calculus (version 0.1.0)

hermite: Hermite Polynomials

Description

Computes univariate and multivariate Hermite polynomials.

Usage

hermite(order, sigma = 1, var = "x")

Arguments

order

integer. The order of the Hermite polynomial.

sigma

the covariance matrix of the Gaussian kernel.

var

character. The variables of the polynomial.

Value

list of Hermite polynomials with components

f

the Hermite polynomial.

order

the order of the Hermite polynomial.

terms

data.frame containing the variables, coefficients and degrees of each term in the Hermite polynomial.

Details

Hermite polynomials are obtained by successive differentiation of the Gaussian kernel $$H_{\nu}(x,\Sigma) = exp \Bigl( \frac{1}{2} x^\dagger \Sigma x \Bigl) (- \partial_x )^\nu exp \Bigl( -\frac{1}{2} x^\dagger \Sigma x \Bigl)$$ where \(\Sigma\) is a d-dimensional square matrix and \(\nu=(\nu_1, ..., \nu_d)\) is the vector representing the order of differentiation for each variable.

Examples

# NOT RUN {
# univariate Hermite polynomials up to order 3
hermite(3)

# univariate Hermite polynomials with variable z
hermite(3, var = 'z')

# multivariate Hermite polynomials up to order 2
hermite(order = 2, 
        sigma = matrix(c(1,0,0,1), nrow = 2), 
        var = c('z1', 'z2'))

# }