hermite() R function from [calculus]

Hermite Polynomials

Computes univariate and multivariate Hermite polynomials.


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

Arguments

order: the order of the Hermite polynomial.
sigma: the covariance matrix of the Gaussian kernel.
var: character vector giving the variables of the polynomial.
transform: character vector representing a change of variables. See details.

Returns

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 differentiation of the Gaussian kernel:

H_{\nu}(x,\Sigma) = exp \Bigl( \frac{1}{2} x_i \Sigma_{ij} x_j \Bigl) (- \partial_x )^\nu exp \Bigl( -\frac{1}{2} x_i \Sigma_{ij} x_j \Bigl)

where $\Sigma$ is a $d$ -dimensional square matrix and $\nu=(\nu_1 \dots \nu_d)$ is the vector representing the order of differentiation for each variable $x = (x_1\dots x_d)$ . In the case where $\Sigma=1$ and $x=x_1$ the formula reduces to the standard univariate Hermite polynomials:

H_{\nu}(x) = e^{\frac{x^2}{2}}(-1)^\nu \frac{d^\nu}{dx^\nu}e^{-\frac{x^2}{2}}

If transform is not NULL, the variables var $x$ are replaced with transform $f(x)$ to compute the polynomials $H_{\nu}(f(x),\Sigma)$

Examples


### univariate Hermite polynomials up to order 3
hermite(3)

### multivariate Hermite polynomials up to order 2
hermite(order = 2, 
        sigma = matrix(c(1,0,0,1), nrow = 2), 
        var = c('z1', 'z2'))
        
### multivariate Hermite polynomials with transformation of variables
hermite(order = 2, 
        sigma = matrix(c(1,0,0,1), nrow = 2), 
        var = c('z1', 'z2'),
        transform = c('z1+z2','z1-z2'))

References

Guidotti E (2022). "calculus: High-Dimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 1-37. tools:::Rd_expr_doi("10.18637/jss.v104.i05")

hermite function