Quadrature Rules

Latin Hypercube Centered

**QUADRATURE_RULES_LATIN_CENTER**
is a dataset directory which
contains examples of Latin Center quadrature rules.

A Latin Square can be generalized to a multidimensional "Latin Hypercube". Latin hypercubes are often used for sparse sampling of multidimensional data, in cases where it is important that each coordinate be sampled at regularly spaced values.

A "centered" Latin Hypercube of order N divides the M-dimensional
unit hypercube into N^{M} smaller hypercubes, and chooses
"randomly" the centers of N of these hypercubes, in such a way
that the projections of the samples onto any of the M coordinates
axes is N equally spaced values.

This sampling method can be used as a simple quadrature method, simply by evaluating the function to be integrated at each of the sample points, and dividing by the number of points. It might be expected that the estimate of the integral is somewhat more accurate than a Monte Carlo estimate, since the sample points are more carefully scattered.

The Centered Latin Hypercube datasets stored here have been packaged as quadrature rules - that is, there is a file of abscissas, a file of weights (all 1/N) and a file that defines the corners of the region. This format allows these datasets to be passed into various programs that expect such a file structure for quadrature rules.

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

LATIN_CENTER, a C++ library which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_CENTER_DATASET, a C++ program which creates a Latin Center Hypercube dataset;

Latin Hypercube Centered Quadrature Rules in 6D, defined on the [-1,1] hypercube:

- lc_d6_00001_x.txt, the abscissas for the level 0, order 1 sparse rule.
- lc_d6_00001_w.txt, the weights.
- lc_d6_00001_r.txt, the region.
- lc_d6_00001.out quadrature estimates for Genz test functions.
- lc_d6_00013_x.txt, the abscissas for the level 1, order 13 sparse rule.
- lc_d6_00013_w.txt, the weights.
- lc_d6_00013_r.txt, the region.
- lc_d6_00013.out quadrature estimates for Genz test functions.
- lc_d6_00085_x.txt, the abscissas for the level 2, order 85 sparse rule.
- lc_d6_00085_w.txt, the weights.
- lc_d6_00085_r.txt, the region.
- lc_d6_00085.out quadrature estimates for Genz test functions.
- lc_d6_00389_x.txt, the abscissas for the level 3, order 389 sparse rule.
- lc_d6_00389_w.txt, the weights.
- lc_d6_00389_r.txt, the region.
- lc_d6_00389.out quadrature estimates for Genz test functions.
- lc_d6_01457_x.txt, the abscissas for the level 4, order 1457 sparse rule.
- lc_d6_01457_w.txt, the weights.
- lc_d6_01457_r.txt, the region.
- lc_d6_01457.out quadrature estimates for Genz test functions.
- lc_d6_04865_x.txt, the abscissas for the level 5, order 4865 sparse rule.
- lc_d6_04865_w.txt, the weights.
- lc_d6_04865_r.txt, the region.
- lc_d6_04865.out quadrature estimates for Genz test functions.

Latin Hypercube Centered Quadrature Rules in 10D, defined on the [-1,1] hypercube:

- lc_d10_00001_x.txt, the abscissas for the level 0, order 1 sparse rule.
- lc_d10_00001_w.txt, the weights.
- lc_d10_00001_r.txt, the region.
- lc_d10_00001.out quadrature estimates for Genz test functions.
- lc_d10_00021_x.txt, the abscissas for the level 1, order 21 sparse rule.
- lc_d10_00021_w.txt, the weights.
- lc_d10_00021_r.txt, the region.
- lc_d10_00021.out quadrature estimates for Genz test functions.
- lc_d10_00221_x.txt, the abscissas for the level 2, order 221 sparse rule.
- lc_d10_00221_w.txt, the weights.
- lc_d10_00221_r.txt, the region.
- lc_d10_00221.out quadrature estimates for Genz test functions.
- lc_d10_01581_x.txt, the abscissas for the level 3, order 1581 sparse rule.
- lc_d10_01581_w.txt, the weights.
- lc_d10_01581_r.txt, the region.
- lc_d10_01581.out quadrature estimates for Genz test functions.
- lc_d10_08801_x.txt, the abscissas for the level 4, order 8801 sparse rule.
- lc_d10_08801_w.txt, the weights.
- lc_d10_08801_r.txt, the region.
- lc_d10_08801.out quadrature estimates for Genz test functions.
- lc_d10_41265_x.txt, the abscissas for the level 5, order 41265 sparse rule.
- lc_d10_41265_w.txt, the weights.
- lc_d10_41265_r.txt, the region.
- lc_d10_41265.out quadrature estimates for Genz test functions.

You can go up one level to the DATASETS page.