HERMITE_RULE
Gauss-Hermite Quadrature Rules

HERMITE_RULE is a FORTRAN77 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:

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

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

The value of C in front of the integral depends on the user's choice of the SCALE parameter:

• scale=0, then C = 1; this is the standard choice for Gauss-Hermite quadrature.
• scale=1, then C is a normalization factor so that f(x)=1 will integrate to 1. This implies in turn that the weights will sum to 1. This choice is appropriate when using the rule to compute probabilities.

Usage:

hermite_rule n a b scale filename

• n is the number of points in the quadrature rule.
• a is the center point (default 0);
• b is the scale factor (default 1);
• scale is the normalization option (0/1). If 1, then the weights are normalized to have unit sum;
• 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 made available on this web page are distributed under the GNU LGPL license.

Languages:

HERMITE_RULE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CCN_RULE, a FORTRAN77 program which defines one of a set of nested Clenshaw Curtis quadrature rules of any order.

CLENSHAW_CURTIS_RULE, a FORTRAN77 program which defines a Clenshaw Curtis quadrature rule.

HERMITE_EXACTNESS, a FORTRAN77 program which tests the polynomial exactness of Gauss-Hermite quadrature rules for estimating the integral of a function with density exp(-x^2) over the interval (-oo,+oo).

HERMITE_POLYNOMIAL, a FORTRAN77 library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

HERMITE_TEST_INT, a FORTRAN77 library which defines test integrands for Hermite integrals with interval (-oo,+oo) and density exp(-x^2).

LAGUERRE_RULE, a FORTRAN77 program which can compute and print a Gauss-Laguerre quadrature rule for estimating the integral of a function with density exp(-x) over the interval [0,+oo).

LEGENDRE_RULE, a FORTRAN77 program which computes a Gauss-Legendre quadrature rule.

PATTERSON_RULE, a FORTRAN77 program which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

QUADRATURE_RULES_HERMITE_PHYSICIST, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2).

QUADRATURE_RULES_HERMITE_PROBABILIST, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2/2).

QUADRATURE_RULES_HERMITE_UNWEIGHTED, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function 1.

QUADRULE, a FORTRAN77 library which contains 1-dimensional quadrature rules.

TRUNCATED_NORMAL_RULE, a FORTRAN77 program which computes a quadrature rule for a normal distribution that has been truncated to [A,+oo), (-oo,B] or [A,B].

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.f, the source code.
• hermite_rule.sh, commands to compile the source code.

Examples and Tests:

HERM_O4 is a Hermite rule of order 4, created by the command

        hermite_rule 4 0.0 1.0 0 herm_o4
      

• herm_o4_r.txt, the region file;
• herm_o4_w.txt, the weight file;
• herm_o4_x.txt, the abscissa file;

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.
• 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.
• CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
• GET_UNIT returns a free FORTRAN unit number.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• PARCHK checks parameters ALPHA and BETA for classical weight functions.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HUGE returns a very large R8.
• R8MAT_WRITE writes an R8MAT file.
• RULE_WRITE writes a quadrature rule to a file.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 from a string.
• 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 FORTRAN77 source codes.