Sparse multivariate polynomials in Julia
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Updated Last
2 Years Ago
Started In
March 2014

Sparse multivariate polynomials

This package provides support for working with sparse multivariate polynomials in Julia.

This package is superseded by MultivariatePolynomials.jl and is no longer maintained.


In the Julia REPL run


The MPoly type

Multivariate polynomials are stored in the type

struct MPoly{T}

Here each item in the dictionary terms corresponds to a term in the polynomial, where the key represents the monomial powers and the value the coefficient of the monomial. Each of the keys in terms should be a vector of integers whose length equals length(vars).

Constructing polynomials

For constructing polynomials you can use the generators of the polynomial ring:

julia> using MultiPoly

julia> x, y = generators(MPoly{Float64}, :x, :y);

julia> p = (x+y)^3
MultiPoly.MPoly{Float64}(x^3 + 3.0x^2*y + 3.0x*y^2 + y^3)

For the zero and constant one polynomials use


where you can optionally supply the variables of the polynomials with vars = [:x, :y].

Alternatively you can construct a polynomial using a dictionary for the terms:

MPoly{Float64}(terms, vars)

For example, to construct the polynomial 1 + x^2 + 2x*y^3 use

julia> using MultiPoly, DataStructures

julia> MPoly{Float64}(OrderedDict([0,0] => 1.0, [2,0] => 1.0, [1,3] => 2.0), [:x, :y])
MultiPoly.MPoly{Float64}(1.0 + x^2 + 2.0x*y^3)

Laurent polynomials may be constructed too:

x^1 * y^2 + x^1 * y^(-2) + x^(-1) * y^2 + x^(-1) * y^(-2)

Polynomial arithmetic

The usual ring arithmetic is supported and MutliPoly will automatically deal with polynomials in different variables or having a different coefficient type. Examples:

julia> using MultiPoly

julia> x, y = generators(MPoly{Float64}, :x, :y);

julia> z = generator(MPoly{Int}, :z)

julia> x+z
MPoly{Float64}(x + z)

julia> vars(x+z)
3-element Array{Symbol,1}:

Evaluating a polynomial

To evaluate a polynomial P(x,y, ...) at a point (x0, y0, ...) the evaluate function is used. Example:

julia> p = (x+x*y)^2
MultiPoly.MPoly{Float64}(x^2 + 2.0x^2*y + x^2*y^2)

julia> evaluate(p, 3.0, 2.0)


MultiPoly supports integration and differentiation. Currently the integrating constant is set to 0. Examples:

julia> p = x^4 + y^4
MultiPoly.MPoly{Float64}(x^4 + y^4)

julia> diff(p, :x)

julia> diff(p, :y, 3)

julia> integrate(p, :x, 2)
MultiPoly.MPoly{Float64}(0.03333333333333333x^6 + 0.5x^2*y^4)

Integrations which would involve integrating a term with a -1 power raise an error. This example can be intergrated once, but not twice, in :x and can't be integrated in :y:

julia> q = x^(-2) * y^(-1);
julia> integrate(q, :y)  
ERROR: ArgumentError: can't integrate 1 times in y as it would involve a -1 power requiring a log term