HERMITE_RULE
Gauss-Hermite Quadrature Rules

HERMITE_RULE is a C++ program which generates a specific Gauss-Hermite quadrature rule, based on user input.

The rule is written to three files for easy use as input to other programs.

The Gauss Hermite quadrature rule is used as follows:

        Integral ( -oo < x < +oo ) f(x) exp ( - b * ( x - a )^2 ) dx
      

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

Usage:

hermite_rule order a b filename

• order is the number of points in the quadrature rule.
• a is the center point (default 0);
• b is the scale factor (default 1);
• filename specifies the output filenames: filename_w.txt, filename_x.txt, and filename_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:

CHEBYSHEV1_RULE, is a C++ program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, is a C++ program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE is a C++ program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, is a C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, is a C++ program which can compute and print a generalized Gauss-Hermite quadrature rule.

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

HERMITE_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

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

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

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

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

PATTERSON_RULE, is a C++ program which computes a Gauss-Patterson quadrature rule.

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

PRODUCT_RULE is a 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.

TANH_SINH_RULE, a C++ program which computes and writes out a tanh-sinh quadrature rule of given order.

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. Sylvan Elhay, Jaroslav Kautsky,
   Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature,
   ACM Transactions on Mathematical Software,
   Volume 13, Number 4, December 1987, pages 399-415.
4. Jaroslav Kautsky, Sylvan Elhay,
   Calculation of the Weights of Interpolatory Quadratures,
   Numerische Mathematik,
   Volume 40, 1982, pages 407-422.
5. Roger Martin, James Wilkinson,
   The Implicit QL Algorithm,
   Numerische Mathematik,
   Volume 12, Number 5, December 1968, pages 377-383.
6. 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:

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

  
              hermite_rule 4 0.0 1.0 herm_o4
            

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

  
              hermite_rule 4 0.0 1.0 herm_o4
            

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

  
              hermite_rule 4 0.0 1.0 herm_o4
            

List of Routines:

• MAIN is the main program for HERMITE_RULE.
• CDGQF computes a Gauss quadrature formula with default A, B and simple knots.
• CGQF computes knots and weights of a Gauss quadrature formula.
• CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• PARCHK checks parameters ALPHA and BETA for classical weight functions.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8_HUGE returns a "huge" R8.
• R8_SIGN returns the sign of an R8.
• R8MAT_WRITE writes an R8MAT file with no header.
• RULE_WRITE writes a quadrature rule to three files.
• SCQF scales a quadrature formula to a nonstandard interval.
• SGQF computes knots and weights of a Gauss Quadrature formula.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.