QUADRATURE_RULES_HERMITE_UNWEIGHTED is a dataset directory which contains examples of quadrature rules of Gauss-Hermite type using weight 1.
Gauss-Hermite quadrature rules are designed to approximate integrals on the infinite interval (-oo,+oo).
The Gauss Hermite quadrature assumes that the integrand we are considering has a form like:
Integral ( -oo < x < +oo ) w(x) * f(x) dxwhere the factor w(x) is regarded as a weight factor.
There are three common variations of the rule, depending on the form of the weight factor w(x):
Integral ( -oo < x < +oo ) f(x) dx
Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx
Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx
The corresponding Gauss-Hermite rule that uses order points will approximate the integral by
sum ( 1 <= i <= order ) w(i) * f(x(i))where, confusingly, w(i) is a vector of quadrature weights, which has no connection with the w(x) weight function.
For this directory, a quadrature rule is stored as three files, containing the weights, the points, and a file containing two points defining the endpoints of the region. Since the Hermite rules are defined on an infinite region, we set the endpoints to a very large negative and positive values respectively, and hope the program will understand what we mean.
If an unweighted Gauss-Hermite rule of order n is computed and evaluated with no numerical error, then it should produce the exact integral of any integrand, having the form of the product of a polynomial of degree 2*n-1 or less times exp(-x^2).
We consider an unweighted weighted Gauss-Hermite quadrature rule of order 4.
Here is the text of the "W" file storing the weights of such a rule:
1.240225817691956 1.059964482891671 1.059964482891671 1.240225817691956
Here is the text of the "X" file storing the abscissas of such a rule:
-1.650680123885785 -0.5246476232752904 0.5246476232752904 1.650680123885785
Here is the text of the "R" file storing the lower and upper limits of the region. These are formally -oo and +oo, but here we simply give them as large values.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
HERMITE_POLYNOMIAL, a C++ library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.
HERMITE_RULE, a C++ program which can compute and print a Gauss-Hermite quadrature rule.
INT_EXACTNESS_HERMITE, a C++ program which tests the polynomial exactness of Gauss-Hermite quadrature rules.
QUADRATURE_RULES_GEN_HERMITE, a dataset directory which contains quadrature rules for integration on an infinite interval, using a generalized Gauss-Hermite rule.
QUADRATURE_RULES_HERMITE_PHYSICIST, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2).
QUADRATURE_RULES_HERMITE_PROBABILIST, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2/2).
SPARSE_GRID_HERMITE, a dataset directory which contains M-dimensional Smolyak sparse grids based on the 1D Gauss-Hermite rule;
TEST_INT_HERMITE, a C++ library which defines test integrands for integration over (-oo,+oo).
Gauss-Hermite Quadrature Rule, Unweighted, Order 1:
Gauss-Hermite Quadrature Rule, Unweighted, Order 2:
Gauss-Hermite Quadrature Rule, Unweighted, Order 4:
Gauss-Hermite Quadrature Rule, Unweighted, Order 8:
Gauss-Hermite Quadrature Rule, Unweighted, Order 16:
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