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:

MONTE_CARLO_RULE, a FORTRAN90 program which generates a dataset of N random M-dimensional points, regards it as a quadrature rule for the unit hypercube, and writes out three files of information.

NINT_EXACTNESS, a C++ program which measures the polynomial exactness of an M-dimensional quadrature rule defined over a finite rectangular product region.

QUADRATURE_RULES, a dataset directory which contains quadrature rules for 1-dimensional intervals, 2D rectangles or M-dimensional rectangular regions, stored as a file of abscissas, a file of weights, and a file of region limits.

QUADRATURE_TEST, a C++ program which reads files defining a M-dimensional quadrature rule, and applies them to all the test integrals defined by TEST_NINT.

QUADRATURE_TEST_GENZ, a FORTRAN90 program which reads the definition of a M-dimensional quadrature rule from three files, applies the rule to the Genz test integrals, and prints the results. (This is a version of QUADRATURE_TEST that is restricted to the Genz problems).

TEST_NINT, a C++ library which defines test functions for M-dimensional quadrature routines.

UNIFORM_DATASET, a C++ program which generates a dataset of uniform pseudorandom values and writes them to a file.

Sample Files:

uniform_d6, "series a", is a family of quadrature rules in 6D, defined on the [0,1] hypercube, generated by a uniform random number generator with seed 123456789 and assigned equal weights. The number of points in the rules was chosen to match the number in the first six sparse grids based on the Clenshaw Curtis rule.

• uniform_d6_00001a_x.txt, the abscissas for an order 1 rule.
• uniform_d6_00001a_w.txt, the weights.
• uniform_d6_00001a_r.txt, the range.
• uniform_d6_00013a_x.txt, the abscissas for an order 13 rule.
• uniform_d6_00013a_w.txt, the weights.
• uniform_d6_00013a_r.txt, the range.
• uniform_d6_00085a_x.txt, the abscissas for an order 85 rule.
• uniform_d6_00085a_w.txt, the weights.
• uniform_d6_00085a_r.txt, the range.
• uniform_d6_00389a_x.txt, the abscissas for an order 389 rule.
• uniform_d6_00389a_w.txt, the weights.
• uniform_d6_00389a_r.txt, the range.
• uniform_d6_01457a_x.txt, the abscissas for an order 1457 rule.
• uniform_d6_01457a_w.txt, the weights.
• uniform_d6_01457a_r.txt, the range.
• uniform_d6_04865a_x.txt, the abscissas for an order 4865 rule.
• uniform_d6_04865a_w.txt, the weights.
• uniform_d6_04865a_r.txt, the range.

uniform_d6, "series b", is a family of quadrature rules in 6D, defined on the [0,1] hypercube, generated by a uniform random number generator with seed 234567891 and assigned equal weights. The number of points in the rules was chosen to match the number in the first six sparse grids based on the Clenshaw Curtis rule.

• uniform_d6_00001b_x.txt, the abscissas for an order 1 rule.
• uniform_d6_00001b_w.txt, the weights.
• uniform_d6_00001b_r.txt, the range.
• uniform_d6_00013b_x.txt, the abscissas for an order 13 rule.
• uniform_d6_00013b_w.txt, the weights.
• uniform_d6_00013b_r.txt, the range.
• uniform_d6_00085b_x.txt, the abscissas for an order 85 rule.
• uniform_d6_00085b_w.txt, the weights.
• uniform_d6_00085b_r.txt, the range.
• uniform_d6_00389b_x.txt, the abscissas for an order 389 rule.
• uniform_d6_00389b_w.txt, the weights.
• uniform_d6_00389b_r.txt, the range.
• uniform_d6_01457b_x.txt, the abscissas for an order 1457 rule.
• uniform_d6_01457b_w.txt, the weights.
• uniform_d6_01457b_r.txt, the range.
• uniform_d6_04865b_x.txt, the abscissas for an order 4865 rule.
• uniform_d6_04865b_w.txt, the weights.
• uniform_d6_04865b_r.txt, the range.

uniform_d10, "series a", is a family of quadrature rules in 10D, defined on the [0,1] hypercube, generated by a uniform random number generator with seed 123456789 and assigned equal weights. The number of points in the rules was chosen to match the number in the first six sparse grids based on the Clenshaw Curtis rule.

• uniform_d10_00001a_x.txt, the abscissas for the order 1 rule.
• uniform_d10_00001a_w.txt, the weights.
• uniform_d10_00001a_r.txt, the region.
• uniform_d10_00021a_x.txt, the abscissas for the order 21 rule.
• uniform_d10_00021a_w.txt, the weights.
• uniform_d10_00021a_r.txt, the region.
• uniform_d10_00221a_x.txt, the abscissas for the order 221 rule.
• uniform_d10_00221a_w.txt, the weights.
• uniform_d10_00221a_r.txt, the region.
• uniform_d10_01581a_x.txt, the abscissas for the order 1581 rule.
• uniform_d10_01581a_w.txt, the weights.
• uniform_d10_01581a_r.txt, the region.
• uniform_d10_08801a_x.txt, the abscissas for the order 8801 rule.
• uniform_d10_08801a_w.txt, the weights.
• uniform_d10_08801a_r.txt, the region.
• uniform_d10_41265a_x.txt, the abscissas for the order 41265 rule.
• uniform_d10_41265a_w.txt, the weights.
• uniform_d10_41265a_r.txt, the region.

uniform_d10, "series a", is a family of quadrature rules in 10D, defined on the [0,1] hypercube, generated by a uniform random number generator with seed 234567891 and assigned equal weights. The number of points in the rules was chosen to match the number in the first six sparse grids based on the Clenshaw Curtis rule.

• uniform_d10_00001b_x.txt, the abscissas for the order 1 rule.
• uniform_d10_00001b_w.txt, the weights.
• uniform_d10_00001b_r.txt, the region.
• uniform_d10_00021b_x.txt, the abscissas for the order 21 rule.
• uniform_d10_00021b_w.txt, the weights.
• uniform_d10_00021b_r.txt, the region.
• uniform_d10_00221b_x.txt, the abscissas for the order 221 rule.
• uniform_d10_00221b_w.txt, the weights.
• uniform_d10_00221b_r.txt, the region.
• uniform_d10_01581b_x.txt, the abscissas for the order 1581 rule.
• uniform_d10_01581b_w.txt, the weights.
• uniform_d10_01581b_r.txt, the region.
• uniform_d10_08801b_x.txt, the abscissas for the order 8801 rule.
• uniform_d10_08801b_w.txt, the weights.
• uniform_d10_08801b_r.txt, the region.
• uniform_d10_41265b_x.txt, the abscissas for the order 41265 rule.
• uniform_d10_41265b_w.txt, the weights.
• uniform_d10_41265b_r.txt, the region.

mc_d20 is a family of quadrature rules in 20D, defined on the [0,1] hypercube, generated by a uniform random number generator with seed 123456789 and assigned equal weights. The number of points in the rules was chosen to match the number in the sparse grids of levels 0 through 4 based on the Clenshaw Curtis rule.

• mc_d20_n1_s123456789_x.txt, the abscissas for the order 1 rule.
• mc_d20_n1_s123456789_w.txt, the weights.
• mc_d20_n1_s123456789_r.txt, the region.
• mc_d20_n41_s123456789_x.txt, the abscissas for the order 41 rule.
• mc_d20_n41_s123456789_w.txt, the weights.
• mc_d20_n41_s123456789_r.txt, the region.
• mc_d20_n841_s123456789_x.txt, the abscissas for the order 841 rule.
• mc_d20_n841_s123456789_w.txt, the weights.
• mc_d20_n841_s123456789_r.txt, the region.
• mc_d20_n11561_s123456789_x.txt, the abscissas for the order 11561 rule.
• mc_d20_n11561_s123456789_w.txt, the weights.
• mc_d20_n11561_s123456789_r.txt, the region.
• mc_d20_n120401_s123456789_x.txt, the abscissas for the order 120401 rule.
• mc_d20_n120401_s123456789_w.txt, the weights.
• mc_d20_n120401_s123456789_r.txt, the region.

You can go up one level to the DATASETS page.