QUADRATURE_RULES_UNIFORM
Quadrature Rules Using Uniform Pseudorandom Points

QUADRATURE_RULES_UNIFORM is a dataset directory which contains examples of "quadrature rules" based on multidimensional uniform pseudorandom values.

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:

• NINT_EXACTNESS is a FORTRAN90 program which can read a set of files defining a quadrature rule in 1D or multidimensions, and test it for exactness against monomial integrands.
• PRODUCT_FACTOR is a FORTRAN90 program which creates a multidimensional product rule whose factors are distinct 1D quadrature rules.
• PRODUCT_RULE is a FORTRAN90 program which creates a multidimensional product rule whose factors are identical 1D quadrature rules.
• QUADRATURE_RULES_HALTON is a collection of quadrature rules generated by the Halton quasirandom sequence.
• QUADRATURE_TEST is an executable MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.
• QUADRULE is a FORTRAN90 library of routines which defines various quadrature rules.
• TABLE is the file format used to store this data;
• UNIFORM is a FORTRAN90 library of routines for generating a multidimensional dataset of pseudorandom values.

Sample Files:

"UNIFORM" (pseudorandom) Quadrature Rules in 6D, defined on the [0,1] hypercube: (we're regarding the Monte Carlo method as a sort of quadrature rule with all weights equal)

• uniform_d6_00001_x.txt, the abscissas for an order 1 rule.
• uniform_d6_00001_w.txt, the weights.
• uniform_d6_00001_r.txt, the range.
• uniform_d6_00013_x.txt, the abscissas for an order 13 rule.
• uniform_d6_00013_w.txt, the weights.
• uniform_d6_00013_r.txt, the range.
• uniform_d6_00085_x.txt, the abscissas for an order 85 rule.
• uniform_d6_00085_w.txt, the weights.
• uniform_d6_00085_r.txt, the range.
• uniform_d6_00389_x.txt, the abscissas for an order 389 rule.
• uniform_d6_00389_w.txt, the weights.
• uniform_d6_00389_r.txt, the range.
• uniform_d6_01457_x.txt, the abscissas for an order 1457 rule.
• uniform_d6_01457_w.txt, the weights.
• uniform_d6_01457_r.txt, the range.
• uniform_d6_04865_x.txt, the abscissas for an order 4865 rule.
• uniform_d6_04865_w.txt, the weights.
• uniform_d6_04865_r.txt, the range.

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

• uniform_d10_level0_x.txt, the abscissas for the order 1 rule.
• uniform_d10_level0_w.txt, the weights.
• uniform_d10_level0_r.txt, the region.
• uniform_d10_level1_x.txt, the abscissas for the order 21 rule.
• uniform_d10_level1_w.txt, the weights.
• uniform_d10_level1_r.txt, the region.
• uniform_d10_level2_x.txt, the abscissas for the order 221 rule.
• uniform_d10_level2_w.txt, the weights.
• uniform_d10_level2_r.txt, the region.
• uniform_d10_level3_x.txt, the abscissas for the order 1581 rule.
• uniform_d10_level3_w.txt, the weights.
• uniform_d10_level3_r.txt, the region.
• uniform_d10_level4_x.txt, the abscissas for the order 8801 rule.
• uniform_d10_level4_w.txt, the weights.
• uniform_d10_level4_r.txt, the region.
• uniform_d10_level5_x.txt, the abscissas for the order 41265 rule.
• uniform_d10_level5_w.txt, the weights.
• uniform_d10_level5_r.txt, the region.

You can go up one level to the DATASETS page.