Quadrature Rules Using Halton Points

These are some 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.

Related Data and Programs:

• HALTON is a FORTRAN90 library of routines for generating a multidimensional Halton sequence.
• 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_TRI is a collection of quadrature rules to be applied to triangular regions.
• 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.
• SPARSE_GRID_CC is a dataset collection of quadrature rules derived from the Clenshaw Curtis rule.
• SPARSE_GRID_F1 is a dataset collection of quadrature rules derived from the Fejer Type 1 rule.
• SPARSE_GRID_F2 is a dataset collection of quadrature rules derived from the Fejer Type 2 rule.
• SPARSE_GRID_GP is a dataset collection of quadrature rules derived from the Gauss-Patterson rule.
• SPARSE_GRID_NCC is a dataset collection of quadrature rules derived from the Newton Cotes Closed rule.
• SPARSE_GRID_NCO is a dataset collection of quadrature rules derived from the Newton Cotes Open rule.
• SPARSE_GRID_NCOH is a dataset collection of quadrature rules derived from the Newton Cotes Open Half rule.
• TABLE is the file format used to store this data;

Sample Files:

"HALTON" (quasirandom) Quadrature Rules in 6D, defined on the [0,1] square: (we're regarding the QuasiMonte Carlo 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.

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