MultivariatePolynomialBasis Class

public sealed class MultivariatePolynomialBasis

Public NotInheritable Class MultivariatePolynomialBasis

public ref class MultivariatePolynomialBasis sealed

[<SealedAttribute>]
type MultivariatePolynomialBasis = class end

Inheritance

Object  →  MultivariatePolynomialBasis

This class represents a polynomial basis for functions on ℝᵈ, where the basis consists of all monomials x^α = x₁^α₁ · x₂^α₂ · ... · xₐ^αₐ with total degree |α| = α₁ + α₂ + ... + αₐ ≤ MaxDegree.

The monomials are enumerated in a canonical order:

• First, by total degree: 0, 1, 2, ..., MaxDegree
• Within each degree, lexicographically by multi-index

1 (degree 0)
x, y, z (degree 1)
x², xy, xz, y², yz, z² (degree 2)

This basis is used for polynomial augmentation in radial basis function interpolation with conditionally positive definite (CPD) kernels, where a polynomial tail is required to ensure well-posedness and exact reproduction of low-degree polynomials.

Reference