Introduction

AlgebraicInterpolation.jl is a Julia package that provides methods for interpolating sparse and dense multivariate polynomial and rational functions from samples taken either from an algebraic variety or an affine space.

Quick start

julia> using AlgebraicInterpolation
julia> @var R[1:3,1:3] t[1:3] E[1:3,1:3](Variable[R₁₋₁ R₁₋₂ R₁₋₃; R₂₋₁ R₂₋₂ R₂₋₃; R₃₋₁ R₃₋₂ R₃₋₃], Variable[t₁, t₂, t₃], Variable[E₁₋₁ E₁₋₂ E₁₋₃; E₂₋₁ E₂₋₂ E₂₋₃; E₃₋₁ E₃₋₂ E₃₋₃])
julia> X = AlgebraicVariety([R'*R-I, det(R)-1]; variables=[R, t]);
julia> tₓ = [0 -t[3] t[2]; t[3] 0 -t[1]; -t[2] t[1] 0] # skew-symmetric matrix3×3 Matrix{Expression}:
   0  -t₃   t₂
  t₃    0  -t₁
 -t₂   t₁    0
julia> φ = ExpressionMap(X, expressions=Pair(E, tₓ*R)); # map to essential matrices
julia> image_dimension(φ)6
julia> Γ = MapGraph(φ)MapGraph Γ ⊂ ℂ¹² × ℂ⁹
 domain part:
  AlgebraicVariety X ⊂ ℂ¹²
   12 variables: R₁₋₁, R₂₋₁, R₃₋₁, R₁₋₂, R₂₋₂, R₃₋₂, R₁₋₃, R₂₋₃, R₃₋₃, t₁, t₂, t₃
   10 expressions:
    -1 + R₁₋₁^2 + R₂₋₁^2 + R₃₋₁^2
    R₁₋₁*R₁₋₂ + R₂₋₁*R₂₋₂ + R₃₋₂*R₃₋₁
    R₁₋₁*R₁₋₃ + R₂₋₁*R₂₋₃ + R₃₋₃*R₃₋₁
    R₁₋₁*R₁₋₂ + R₂₋₁*R₂₋₂ + R₃₋₂*R₃₋₁
    -1 + R₁₋₂^2 + R₂₋₂^2 + R₃₋₂^2
    R₁₋₂*R₁₋₃ + R₂₋₂*R₂₋₃ + R₃₋₃*R₃₋₂
    R₁₋₁*R₁₋₃ + R₂₋₁*R₂₋₃ + R₃₋₃*R₃₋₁
    R₁₋₂*R₁₋₃ + R₂₋₂*R₂₋₃ + R₃₋₃*R₃₋₂
    -1 + R₁₋₃^2 + R₂₋₃^2 + R₃₋₃^2
    -1 + (R₁₋₂*R₂₋₃ - R₁₋₃*R₂₋₂)*R₃₋₁ - (R₁₋₂*R₃₋₃ - R₁₋₃*R₃₋₂)*R₂₋₁ + (-R₃₋₂*R₂₋₃ + R₃₋₃*R₂₋₂)*R₁₋₁
 image part:
  E₁₋₁ = t₂*R₃₋₁ - t₃*R₂₋₁
  E₂₋₁ = -t₁*R₃₋₁ + t₃*R₁₋₁
  E₃₋₁ = t₁*R₂₋₁ - t₂*R₁₋₁
  E₁₋₂ = t₂*R₃₋₂ - t₃*R₂₋₂
  E₂₋₂ = -t₁*R₃₋₂ + t₃*R₁₋₂
  E₃₋₂ = t₁*R₂₋₂ - t₂*R₁₋₂
  E₁₋₃ = t₂*R₃₋₃ - t₃*R₂₋₃
  E₂₋₃ = -t₁*R₃₋₃ + t₃*R₁₋₃
  E₃₋₃ = t₁*R₂₋₃ - t₂*R₁₋₃

Contents