HERMITE_POLYNOMIAL
Hermite Polynomials

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

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

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

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

        Integral ( -oo < X < +oo ) exp ( - X^2 ) * Hn(M,X) Hn(N,X) dX = delta ( N, M )
      

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

        He(n,x) = H(n,x/sqrt(2)) / sqrt ( 2^n )
      

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

        Integral ( -oo < X < +oo ) exp ( - 0.5*X^2 ) * Hen(M,X) Hen(N,X) dX = delta ( N, M )
      

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

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

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

        Integral ( -oo < X < +oo ) Hf(M,X) Hf(N,X) dX = delta ( N, M )
      

Licensing:

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

Languages:

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

Related Data and Programs:

CHEBYSHEV_POLYNOMIAL, a MATLAB library which evaluates the Chebyshev polynomial and associated functions.

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

HERMITE_PRODUCT_DISPLAY, a MATLAB program 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_RULE, a MATLAB program which can compute and print a Gauss-Hermite quadrature rule.

JACOBI_POLYNOMIAL, a MATLAB library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a MATLAB library which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a MATLAB library which evaluates the Legendre polynomial and associated functions.

PCE_BURGERS, a MATLAB program which defines and solves a version of the time-dependent viscous Burgers equation, with uncertain viscosity, using a polynomial chaos expansion in terms of Hermite polynomials, by Gianluca Iaccarino.

PCE_ODE_HERMITE, a MATLAB program 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 library which evaluates a variety of mathematical functions.

TEST_VALUES, a MATLAB library 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 a monomial physicist's Hermite integral for H(n,x).
• h_polynomial.m, evaluates the physicist's Hermite polynomial H(n,x).
• h_polynomial_coefficients.m, evaluates the coefficients of the physicist's Hermite polynomial H(n,x).
• h_polynomial_values.m, a few tabulated values of the physicist's Hermite polynomial H(n,x).
• h_polynomial_zeros.m, returns zeros of the physicist's Hermite polynomial H(n,x).
• h_quadrature_rule.m, returns quadrature rules associated with the physicist's Hermite polynomial H(n,x).
• he_double_product_integral.m, integral of He(i,x)*He(j,x)*e^(-0.5*x^2).
• he_integral.m, evaluates a monomial probabilist's Hermite integral for He(n,x).
• he_polynomial.m, evaluates the probabilist's Hermite polynomial He(n,x).
• he_polynomial_values.m, a few tabulated values of the probabilist's Hermite polynomial He(n,x).
• he_polynomial_zeros.m, returns zeros of the probabilist's Hermite polynomial He(n,x).
• he_quadrature_rule.m, returns quadrature rules associated with the probabilist's Hermite polynomial He(n,x).
• he_triple_product_integral.m, integral of He(i,x)*He(j,x)*He(k,x)*e^(-0.5*x^2).
• hen_exponential_product.m, tabulates integrals of e^(b*x) Hen(i,x) Hen(j,x) e^(-0.5*x^2).
• hen_polynomial.m, evaluates the normalized probabilist's Hermite polynomial Hen(n,x).
• hen_power_product.m, tabulates integrals of x^e Hen(i,x) Hen(j,x) e^(-0.5*x^2).
• hf_exponential_product.m, tabulates integrals of e^(b*x) Hf(i,x) Hf(j,x).
• hf_function.m, evaluates the Hermite function Hf(n,x);
• hf_function_values.m, a few tabulated values of the Hermite function Hf(n,x);
• hf_plot.m, plots one or more Hermite functions Hf(n,x).
• hf_power_product.m, tabulates integrals of x^e Hf(i,x) Hf(j,x).
• hf_quadrature_rule.m, returns quadrature rules associated with the Hermite function Hf(n,x).
• hn_exponential_product.m, tabulates integrals of e^(b*x) Hn(i,x) Hn(j,x) e^(-x^2).
• hn_polynomial.m, evaluates the normalized physicist's Hermite polynomial Hn(n,x).
• hen_power_product.m, tabulates integrals of x^e Hn(i,x) Hn(j,x) e^(-x^2).
• imtqlx.m, diagonalizes a symmetric tridiagonal matrix;
• r8_factorial.m, computes the factorial function;
• 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;
• timestamp.m, prints the current YMDHMS date as a time stamp.

Examples and Tests:

• hermite_polynomial_test.m, calls all the tests;
• hermite_polynomial_test_output.txt, the output file.
• hermite_polynomial_test01.m, tests h_polynomial;
• hermite_polynomial_test02.m, tests he_polynomial;
• hermite_polynomial_test03.m, tests hf_function;
• hermite_polynomial_test04.m, tests h_polynomial_zeros;
• hermite_polynomial_test05.m, tests he_polynomial_zeros;
• hermite_polynomial_test06.m, tests h_quadrature_rule;
• hermite_polynomial_test07.m, tests he_quadrature_rule;
• hermite_polynomial_test08.m, tests hn_exponential_product;
• hermite_polynomial_test09.m, tests hn_power_product;
• hermite_polynomial_test10.m, tests hen_exponential_product;
• hermite_polynomial_test11.m, tests hen_power_product;
• hermite_polynomial_test12.m, tests hf_exponential_product;
• hermite_polynomial_test13.m, tests hf_power_product;
• hermite_polynomial_test14.m, tests h_polynomial_coefficients;
• hermite_polynomial_plot01.m, tests hf_plot;
• hf_plot.png, a plot of Hermite functions 0 through 5 over [-5,+5];

You can go up one level to the MATLAB source codes.