HermiteN {MultiStatM}

Hermite Polynomials (Univariate and Multivariate)

Description

Computes either univariate or multivariate Hermite polynomials up to a specified order.

Usage

HermiteN(x, N, Type, sigma2 = 1, Sig2 = diag(length(x)))

Arguments

x

A scalar (for univariate) or a vector (for multivariate) at which to evaluate the Hermite polynomials.

N

The maximum order of the polynomials.

Type

A character string specifying the type of Hermite polynomials to compute. Can be either "Univariate" or "Multivariate".

sigma2

The variance for univariate Hermite polynomials. Default is 1. (Only used if Type is "Univariate").

Sig2

The covariance matrix for multivariate Hermite polynomials. Default is the unit matrix diag(length(x)). (Only used if Type is "Multivariate").

Details

Depending on the value of the 'Type' parameter, this function computes either the univariate or the multivariate Hermite polynomials.

Value

Depending on the type, the function returns:

• Univariate: A vector of univariate Hermite polynomials with degrees from 1 to N evaluated at x.

• Multivariate: A list of multivariate polynomials of order from 1 to N evaluated at vector x.

References

Gy.Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Section 4.1 (for univariate), Section 4.6.2, (4.73), p.223 (for multivariate).

See Also

Other Hermite Polynomials: HermiteCoeff(), HermiteCov12(), HermiteN2X()

Examples

# Univariate example
HermiteN(x = 1, N = 3, Type = "Univariate")

# Multivariate example
HermiteN(x = c(1, 3), N = 3, Type = "Multivariate", Sig2 = diag(2))