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Hermite Polynomials and Related Functions

Source: R/polynomials.R
hermite_polynomials.Rd

Functions to compute Hermite polynomials using three-term recurrence relations.

Hermite polynomials are orthogonal polynomials that appear in probability theory (as derivatives of the Gaussian function), quantum mechanics (quantum harmonic oscillator), and numerical analysis.

Hermite Polynomials H_n(x):

  • hermite(n, x): Evaluates the Hermite polynomial of degree n at point x

  • Orthogonal with respect to the weight function $$e^{-x^2}$$ on (-Inf, Inf)

  • Appear as eigenfunctions of the quantum harmonic oscillator

Recurrence Relation:

  • hermite_next(n, x, Hn, Hnm1): Computes $$H_{n+1}(x)$$ from H_n and $$H_{n-1}$$

  • Uses stable three-term recurrence for sequential computation

Implementation Notes:

  • Guarantees low absolute error but not low relative error near polynomial roots

  • Values greater than ~120 for n are unlikely to produce sensible results

  • Relative errors may grow arbitrarily large when the function is very close to a root

Usage

hermite(n, x)

hermite_next(n, x, Hn, Hnm1)

Arguments

n

Degree of the polynomial (practical limit ~120)

x

Argument of the polynomial

Hn

Value of the Hermite polynomial H_n(x)

Hnm1

Value of the Hermite polynomial $$H_{n-1}(x)$$

Value

A single numeric value with the computed Hermite polynomial.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Hermite polynomial H_2(0.5)
hermite(2, 0.5)
#> [1] -1
# Next Hermite polynomial H_3(0.5) using H_2(0.5) and H_1(0.5)
hermite_next(2, 0.5, hermite(2, 0.5), hermite(1, 0.5))
#> [1] -5