📂Julia

How to Use Polynomials in Julia

Overview

Polynomials.jl is a package that includes the representation and computation of polynomial functions. Since polynomials are mathematically simple, there’s a tendency to think their coding should be straightforward, too. However, once you start implementing the necessary features, it can become quite cumbersome. Of course, it’s not extremely difficult, but it’s generally better to use a package whenever possible.

This is not an exhaustive list of all features of the package, but rather a selection of potentially useful ones, so check the repository for more details¹.

General Polynomial Functions

Define a Polynomial Function by Coefficients Polynomial()

julia> using Polynomials

julia> p = Polynomial([1,0,0,1])
Polynomial(1 + x^3)

julia> q = Polynomial([1,1])
Polynomial(1 + x)

julia> r = Polynomial([1,1], :t)
Polynomial(1 + t)

julia> p(0)
1

julia> p(2)
9

Polynomial{T, X}(coeffs::AbstractVector{T}, [var = :x])

• It returns the polynomial function itself. coeffs is an array of coefficients, with the constant term at the beginning and higher-order terms towards the end.
• var specifies the polynomial’s variable. As shown in the example, entering the symbol :t makes it a polynomial in t.

Define a Polynomial Function by Data fit()

julia> fit([-1,0,1], [2,0,2])
Polynomial(2.0*x^2)

fit(::Type{RationalFunction}, xs::AbstractVector{S}, ys::AbstractVector{T}, m, n; var=:x)

• It returns the polynomial function that passes through points with coordinates $x$ at xs and $y$ at ys.

Define a Polynomial Function by Roots roots()

julia> fromroots([-2,2])
Polynomial(-4 + x^2)

fromroots(::AbstractVector{<:Number}; var=:x)

• It returns the polynomial function with the provided array’s elements as roots.

Operations +, -, *, ÷

julia> p + 1
Polynomial(2 + x^3)

julia> 2p
Polynomial(2 + 2*x^3)

Adding or multiplying by a scalar results in the intuitive operation as above.

julia> p + q
Polynomial(2 + x + x^3)

julia> p - q
Polynomial(-x + x^3)

julia> p * q
Polynomial(1 + x + x^3 + x^4)

julia> p ÷ q
Polynomial(1.0 - 1.0*x + 1.0*x^2)

Arithmetic operations between polynomial functions are overridden with +, -, *, ÷.

Finding Roots roots()

julia> roots(p)
3-element Vector{ComplexF64}:
               -1.0 + 0.0im
 0.4999999999999998 - 0.8660254037844383im
 0.4999999999999998 + 0.8660254037844383im

roots(f::AbstractPolynomial)

• It returns the roots of f as a vector.

Differentiation derivative()

julia> derivative(p, 3)
Polynomial(6)

derivative(f::AbstractPolynomial, order::Int = 1)

• It returns the orderth derivative of f.

Integration integrate()

julia> integrate(p, 7)
Polynomial(7.0 + 1.0*x + 0.25*x^4)

integrate(f::AbstractPolynomial, C = 0)

• It returns the indefinite integral of f with an integration constant of C.

Special Polynomial Functions

Laurent Polynomial Functions LaurentPolynomial()

julia> LaurentPolynomial([4,3,2,1], -1)
LaurentPolynomial(4*x⁻¹ + 3 + 2*x + x²)

LaurentPolynomial{T,X}(coeffs::AbstractVector, [m::Integer = 0], [var = :x])

• It returns the Laurent polynomial function extended to integers for degree.
• The degree of the smallest term is given as m.

Chebyshev Polynomial Functions ChebyshevT()

julia> ChebyshevT([3,2,1])
ChebyshevT(3⋅T_0(x) + 2⋅T_1(x) + 1⋅T_2(x))

ChebyshevT{T, X}(coeffs::AbstractVector)

• It returns the Type I Chebyshev polynomial function. The formula’s T_n(x) is represented by $T_{n}(x) = \cos \left( n \cos^{-1} x \right)$.

Environment

• OS: Windows
• julia: v1.6.2