Quadrature Rules Using Halton Points

**QUADRATURE_RULES_HALTON**
is a dataset directory which
contains examples of "quadrature rules" based on multidimensional
quasirandom Halton sequences.

A quadrature rule
is a set of **n** points **x** and associated weights **w**
so that the integral of a function **f(x)** over some particular
region can be approximated by:

Integral f(x) dx = Sum ( 1 <= i <= n ) w(i) * f(x(i))

Using a random, pseudorandom, or quasirandom sequence can be regarded as a kind of quadrature rule in which the weight vector is 1/N.

For this directory, a quadrature rule is stored as three files, containing the weights, the points, and a file containing two points defining the corners of the rectangular region. The dimension of the region is deduced implicitly from the dimension of the points.

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

HALTON, a C++ library which computes elements of a Halton quasirandom sequence.

INT_EXACTNESS, a C++ program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

HALTON (quasirandom) Quadrature Rules in 6D, defined on the [0,1] square: (we're regarding the method as a sort of quadrature rule with all weights equal)

- halton_d6_00001_x.txt, the abscissas for an order 1 rule.
- halton_d6_00001_w.txt, the weights.
- halton_d6_00001_r.txt, the range.
- halton_d6_00013_x.txt, the abscissas for an order 13 rule.
- halton_d6_00013_w.txt, the weights.
- halton_d6_00013_r.txt, the range.
- halton_d6_00085_x.txt, the abscissas for an order 85 rule.
- halton_d6_00085_w.txt, the weights.
- halton_d6_00085_r.txt, the range.
- halton_d6_00389_x.txt, the abscissas for an order 389 rule.
- halton_d6_00389_w.txt, the weights.
- halton_d6_00389_r.txt, the range.
- halton_d6_01457_x.txt, the abscissas for an order 1457 rule.
- halton_d6_01457_w.txt, the weights.
- halton_d6_01457_r.txt, the range.
- halton_d6_04865_x.txt, the abscissas for an order 4865 rule.
- halton_d6_04865_w.txt, the weights.
- halton_d6_04865_r.txt, the range.

"HALTON" (quasirandom) Quadrature Rules in 10D, defined on the [0,1] square: (we're regarding the QuasiMonte Carlo method as a sort of quadrature rule with all weights equal)

- halton_d10_level0_x.txt, the abscissas for the order 1 sparse rule.
- halton_d10_level0_w.txt, the weights.
- halton_d10_level0_r.txt, the region.
- halton_d10_level1_x.txt, the abscissas for the order 21 sparse rule.
- halton_d10_level1_w.txt, the weights.
- halton_d10_level1_r.txt, the region.
- halton_d10_level2_x.txt, the abscissas for the order 221 sparse rule.
- halton_d10_level2_w.txt, the weights.
- halton_d10_level2_r.txt, the region.
- halton_d10_level3_x.txt, the abscissas for the order 1581 sparse rule.
- halton_d10_level3_w.txt, the weights.
- halton_d10_level3_r.txt, the region.
- halton_d10_level4_x.txt, the abscissas for the order 8801 sparse rule.
- halton_d10_level4_w.txt, the weights.
- halton_d10_level4_r.txt, the region.
- halton_d10_level5_x.txt, the abscissas for the order 41265 sparse rule.
- halton_d10_level5_w.txt, the weights.
- halton_d10_level5_r.txt, the region.

You can go up one level to the DATASETS page.