R16_INT_EXACTNESS_GEN_HERMITE
Exactness of Generalized Gauss-Hermite Quadrature Rules

R16_INT_EXACTNESS_GEN_HERMITE is a FORTRAN90 program which investigates the polynomial exactness of a generalized Gauss-Hermite quadrature rule for the infinite interval (-oo,+oo), using "quadruple real precision" arithmetic.

The use of quadruple real precision arithmetic is motivated here by the extreme nature of the calculations being performed. A "reasonable" calculation might require checking a generalized Gauss-Hermite quadrature rule of order 16, with ALPHA = 1, against monomials up to degree 35. In effect, this requires us to sum quantities involving some terms like |x| exp (-x^2) x^35 for values of x on the order of 4. Even with double precision, a symmetric quadrature rule did not necessary give a value of 0, or close to zero, when handling odd functions (when the monomial exponent is odd). And results were also unsatisfactory for monomials with a large even exponent. However, once we moved to quadruple precision, results were vastly improved, even though we made no numerical changes to the code. For instance, the Gamma function is still computed using constants that were suitable for a good double precision value; one might have thought an improved Gamma evaluator would have been needed, but apparently it's primarily a simple arithmetic problem.

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:

r16_int_exactness_gen_hermite prefix degree_max alpha option

• prefix is the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;
• degree_max is the maximum monomial degree to check. This would normally be a relatively small nonnegative number, such as 5, 10 or 15.
• alpha is 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.

Languages:

R16_INT_EXACTNESS_GEN_HERMITE is available in a FORTRAN90 version.

Related Data and Programs:

GEN_HERMITE_RULE, a FORTRAN90 program which can generate a generalized Gauss-Hermite quadrature rule on request.

INT_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of a quadrature rule for a finite interval.

INT_EXACTNESS_GEN_HERMITE, a FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules. It is the double precision program upon which this quadruple precision program is based.

INT_EXACTNESS_GEN_LAGUERRE, a FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.

INT_EXACTNESS_HERMITE, a FORTRAN90 program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

INT_EXACTNESS_JACOBI, a FORTRAN90 program which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

INT_EXACTNESS_LAGUERRE, a FORTRAN90 program which tests the polynomial exactness of Gauss-Laguerre quadrature rules.

INT_EXACTNESS_LEGENDRE, a FORTRAN90 program which tests the polynomial exactness of Gauss-Legendre quadrature rules.

INTEGRAL_TEST, a FORTRAN90 program which uses test integrals to measure the effectiveness of certain sets of quadrature rules.

INTLIB, a FORTRAN90 library which numerically estimate integrals in one dimension.

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

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

QUADRULE, a FORTRAN90 library which define quadrature rules on a variety of intervals with different weight functions.

R16_HERMITE_RULE, a FORTRAN90 program which can compute and print a Gauss-Hermite quadrature rule, using "quadruple precision real" arithmetic.

STROUD, 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, a FORTRAN90 library which define 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:

• r16_int_exactness_gen_hermite.f90, the source code.
• r16_int_exactness_gen_hermite.sh, 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

  r16_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

  r16_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

  r16_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

  r16_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

  r16_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

  r16_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

  r16_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

  r16_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

  r16_int_exactness_gen_hermite gen_herm_o16_a1.0_modified 35 1.0 1

List of Routines:

• MAIN is the main program for R16_INT_EXACTNESS_GEN_HERMITE.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• 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.
• GET_UNIT returns a free FORTRAN unit number.
• HERMITE_INTEGRAL2 returns the value of a Hermite integral.
• MONOMIAL_QUADRATURE_GEN_HERMITE applies a quadrature rule to a monomial.
• R16MAT_DATA_READ reads data from an R16MAT file.
• R16MAT_HEADER_READ reads the header from an R16MAT file.
• S_TO_I4 reads an I4 from a string.
• S_TO_R16 reads an R16 from a string.
• S_TO_R16VEC reads an R16VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.