hermite_rule, a Python code which returns Gauss-Hermite quadrature rule.
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 ) dxis to be approximated by
Sum ( 1 <= i <= order ) w(i) * f(x(i))Generally, a Gauss-Hermite quadrature rule of n points will produce the exact integral when f(x) is a polynomial of degree 2n-1 or less.
The value of C in front of the integral depends on the user's choice of the SCALE parameter:
hermite_rule ( order, a, b, scale 'filename' )where
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
hermite_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and a Octave version and a Python version.
alpert_rule, a Python code which returns an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.
ccn_rule, a Python code which returns a nested Clenshaw Curtis quadrature rule.
chebyshev1_rule, a Python code which returns a Gauss-Chebyshev type 1 quadrature rule.
chebyshev2_rule, a Python code which returns a Gauss-Chebyshev type 2 quadrature rule.
clenshaw_curtis_rule, a Python code which returns a Clenshaw Curtis quadrature rule.
gegenbauer_rule, a Python code which returns a Gauss-Gegenbauer quadrature rule.
gen_hermite_rule, a Python code which returns a generalized Gauss-Hermite quadrature rule.
gen_laguerre_rule, a Python code which returns a generalized Gauss-Laguerre quadrature rule.
hermite_exactness, a Python code which tests the exactness of Gauss-Hermite quadrature rules.
hermite_polynomial, a Python code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.
jacobi_rule, a Python code which returns a Gauss-Jacobi quadrature rule.
laguerre_rule, a Python code which returns a Gauss-Laguerre quadrature rule.
legendre_rule, a Python code which returns a Gauss-Legendre quadrature rule.
line_felippa_rule, a Python code which returns a Felippa quadrature rule over the interior of a line segment in 1D.
patterson_rule, a Python code which computes a Gauss-Patterson quadrature rule.
quad_rule, a Python code which contains 1-dimensional quadrature rules.
truncated_normal_rule, a Python code which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian distribution, that has been truncated to [A,+oo), (-oo,B] or [A,B].
HERM_O4 is a Hermite rule of order 4, created by the command
hermite_rule ( 4, 0.0, 1.0, 0, 'herm_o4' )