Generation of multivariate polynomial basis with given degree

New to Julia
1

Hi,

I would like to make a multivariate polynomial with given degree.
For example, the bivariate polynomial basis of a vector x = [x1, y2] with degree of 2 would be
x -> [1, x1, x1^2, x2, x2^2, x1*x2].

It seems that there are some nice packages for handling multivariate polynomials such as MultivariatePolynomials.jl but I don’t know how to create a polynomial basis functions easily.

Any thoughts?

Notes

2 
julia> using DynamicPolynomials

julia> x,y = @polyvar x y
(x, y)

julia> monomials([x,y], 2)
3-element MonomialVector{true}:
 x²
 xy
 y²

julia> monomials([x,y], 1:2)
5-element MonomialVector{true}:
 x²
 xy
 y²
 x
 y
3

Well, I don’t wanna define polyvar objects in this case.

I actually find a way as follows:

using Combinatorics, Transducers

"""
n ∈ N: length of array, i.e., x ∈ Rⁿ
d ∈ N: degree
"""
function polynomial_basis(n, d; with_bias=true)
    _n = with_bias ? n+1 : n
    exponents = multiexponents(_n, d)
    return function (x)
        _x = with_bias ? [x..., 1] : x
        exponents |> Map(exponent -> prod(_x.^exponent)) |> collect
    end
end
2 Likes
4

so you don’t want a polynomial basis, but a function which evaluates such basis at x?

MultivariatePolynomials or DynamicPolynomials are more about creating and manipulating polynomial objects

1 Like
5

For anyone in the future, here is a type-stable version that is also 2x faster:

using Combinatorics, Transducers

"""
n ∈ N: length of array, i.e., x ∈ Rⁿ
d ∈ N: degree
"""
function polynomial_basis(n, d; with_bias=true)
    _n = with_bias ? n+1 : n
    exponents = multiexponents(_n, d)
    return function (x)
        _x = with_bias ? vcat(x, one(eltype(x))) : x
        exponents |> Map(exponent -> prod(_x.^exponent)) |> collect
    end
end
1 Like