DynamicPolynomials.jl

Multivariate polynomials implementation of commutative and non-commutative variables
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Started In
April 2017

Dynamic Polynomials

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Sparse dynamic representation of multivariate polynomials that can be used with MultivariatePolynomials (see the documentation there for more information). Both commutative and non-commutative variables are supported. The following types are defined:

• `PolyVar{C}`: A variable which is commutative with `*` when `C` is `true`. Commutative variables are created using the `@polyvar` macro, e.g. `@polyvar x y`, `@polyvar x[1:8]` and non-commutative variables are created likewise using the `@ncpolyvar` macro.
• `Monomial{C}`: A product of variables: e.g. `x*y^2`.
• `Term{C, T}`: A product between an element of type `T` and a `Monomial{C}`, e.g `2x`, `3.0x*y^2`.
• `Polynomial{C, T}`: A sum of `Term{C, T}`, e.g. `2x + 3.0x*y^2 + y`.

All common algebraic operations between those types are designed to be as efficient as possible without doing any assumption on `T`. Typically, one imagine `T` to be a subtype of `Number` but it can be anything. This is useful for example in the package PolyJuMP where `T` is often an affine expression of JuMP decision variables. The commutativity of `T` with `*` is not assumed, even if it is the coefficient of a monomial of commutative variables. However, commutativity of `T` and of the variables `+` is always assumed. This allows to keep the terms sorted (Graded Lexicographic order is used) in polynomial and measure which enables more efficient operations.

Below is a simple usage example

```julia> using DynamicPolynomials

julia> @polyvar x y # assigns x (resp. y) to a variable of name x (resp. y)
(x, y)

julia> p = 2x + 3.0x*y^2 + y # define a polynomial in variables x and y
3.0xy² + 2.0x + y

julia> differentiate(p, x) # compute the derivative of p with respect to x
3.0y² + 2.0

julia> differentiate.(p, (x, y)) # compute the gradient of p
(3.0y² + 2.0, 6.0xy + 1.0)

julia> p((x, y)=>(y, x)) # replace any x by y and y by x
3.0x²y + x + 2.0y

julia> subs(p, y=>x^2) # replace any occurence of y by x^2
3.0x⁵ + x² + 2.0x

julia> p(x=>1, y=>2) # evaluate p at [1, 2]
16.0```

Below is an example with `@polyvar x[1:n]`

```julia> n = 3;

julia> @polyvar x[1:n] # assign x to a tuple of variables x1, x2, x3
(PolyVar{true}[x₁, x₂, x₃],)

julia> p = sum(x .* x) # compute the sum of squares
x₁² + x₂² + x₃²

julia> subs(p, x[1]=>2, x[3]=>3) # make a partial substitution
x₂² + 13

julia> A = reshape(1:9, 3, 3);

julia> p(x => A * vec(x))  # corresponds to dot(A*x, A*x), need vec to convert the tuple to a vector
14x₁² + 64x₁x₂ + 100x₁x₃ + 77x₂² + 244x₂x₃ + 194x₃²```

Note that, when doing substitution, it is required to give the `PolyVar` ordering that is meant. Indeed, the ordering between the `PolyVar` is not alphabetical but rather by order of creation which can be undeterministic with parallel computing. Therefore, this order cannot be used for substitution, even as a default (see here for a discussion about this).

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