QUADRATURE_RULES_HALTON
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.

Licensing:

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

Related Data and Programs:

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.

Sample Files:

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.