MultivariatePolynomials implementation using typed variables in Julia
Author JuliaAlgebra
17 Stars
Updated Last
2 Years Ago
Started In
March 2017


Build Status codecov.io

TypedPolynomials.jl provides an implementation of the multivariate polynomial interface from MultivariatePolynomials.jl using strongly typed variables. That is, in this package, the identity of a variable is encoded by its type, so variables x and y are of different types. This allows us to use the type system to handle certain operations, like computing the intersection of two monomials' variables, at compile-time.


  • Handling variables at the type level makes constructing variables, monomials, and terms more efficient than with DynamicPolynomials.jl.
  • Despite the heavy use of the type system, this package has no @generated functions and should be compatible with static compilation (though this has not yet been tested).


  • There is no distinction in this package between a variable's name and its identity. That is, two variables named x are exactly the same object, regardless of how they were created.
  • For problems with large numbers of variables, or for which the set of variables is not known at compile-time, you may see better performance overall with DynamicPolynomials.jl, e.g. #32. This may change in the future.


The easiest way to create variables is the @polyvar macro:

julia> @polyvar x y z  # Declare three `Variable`s named x, y, and z and assign local variables with the same names
(x, y, z)

julia> typeof(x)

julia> typeof(y)

Multiplying variables creates a Monomial{V} where V is the vector of variables contained in the monomial:

julia> x * y

julia> typeof(x * y)
TypedPolynomials.Monomial{(x, y),2}

julia> typeof(x^2)

Multiplying a monomial (or variable) by anything other than another Variable or Monomial creates a Term:

julia> 3 * x

julia> typeof(3 * x)

julia> typeof(3.0 * x^2 * y)
TypedPolynomials.Term{Float64,TypedPolynomials.Monomial{(x, y),2}}

Addition or subtraction of terms, monomials, and/or variables creates a Polynomial:

julia> x + y
x + y

julia> typeof(x + y) <: Polynomial

julia> x + 3y^2 + z/2 + x^3
x^3 + 3.0y^2 + x + 0.5z

More Examples

Differentiation and Substitution

using TypedPolynomials
using Test
@polyvar x y # assigns x (resp. y) to a variable of name x (resp. y)
p = 2x + 3.0x*y^2 + y
@test differentiate(p, x) == 3y^2 + 2 # compute the derivative of p with respect to x
@test differentiate.(p, (x, y)) == (3y^2 + 2, 6*x*y + 1) # compute the gradient of p
@test p((x, y)=>(y, x)) == 2y + 3y*x^2 + x  # replace any x by y and y by x
@test p(y=>x^2) == 2x + 3x*(x^4) + x^2 # replace any occurence of y by x^2
@test p(x=>1, y=>2) == 2 * 1 + 3 * 1 * 2^2 + 2 # evaluate p at [1, 2]

Vectors of Variables

The @polyvar macro can also create a tuple of variables automatically:

using TypedPolynomials
A = rand(3, 3)
@polyvar x[1:3] # assign x to a tuple of variables x1, x2, x3
p = sum(x .* x) # x_1^2 + x_2^2 + x_3^2
p(x[1]=>2, x[3]=>3) # x_2^2 + 13
p(x=>A*vec(x)) # corresponds to dot(A*x, A*x), need vec to convert the tuple to a vector