QUADRATURE_RULES_HERMITE
Quadrature Rules of Gauss-Hermite Type

QUADRATURE_RULES_HERMITE is a dataset directory which contains examples of quadrature rules of Gauss-Hermite type.

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) dx
      

We consider three variations of the rule, depending on the form of the weight factor w(x):

• option = 0, the unweighted rule:

              Integral ( -oo < x < +oo ) f(x) dx
            

• option = 1, the physicist weighted rule:

              Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx
            

• option = 2, the probabilist weighted rule:

              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)) 
      

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.

Example:

We consider a physicist weighted Gauss-Hermite quadrature rule of order 4.

Here is the text of the "W" file storing the weights of such a rule:

        0.8131283544699208E-01
        0.8049140900030080    
        0.8049140900030080    
        0.8131283544699208E-01
       

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:

        -1.0E+30
         1.0E+30
      

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:

• HERMITE_RULE is a FORTRAN90 program which generates a Hermite rule of the desired order.
• INT_EXACTNESS_HERMITE is a FORTRAN90 program which can read a set of files defining a Gauss-Hermite quadrature rule, 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_GEN_HERMITE is a dataset directory which contains generalized Gauss-Hermite quadrature rules, for the interval (-oo,+oo) with weight |x|^(2*alpha)*exp(-x*x).
• QUADRATURE_TEST is a 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 which defines various quadrature rules.
• TABLE is a data file format which is used to store this data;
• TEST_INT_HERMITE is a FORTRAN90 library which defines several test integrands which may be integrated using a Gauss-Hermite rule.

Sample Files:

Unweighted Gauss-Hermite Rule, Order 1:

• herm_o1_unweighted_x.txt, the abscissas.
• herm_o1_unweighted_w.txt, the weights.
• herm_o1_unweighted_r.txt, the range.

Unweighted Gauss-Hermite Rule, Order 2:

• herm_o2_unweighted_x.txt, the abscissas.
• herm_o2_unweighted_w.txt, the weights.
• herm_o2_unweighted_r.txt, the range.

Unweighted Gauss-Hermite Rule, Order 4:

• herm_o4_unweighted_x.txt, the abscissas.
• herm_o4_unweighted_w.txt, the weights.
• herm_o4_unweighted_r.txt, the range.

Unweighted Gauss-Hermite Rule, Order 8:

• herm_o8_unweighted_x.txt, the abscissas.
• herm_o8_unweighted_w.txt, the weights.
• herm_o8_unweighted_r.txt, the range.

Unweighted Gauss-Hermite Rule, Order 16:

• herm_o16_unweighted_x.txt, the abscissas.
• herm_o16_unweighted_w.txt, the weights.
• herm_o16_unweighted_r.txt, the range.

Unweighted Gauss-Hermite Rule, Order 32:

• herm_o32_unweighted_x.txt, the abscissas.
• herm_o32_unweighted_w.txt, the weights.
• herm_o32_unweighted_r.txt, the range.

Unweighted Gauss-Hermite Rule, Order 64:

• herm_o64_unweighted_x.txt, the abscissas.
• herm_o64_unweighted_w.txt, the weights.
• herm_o64_unweighted_r.txt, the range.

Physicist weighted Gauss-Hermite Rule, Order 1:

• herm_o1_physicist_x.txt, the abscissas.
• herm_o1_physicist_w.txt, the weights.
• herm_o1_physicist_r.txt, the range.

Physicist weighted Gauss-Hermite Rule, Order 2:

• herm_o2_physicist_x.txt, the abscissas.
• herm_o2_physicist_w.txt, the weights.
• herm_o2_physicist_r.txt, the range.

Physicist weighted Gauss-Hermite Rule, Order 4:

• herm_o4_physicist_x.txt, the abscissas.
• herm_o4_physicist_w.txt, the weights.
• herm_o4_physicist_r.txt, the range.

Physicist weighted Gauss-Hermite Rule, Order 8:

• herm_o8_physicist_x.txt, the abscissas.
• herm_o8_physicist_w.txt, the weights.
• herm_o8_physicist_r.txt, the range.

Physicist weighted Gauss-Hermite Rule, Order 16:

• herm_o16_physicist_x.txt, the abscissas.
• herm_o16_physicist_w.txt, the weights.
• herm_o16_physicist_r.txt, the range.

Physicist weighted Gauss-Hermite Rule, Order 32:

• herm_o32_physicist_x.txt, the abscissas.
• herm_o32_physicist_w.txt, the weights.
• herm_o32_physicist_r.txt, the range.

Physicist weighted Gauss-Hermite Rule, Order 64:

• herm_o64_physicist_x.txt, the abscissas.
• herm_o64_physicist_w.txt, the weights.
• herm_o64_physicist_r.txt, the range.

Probabilist weighted Gauss-Hermite Rule, Order 1:

• herm_o1_probabilist_x.txt, the abscissas.
• herm_o1_probabilist_w.txt, the weights.
• herm_o1_probabilist_r.txt, the range.

Probabilist weighted Gauss-Hermite Rule, Order 2:

• herm_o2_probabilist_x.txt, the abscissas.
• herm_o2_probabilist_w.txt, the weights.
• herm_o2_probabilist_r.txt, the range.

Probabilist weighted Gauss-Hermite Rule, Order 4:

• herm_o4_probabilist_x.txt, the abscissas.
• herm_o4_probabilist_w.txt, the weights.
• herm_o4_probabilist_r.txt, the range.

Probabilist weighted Gauss-Hermite Rule, Order 8:

• herm_o8_probabilist_x.txt, the abscissas.
• herm_o8_probabilist_w.txt, the weights.
• herm_o8_probabilist_r.txt, the range.

Probabilist weighted Gauss-Hermite Rule, Order 16:

• herm_o16_probabilist_x.txt, the abscissas.
• herm_o16_probabilist_w.txt, the weights.
• herm_o16_probabilist_r.txt, the range.

Probabilist weighted Gauss-Hermite Rule, Order 32:

• herm_o32_probabilist_x.txt, the abscissas.
• herm_o32_probabilist_w.txt, the weights.
• herm_o32_probabilist_r.txt, the range.

Probabilist weighted Gauss-Hermite Rule, Order 64:

• herm_o64_probabilist_x.txt, the abscissas.
• herm_o64_probabilist_w.txt, the weights.
• herm_o64_probabilist_r.txt, the range.

You can go up one level to the DATASETS page.