INT_EXACTNESS_GEN_HERMITE
Exactness of Generalized Gauss-Hermite Quadrature Rules

INT_EXACTNESS_GEN_HERMITE is a C++ program, using double precision arithmetic, which investigates the polynomial exactness of a generalized Gauss-Hermite quadrature rule for the infinite interval (-oo,oo).

Note that the arithmetic accuracy of this program may be insufficient to discriminant the accuracy of generalized Gauss-Hermite rules of moderate order - say, order 16. Therefore, a FORTRAN90 version of this program has been prepared, using "quadruple precision", which should give more satisfactory results for a wider range of orders. This program can be found cataloged below as INT_EXACTNESS_GEN_HERMITE_R16.

Standard generalized Gauss-Hermite quadrature assumes that the integrand we are considering has a form like:

        Integral ( -oo < x < +oo ) |x|^alpha * exp(-x^2) * f(x) dx
      

A standard generalized Gauss-Hermite quadrature rule is a set of n positive weights w and abscissas x so that

        Integral ( -oo < x < +oo ) |x|^alpha * exp(-x^2) * f(x) dx
      

        Sum ( 1 <= I <= N ) w(i) * f(x(i))
      

It is often convenient to consider approximating integrals in which the weighting factor |x|^alpha * exp(-x^2) is implicit. In that case, we are looking at approximating

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

A modified generalized Gauss-Hermite quadrature rule is a set of n positive weights w and abscissas x so that

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

        Sum ( 1 <= I <= N ) w(i) * f(x(i))
      

When using a generalized Gauss-Hermite quadrature rule, it's important to know whether the rule has been developed for the standard or modified cases. Basically, the only change is that the weights of the modified rule have been divided by the weighting function evaluated at the corresponding abscissa.

For a standard generalized Gauss-Hermite rule, polynomial exactness is defined in terms of the function f(x). That is, we say the rule is exact for polynomials up to degree DEGREE_MAX if, for any polynomial f(x) of that degree or less, the quadrature rule will produce the exact value of

        Integral ( -oo < x < +oo ) |x|^alpha * exp(-x^2) * f(x) dx
      

For a modified generalized Gauss-Hermite rule, polynomial exactness is defined in terms of the function f(x) divided by the implicit weighting function. That is, we say a modified generalized Gauss-Hermite rule is exact for polynomials up to degree DEGREE_MAX if, for any integrand f(x) with the property that f(x)/(|x|^alpha*exp(-x^2)) is a polynomial of degree no more than DEGREE_MAX, the quadrature rule will product the exact value of:

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

The program starts at DEGREE = 0, and then proceeds to DEGREE = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of DEGREE, the program generates the corresponding monomial term, applies the quadrature rule to it, and determines the quadrature error. The program uses a scaling factor on each monomial so that the exact integral should always be 1; therefore, each reported error can be compared on a fixed scale.

If the program understands that the rule being considered is a modified rule, then the monomials are multiplied by |x|^alpha * exp(-x^2) when performing the exactness test.

The program is very flexible and interactive. The quadrature rule is defined by three files, to be read at input, and the maximum degree top be checked is specified by the user as well.

Note that the three files that define the quadrature rule are assumed to have related names, of the form

• prefix_x.txt
• prefix_w.txt
• prefix_r.txt

For information on the form of these files, see the QUADRATURE_RULES directory listed below.

The exactness results are written to an output file with the corresponding name:

• prefix_exact.txt

Usage:

int_exactness_gen_hermite prefix degree_max alpha option

prefix

the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;

degree_max

the maximum monomial degree to check. This would normally be a relatively small nonnegative number, such as 5, 10 or 15.

alpha

the value of the parameter, which should be a real number greater than -1.0. Setting alpha to 0.0 results in the basic (non-generalized) Gauss-Hermite rule.

option

0 indicates a standard rule for integrating |x|^alpha * exp(-x^2)*f(x).
1 indicates a modified rule for integrating f(x).

If the arguments are not supplied on the command line, the program will prompt for them.

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:

GEN_HERMITE_RULE is an executable C++ program which can generate a generalized Gauss-Hermite quadrature rule on request.

INT_EXACTNESS is an executable C++ program which tests the polynomial exactness of a quadrature rule for a finite interval.

INT_EXACTNESS_GEN_HERMITE is also available in a FORTRAN90 version and a MATLAB version.

INT_EXACTNESS_GEN_HERMITE_R16 is an executable FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules. It is a version of INT_EXACTNESS_GEN_HERMITE that uses "double double precision" real arithmetic, and gives much improved results.

INT_EXACTNESS_GEN_LAGUERRE is an executable C++ program which tests the polynomial exactness of a generalized Gauss-Laguerre quadrature rule.

INT_EXACTNESS_HERMITE is an executable C++ program which tests the polynomial exactness of a Gauss-Hermite quadrature rule.

INT_EXACTNESS_JACOBI is an executable C++ program which tests the polynomial exactness of a Gauss-Jacobi quadrature rule.

INT_EXACTNESS_LAGUERRE is an executable C++ program which tests the polynomial exactness of a Gauss-Laguerre quadrature rule.

INT_EXACTNESS_LEGENDRE is an executable C++ program which tests the polynomial exactness of a Gauss-Legendre quadrature rule.

INTEGRAL_TEST is an executable C++ program which uses test integrals to measure the effectiveness of certain sets of quadrature rules.

INTLIB is a FORTRAN90 library which numerically estimates integrals in one dimension.

QUADRATURE_RULES is a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_GEN_HERMITE is a dataset directory which contains sets of files that define generalized Gauss-Hermite quadrature rules.

QUADRULE is a C++ library which define quadrature rules on a variety of intervals with different weight functions.

STROUD is a FORTRAN90 library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and multiple dimensions.

TEST_INT_HERMITE is a C++ library which defines integrand functions that can be approximately integrated by a Gauss-Hermite rule.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
2. Arthur Stroud, Don Secrest,
   Gaussian Quadrature Formulas,
   Prentice Hall, 1966,
   LC: QA299.4G3S7.

Source Code:

• int_exactness_gen_hermite.C, the source code.
• int_exactness_gen_hermite.csh, commands to compile the source code.

Examples and Tests:

GEN_HERM_O1_A1.0 is a standard generalized Gauss-Hermite order 1 rule with ALPHA = 1.0.

• gen_herm_o1_a1.0_x.txt, the abscissas of the rule.
• gen_herm_o1_a1.0_w.txt, the weights of the rule.
• gen_herm_o1_a1.0_r.txt, defines the region for the rule.
• gen_herm_o1_a1.0_exact.txt, the results of the command

  int_exactness_gen_hermite gen_herm_o1_a1.0 5 1.0 0

GEN_HERM_O2_A1.0 is a standard generalized Gauss-Hermite order 2 rule with ALPHA = 1.0.

• gen_herm_o2_a1.0_x.txt, the abscissas of the rule.
• gen_herm_o2_a1.0_w.txt, the weights of the rule.
• gen_herm_o2_a1.0_r.txt, defines the region for the rule.
• gen_herm_o2_a1.0_exact.txt, the results of the command

  int_exactness_gen_hermite gen_herm_o2_a1.0 5 1.0 0

GEN_HERM_O4_A1.0 is a standard generalized Gauss-Hermite order 4 rule with ALPHA = 1.0.

• gen_herm_o4_a1.0_x.txt, the abscissas of the rule.
• gen_herm_o4_a1.0_w.txt, the weights of the rule.
• gen_herm_o4_a1.0_r.txt, defines the region for the rule.
• gen_herm_o4_a1.0_exact.txt, the results of the command

  int_exactness_gen_hermite gen_herm_o4_a1.0 10 1.0 0

GEN_HERM_O8_A1.0 is a standard generalized Gauss-Hermite order 8 rule with ALPHA = 1.0.

• gen_herm_o8_a1.0_x.txt, the abscissas of the rule.
• gen_herm_o8_a1.0_w.txt, the weights of the rule.
• gen_herm_o8_a1.0_r.txt, defines the region for the rule.
• gen_herm_o8_a1.0_exact.txt, the results of the command

  int_exactness_gen_hermite gen_herm_o8_a1.0 18 1.0 0

GEN_HERM_O16_A1.0 is a standard generalized Gauss-Hermite order 16 rule with ALPHA = 1.0.

• gen_herm_o16_a1.0_x.txt, the abscissas of the rule.
• gen_herm_o16_a1.0_w.txt, the weights of the rule.
• gen_herm_o16_a1.0_r.txt, defines the region for the rule.
• gen_herm_o16_a1.0_exact.txt, the results of the command

  int_exactness_gen_hermite gen_herm_o16_a1.0 35 1.0 0

GEN_HERM_O1_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 1 rule with ALPHA = 1.0.

• gen_herm_o1_a1.0_modified_x.txt, the abscissas of the rule.
• gen_herm_o1_a1.0_modified_w.txt, the weights of the rule.
• gen_herm_o1_a1.0_modified_r.txt, defines the region for the rule.
• gen_herm_o1_a1.0_modified_exact.txt, the results of the command

  int_exactness_gen_hermite gen_herm_o1_a1.0_modified 5 1.0 1

GEN_HERM_O2_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 2 rule with ALPHA = 1.0.

• gen_herm_o2_a1.0_modified_x.txt, the abscissas of the rule.
• gen_herm_o2_a1.0_modified_w.txt, the weights of the rule.
• gen_herm_o2_a1.0_modified_r.txt, defines the region for the rule.
• gen_herm_o2_a1.0_modified_exact.txt, the results of the command

  int_exactness_gen_hermite gen_herm_o2_a1.0_modified 5 1.0 1

GEN_HERM_O4_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 4 rule with ALPHA = 1.0.

• gen_herm_o4_a1.0_modified_x.txt, the abscissas of the rule.
• gen_herm_o4_a1.0_modified_w.txt, the weights of the rule.
• gen_herm_o4_a1.0_modified_r.txt, defines the region for the rule.
• gen_herm_o4_a1.0_modified_exact.txt, the results of the command

  int_exactness_gen_hermite gen_herm_o4_a1.0_modified 10 1.0 1

GEN_HERM_O8_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 8 rule with ALPHA = 1.0.

• gen_herm_o8_a1.0_modified_x.txt, the abscissas of the rule.
• gen_herm_o8_a1.0_modified_w.txt, the weights of the rule.
• gen_herm_o8_a1.0_modified_r.txt, defines the region for the rule.
• gen_herm_o8_a1.0_modified_exact.txt, the results of the command

  int_exactness_gen_hermite gen_herm_o8_a1.0_modified 18 1.0 1

GEN_HERM_O16_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 16 rule with ALPHA = 1.0.

• gen_herm_o16_a1.0_modified_x.txt, the abscissas of the rule.
• gen_herm_o16_a1.0_modified_w.txt, the weights of the rule.
• gen_herm_o16_a1.0_modified_r.txt, defines the region for the rule.
• gen_herm_o16_a1.0_modified_exact.txt, the results of the command

  int_exactness_gen_hermite gen_herm_o16_a1.0_modified 35 1.0 1

List of Routines:

• MAIN is the main program for INT_GEN_EXACTNESS_HERMITE.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DTABLE_DATA_READ reads the data from a DTABLE file.
• DTABLE_HEADER_READ reads the header from a DTABLE file.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• HERMITE_INTEGRAL2 returns the value of a Hermite integral.
• MONOMIAL_QUADRATURE_GEN_HERMITE applies a quadrature rule to a monomial.
• R8_ABS returns the absolute value of an R8.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HUGE returns a "huge" R8.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.

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