HERMITE_RULE
Gauss-Hermite Quadrature Rules

HERMITE_RULE is an executable C++ program, using double precision arithmetic, which can generate a specific Gauss-Hermite quadrature rule, based on user input.

The rule can be output as text in a standard programming language, or the data can be written to three files for easy use as input to other programs.

The Gauss-Hermite quadrature rule is designed to approximate integrals on infinite intervals.

The Gauss Hermite quadrature assumes that the integrand we are considering has a form like:

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

The standard Gauss Hermite quadrature rule is used as follows:

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

        Sum ( 1 <= i <= order ) w(i) * f(x(i)) 
      

Although the standard rule is defined in terms of the product of the integral weight function exp(-x*x) and a function f(x), there may be cases when it is more convenient to think that we are simply approximating

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

The standard rule can easily be modified, by adjusting the weights, so that the computation can be done in this form. The program allows the user to specify, through the parameter OPTION, whether the standard rule is to be computed (OPTION=0), or the modified rule (OPTION=1).

The modified Gauss-Hermite quadrature rule is used as follows:

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

        Sum ( 1 <= i <= order ) w(i) * f(x(i)) 
      

Usage:

hermite_rule order option output

order

the number of points in the quadrature rule. A typical value might be 4, 8, or 16.

option

a 0 value requests the "standard" integration rule for

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

a 1 value requests the "modified" integration rule for

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

output

specifies how the rule is to be reported:

• C++, print as C++ text;
• F77, print as FORTRAN77 text;
• F90, print as FORTRAN90 text;
• MAT, print as MATLAB text;
• file, written to three files, file_w.txt, file_x.txt, and file_r.txt, containing the weights, abscissas, and interval limits.

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 compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, is an executable C++ program which can compute and print a generalized Gauss-Laguerre quadrature rule.

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

INT_EXACTNESS, is an executable C++ program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

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

INTLIB is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

JACOBI_RULE, is an executable C++ program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, is an executable C++ program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, is an executable C++ program which computes a Gauss-Legendre quadrature rule.

PRODUCT_FACTOR is an executable C++ program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is an executable C++ program which constructs a product rule from identical 1D factor rules.

QUADPACK is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

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

QUADRATURE_RULES_HERMITE is a dataset directory of triples of files defining standard Hermite quadrature rules.

QUADRULE is a C++ library which contains 1-dimensional quadrature rules.

TEST_INT is a FORTRAN90 library which defines functions that may be used as test integrands for quadrature rules in 1D.

TEST_INT_HERMITE is a C++ library which defines test integrands for integration over (-oo,+oo).

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
3. Arthur Stroud, Don Secrest,
   Gaussian Quadrature Formulas,
   Prentice Hall, 1966,
   LC: QA299.4G3S7.

Source Code:

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

Examples and Tests:

• output_cpp.txt, printed output from the command

  
              hermite_rule 4 0 C++
            

• output_f77.txt, printed output from the command

  
              hermite_rule 4 0 F77
            

• output_f90.txt, printed output from the command

  
              hermite_rule 4 0 F90
            

• output_f90_modified.txt, printed output from the command

  
              hermite_rule 4 1 F90
            

• output_mat.txt, printed output from the command

  
              hermite_rule 8 0 MAT
            

• herm_o4_r.txt, the region file created by the command

  
              hermite_rule 4 0 herm_o4
            

• herm_o4_w.txt, the weight file created by the command

  
              hermite_rule 4 0 herm_o4
            

• herm_o4_x.txt, the abscissa file created by the command

  
              hermite_rule 4 0 herm_o4
            

• herm_o4_modified_r.txt, the region file created by the command

  
              hermite_rule 4 1 herm_o4
            

• herm_o4_modified_w.txt, the weight file created by the command

  
              hermite_rule 4 1 herm_o4
            

• herm_o4_modified_x.txt, the abscissa file created by the command

  
              hermite_rule 4 1 herm_o4
            

List of Routines:

• MAIN is the main program for HERMITE_RULE.
• 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_CLOSE_WRITE closes a file to which a DTABLE was to be written.
• DTABLE_DATA_WRITE writes data to a DTABLE file.
• DTABLE_HEADER_WRITE writes the header of a DTABLE file.
• DTABLE_WRITE writes information to a DTABLE file.
• HERMITE_COMPUTE computes a Gauss-Hermite quadrature rule.
• HERMITE_HANDLE computes the requested Gauss-Hermite rule and outputs it.
• HERMITE_RECUR finds the value and derivative of a Hermite polynomial.
• HERMITE_ROOT improves an approximate root of a Hermite polynomial.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HUGE returns a "huge" R8.
• S_EQI reports whether two strings are equal, ignoring case.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.

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