# hermite_polynomial

hermite_polynomial, a Fortran90 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 )
```

### Languages:

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

### Related Data and Programs:

bernstein_polynomial, a Fortran90 code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

chebyshev_polynomial, a Fortran90 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 Fortran90 code which evaluates the Gegenbauer polynomial and associated functions.

gen_hermite_rule, a Fortran90 code which computes a generalized Gauss-Hermite quadrature rule.

hermite_exactness, a Fortran90 code which tests the polynomial exactness of Gauss-Hermite quadrature rules.

hermite_product_polynomial, a Fortran90 code which defines Hermite product polynomials, creating a multivariate polynomial as the product of univariate Hermite polynomials.

hermite_rule, a Fortran90 code which computes a Gauss-Hermite quadrature rule.

jacobi_polynomial, a Fortran90 code which evaluates the Jacobi polynomial and associated functions.

laguerre_polynomial, a Fortran90 code which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

legendre_polynomial, a Fortran90 code which evaluates the Legendre polynomial and associated functions.

lobatto_polynomial, a Fortran90 code which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

pce_burgers, a Fortran90 code 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 Fortran90 code which sets up a simple scalar ODE for exponential decay with an uncertain decay rate, using a polynomial chaos expansion (PCE) in terms of Hermite polynomials.

polynomial_conversion, a Fortran90 code which converts representations of a polynomial between monomial, Bernstein, Chebyshev, Hermite, Lagrange, Laguerre and other forms.

polpak, a Fortran90 code which evaluates a variety of mathematical functions.

test_values, a Fortran90 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:

Last revised on 14 July 2020.