hermite_polynomial

hermite_polynomial, a MATLAB 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 MATLAB version.

Related Data and Programs:

bernstein_polynomial, a MATLAB code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

chebyshev_polynomial, a MATLAB code which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

gegenbauer_polynomial, a MATLAB code which evaluates the Gegenbauer polynomial and associated functions.

gen_hermite_rule, a MATLAB code which can compute and print a generalized Gauss-Hermite quadrature rule.

hermite_polynomial_test

hermite_product_display, a MATLAB code which displays an image of a function created by the Cartesian product of two Hermite polynomials, such as f(x,y) = h(3,x) * h(1,y).

hermite_product_polynomial, a MATLAB code which defines Hermite product polynomials, creating a multivariate polynomial as the product of univariate Hermite polynomials.

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

jacobi_polynomial, a MATLAB code which evaluates the Jacobi polynomial and associated functions.

laguerre_polynomial, a MATLAB code which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

legendre_polynomial, a MATLAB code which evaluates the Legendre polynomial and associated functions.

lobatto_polynomial, a MATLAB code which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

pce_ode_hermite, a MATLAB 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, a MATLAB code which evaluates a variety of mathematical functions.

test_values, a MATLAB 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).
• 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;
• 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;