# HERMITE_POLYNOMIAL Hermite Polynomials

HERMITE_POLYNOMIAL is a FORTRAN90 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(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.

### Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a FORTRAN90 library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a FORTRAN90 library 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 library which evaluates the Gegenbauer polynomial and associated functions.

GEN_HERMITE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Hermite quadrature rule.

HERMITE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

HERMITE_PRODUCT_POLYNOMIAL, a FORTRAN90 library which defines Hermite product polynomials, creating a multivariate polynomial as the product of univariate Hermite polynomials.

HERMITE_RULE, a FORTRAN90 program which can compute and print a Gauss-Hermite quadrature rule.

JACOBI_POLYNOMIAL, a FORTRAN90 library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a FORTRAN90 library which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a FORTRAN90 library which evaluates the Legendre polynomial and associated functions.

LOBATTO_POLYNOMIAL, a FORTRAN90 library which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

PCE_BURGERS, a FORTRAN90 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 FORTRAN90 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 FORTRAN90 library which evaluates a variety of mathematical functions.

TEST_VALUES, a FORTRAN90 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.

### List of Routines:

• H_INTEGRAL evaluates the integral of H(i,x).
• H_POLYNOMIAL_COEFFICIENTS: coefficients of H(i,x).
• H_POLYNOMIAL_VALUE evaluates H(i,x).
• H_POLYNOMIAL_VALUES: tabulated values of H(i,x).
• H_POLYNOMIAL_ZEROS: zeros of H(i,x).
• HE_DOUBLE_PRODUCT_INTEGRAL: integral of He(i,x)*He(j,x)*e^(-x^2/2).
• HE_INTEGRAL evaluates He(i,x).
• HE_POLYNOMIAL_COEFFICIENTS: coefficients of He(i,x).
• HE_POLYNOMIAL_VALUE evaluates He(i,x).
• HE_POLYNOMIAL_VALUES: tabulated values of He(i,x).
• HE_POLYNOMIAL_ZEROS: zeros of He(i,x).
• HE_TRIPLE_PRODUCT_INTEGRAL: integral of He(i,x)*He(j,x)*He(k,x)*e^(-x^2/2).
• HEN_EXPONENTIAL_PRODUCT: exponential product of exp(b*x)*Hen(i,x)*Hen(j,x).
• HEN_POLYNOMIAL_VALUE evaluates Hen(i,x).
• HEN_POWER_PRODUCT: power products, x^e*Hen(i,x)*Hen(j,x).
• HF_EXPONENTIAL_PRODUCT: exponential products, exp(b*x)*Hf(i,x)*Hf(j,x).
• HF_FUNCTION_VALUE evaluates the Hermite function Hf(i,x).
• HF_FUNCTION_VALUES: values of the Hermite function Hf(i,x).
• HF_POWER_PRODUCT: power products x^e*Hf(i,x)*Hf(j,x).
• HN_EXPONENTIAL_PRODUCT: exponential products exp(b*x)*Hn(i,x)*Hn(j,x).
• HN_POLYNOMIAL_VALUE evaluates Hn(i,x).
• HN_POWER_PRODUCT: power products x^e*Hn(i,x)*Hn(j,x).
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• R8_FACTORIAL computes the factorial of N.
• R8_FACTORIAL2 computes the double factorial function.
• R8_SIGN returns the sign of an R8.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8VEC_PRINT prints an R8VEC.
• R8VEC2_PRINT prints an R8VEC2.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.

Last revised on 16 March 2012.