Introduction

# Introduction

FixedPolynomials.jl is a library for really fast evaluation of multivariate polynomials. Here are the latest benchmark results.

Since `FixedPolynomials` polynomials are optimised for fast evaluation they are not suited for construction of polynomials. It is recommended to construct a polynomial with an implementation of MultivariatePolynomials.jl, e.g. DynamicPolynomials.jl, and to convert it then into a `FixedPolynomials.Polynomial` for further computations.

## Tutorial

Here is an example on how to create a `Polynomial` with `Float64` coefficients:

``````using FixedPolynomials
import DynamicPolynomials: @polyvar

@polyvar x y z

f = Polynomial{Float64}(x^2+y^3*z-2x*y)``````

To evaluate `f` you simply have to pass in a `Vector{Float64}`

``````x = rand(3)
f(x) # alternatively evaluate(f, x)``````

But this is note the fastest way possible. In order to achieve the best performance we need to precompute some things and also preallocate intermediate storage. For this we have `GradientConfig` and `JacobianConfig`. For single polynomial the API is as follows

``````cfg = GradientConfig(f) # this can be reused!
f(x) == evaluate(f, x, cfg)
# We can also compute the gradient of f at x
map(g -> g(x), ∇f) == gradient(f, x, cfg)``````

We also have support for systems of polynomials:

``````cfg = JacobianConfig([f, f]) # this can be reused!
[f(x), f(x)] == evaluate([f, f] x, cfg)
# We can also compute the jacobian of [f, f] at x
jacobian(f, x, cfg)``````

Make sure to also check out `GradientDiffResult` and `JacobianDiffResult`.

Note

`f` has then the variable ordering as implied by `DynamicPolynomials.variables(x^2+y^3*z-2x*y)`, i.e. `f([1.0, 2.0, 3.0])` will evaluate `f` with `x=1`, `y=2` and `z=3`.

## Safety notes

Warning

For the evaluation multivariate variant of Horner's method is used. Due to that for polynomials with terms of degree over 43 we cannot guarantee an error of less than 1 ULP.