to
the C++ source codes.
Last revised on 26 February 2014.