hermite_polynomial

hermite_polynomial, an Octave code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

The physicist's Hermite polynomial H(i,x) can be defined by:

        H(i,x) = (-1)^i exp(x^2/2) * d^i/dx^i ( exp(-x^2/2) )
      

The normalized physicist's Hermite polynomial Hn(i,x) is scaled so that

        Integral ( -oo < x < +oo ) exp ( - x^2 ) * Hn(i,x) Hn(j,x) dx = delta ( i, j )
      

The probabilist's Hermite polynomial He(i,x) is related to H(i,x) by:

        He(i,x) = H(i,x/sqrt(2)) / sqrt ( 2^in )
      

The normalized probabilist's Hermite polynomial Hen(i,x) is scaled so that

        Integral ( -oo < x < +oo ) exp ( - 0.5*x^2 ) * Hen(i,x) Hen(j,x) dx = delta ( i, j )
      

The Hermite function Hf(i,x) is related to H(i,x) by:

        Hf(i,x) = H(i,x) * exp(-x^2/2) / sqrt ( 2^i * i! * sqrt ( pi ) )
      

The Hermite function Hf(i,x) is scaled so that:

        Integral ( -oo < x < +oo ) Hf(i,x) Hf(j,x) dx = delta ( i, j )
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license.

Languages:

hermite_polynomial is available in a C version and a C++ version and a Fortran90 version and a Fortran77 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

hermite_polynomial_test

hermite_rule, an Octave code which computes and prints a Gauss-Hermite quadrature rule.

octave_polynomial, an Octave code which analyzes a variety of polynomial families, returning the polynomial values, coefficients, derivatives, integrals, roots, or other information.

pce_ode_hermite, an Octave code which sets up a simple scalar ODE for exponential decay with an uncertain decay rate, using a polynomial chaos expansion in terms of Hermite polynomials.

polpak, an Octave code which evaluates a variety of mathematical functions.

polynomial_conversion, an Octave code which converts representations of a polynomial between monomial, Bernstein, Chebyshev, Hermite, Lagrange, Laguerre and other forms.

test_values, an Octave code which supplies test values of various mathematical functions.

Reference:

1. Theodore Chihara,
   An Introduction to Orthogonal Polynomials,
   Gordon and Breach, 1978,
   ISBN: 0677041500,
   LC: QA404.5 C44.
2. Walter Gautschi,
   Orthogonal Polynomials: Computation and Approximation,
   Oxford, 2004,
   ISBN: 0-19-850672-4,
   LC: QA404.5 G3555.
3. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
   NIST Handbook of Mathematical Functions,
   Cambridge University Press, 2010,
   ISBN: 978-0521192255,
   LC: QA331.N57.
4. Gabor Szego,
   Orthogonal Polynomials,
   American Mathematical Society, 1992,
   ISBN: 0821810235,
   LC: QA3.A5.v23.

Source Code:

• h_integral.m, evaluates the integral of H(i,x).
• h_polynomial_coefficients.m, coefficients of H(i,x).
• h_polynomial_value.m, evaluates H(i,x).
• h_polynomial_values.m, tabulated values of H(i,x).
• h_polynomial_zeros.m, zeros of H(i,x).
• h_quadrature_rule.m, quadrature for H(i,x).
• h_to_monomial_matrix.m, physicist's Hermite to monomial conversion matrix.
• he_double_product_integral.m, integral of He(i,x)*He(j,x)*e^(-x^2/2).
• he_integral.m, evaluates the integral of He(i,x).
• he_plot.m, plots one or more Hermite polynomials He(i,x).
• he_polynomial_coefficients.m, coefficients of He(i,x).
• he_polynomial_value.m, evaluates He(i,x).
• he_polynomial_values.m, tabulated values of He(i,x).
• he_polynomial_zeros.m, zeros of He(i,x).
• he_quadrature_rule.m, quadrature for He(i,x).
• he_triple_product_integral.m, integral of He(i,x)*He(j,x)*He(k,x)*e^(-x^2/2).
• hen_exponential_product.m, exponential product of exp(b*x)*Hen(i,x)*Hen(j,x).
• hen_polynomial_value.m, evaluates Hen(i,x).
• hen_power_product.m, power products, x^e*Hen(i,x)*Hen(j,x).
• hen_projection.m, determines the projection coefficients for a function f(x) against Hen(0:n,x).
• hen_projection_data.m, determines the least squares projection coefficients for a function f(x) which is only supplied as M data values (x,fx), against Hen(0:n,x).
• hen_projection_value.m, evaluates the projection of a function against Hen(0:n,x).
• hf_exponential_product.m, exponential products, exp(b*x)*Hf(i,x)*Hf(j,x).
• hf_function_value.m, evaluates the Hermite function Hf(i,x).
• hf_function_values.m, values of the Hermite function Hf(i,x).
• hf_plot.m, plots one or more Hermite functions Hf(i,x).
• hf_power_product.m, power products x^e*Hf(i,x)*Hf(j,x).
• hf_quadrature_rule.m, quadrature for Hf(i,x).
• hn_exponential_product.m, exponential products exp(b*x)*Hn(i,x)*Hn(j,x).
• hn_polynomial_value.m, evaluates Hn(i,x).
• hen_power_product.m, power products x^e*Hn(i,x)*Hn(j,x).
• imtqlx.m, diagonalizes a symmetric tridiagonal matrix;
• monomial_to_h_matrix.m, monomial to physicist's Hermite conversion matrix;
• r8_factorial2.m, computes the double factorial function;
• r8_sign.m, returns the sign of an R8.
• r8mat_print.m, prints an R8MAT;
• r8mat_print_some.m, prints some of an R8MAT;
• r8vec_print.m, prints an R8VEC;
• r8vec2_print.m, prints a pair of R8VEC's;