Add multivariate rational interpolation

Project.toml

@@ -10,6 +10,7 @@ Markdown = "d6f4376e-aef5-505a-96c1-9c027394607a"
 Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
 Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
+REPL = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 msolve_jll = "6d01cc9a-e8f6-580e-8c54-544227e08205"
 
@@ -26,6 +27,7 @@ Markdown = "1.6"
 Nemo = "0.43, 0.44, 0.45, 0.46, 0.47, 0.48, 0.49, 0.50, 0.51, 0.52, 0.53, 0.54"
 Printf = "1.6"
 Random = "1.6"
+REPL = "1.6"
 StaticArrays = "1"
 Test = "1.6"
 julia = "1.10"

src/AlgebraicSolving.jl

@@ -18,6 +18,10 @@ include("algorithms/param-curve.jl")
 include("algorithms/hilbert.jl")
 #= siggb =#
 include("siggb/siggb.jl")
+#= progress =#
+include("progress/main.jl")
+#= interp =#
+include("interp/main.jl")
 #= examples =#
 include("examples/katsura.jl")
 include("examples/cyclic.jl")

src/interp/cuyt_lee.jl

@@ -0,0 +1,271 @@
+mutable struct Atomic{T}
+    @atomic x::T
+end
+
+struct CuytLeeError <: Exception
+    msg::String
+end
+
+Base.show(io::IO, e::CuytLeeError) = print(io, "CuytLeeError: ", e.msg)
+
+@doc Markdown.doc"""
+    _random_point(n::Int)
+
+Generates a random point in $\mathbb{Z}^n$ with coordinates between 1 and 99.
+
+**Note**: This is an internal function.
+"""
+function _random_point(n::Int)::Vector{Int}
+    map(a -> 1 + abs(a) % 99, rand(Int, n))
+end
+
+@doc Markdown.doc"""
+    _estimate_total_degree(R::QQMPolyRing, bb::Function; samples=5, show_progress=false)
+
+Estimate the total degree of the rational function corresponding to the black box function `bb` by evaluating it at random points and performing univariate Thiele interpolation.
+
+**Note**: This is an internal function.
+"""
+function _estimate_total_degree(
+    R::QQMPolyRing,
+    bb::Function;
+    samples::Int=5,
+    show_progress::Bool=false
+)::Tuple{Int,Int}
+    t = length(gens(R))
+    @assert t > 0
+    R_z, _ = polynomial_ring(QQ, :z)
+    total_degree_counts = Dict{Tuple{Int,Int},Int}()
+    total = 0
+    while total < samples
+        x = _random_point(t)
+        try
+            f = thiele(R_z, k -> bb(k .* x); show_progress=show_progress, offset=1)
+            total_degree = (degree(denominator(f)), degree(numerator(f)))
+            total_degree_counts[total_degree] = get(total_degree_counts, total_degree, 0) + 1
+            total += 1
+            if findmax(total_degree_counts)[1] > samples ÷ 2
+                break
+            end
+        catch e
+            if isa(e, ThieleError)
+                continue
+            else
+                rethrow(e)
+            end
+        end
+    end
+    return findmax(total_degree_counts)[2]
+end
+
+@doc Markdown.doc"""
+    _homogenize(f::QQMPolyRingElem, d::Int)
+
+Homogenize the given polynomial `f` to total degree `d` by multiplying each term by the appropriate power of the first variable.
+
+**Note**: This is an internal function.
+"""
+function _homogenize(
+    f::QQMPolyRingElem,
+    d::Int
+)::QQMPolyRingElem
+    R = parent(f)
+    C = MPolyBuildCtx(R)
+    for i in 1:f.length
+        exp = exponent_vector(f, i)
+        @assert exp[1] == 0
+        total_deg = sum(exp)
+        exp[1] += d - total_deg
+        @assert exp[1] >= 0
+        push_term!(C, coeff(f, i), exp)
+    end
+    return finish(C)
+end
+
+@doc Markdown.doc"""
+    cuyt_lee_shifted(R::QQMPolyRing, bb::Function; <keyword arguments>)
+
+Compute the multivariate rational function corresponding to the black box function `bb` using Cuyt and Lee's interpolation algorithm.
+This function assumes that a random shift has already been applied to the input of the black box function, and does not perform any retries if the interpolation fails.
+
+**Note**: This is an internal function. For a user-facing function that automatically applies random shifts, see `cuyt_lee`.
+"""
+function cuyt_lee_shifted(
+    R::QQMPolyRing,
+    bb::Function;
+    retry::Int=10,
+    nr_thrds::Int=1,
+    show_progress::Bool=false,
+    desc::String="Multivariate rational interpolation"
+)::FracFieldElem{QQMPolyRingElem}
+    # https://arxiv.org/pdf/1608.01902
+    t = length(gens(R))
+    if t == 0
+        return R(bb(Vector{QQFieldElem}())) // one(R)
+    end
+    R_z, _ = polynomial_ring(QQ, :z)
+    d_den, d_num = _estimate_total_degree(R, bb; show_progress=show_progress)
+    d = [0; fill(max(d_den, d_num), t - 1)...]
+    x = Vector{Vector{ZZRingElem}}()
+    coeffs_den = []
+    coeffs_num = []
+    data_lock = ReentrantLock()
+    prog = ProgressBar(total=prod(d .+ 1); desc=desc, enabled=show_progress)
+    update!(prog, 0)
+    function populate(cur::Vector{ZZRingElem}, dim::Int; num_threads::Int=1, offset=1)::Bool
+        if dim > t
+            f = nothing
+            try
+                f = thiele(R_z, k -> bb(k .* cur); retry=retry, show_progress=show_progress, offset=offset)
+            catch e
+                if isa(e, BoundsError) || isa(e, ThieleError)
+                    return false
+                else
+                    rethrow(e)
+                end
+            end
+            if degree(denominator(f)) != d_den || degree(numerator(f)) != d_num
+                return false
+            end
+            c = constant_coefficient(denominator(f))
+            if c == 0
+                return false
+            end
+            lock(data_lock) do
+                push!(x, copy(cur))
+                push!(coeffs_den, collect(coefficients(denominator(f))) ./ c)
+                push!(coeffs_num, collect(coefficients(numerator(f))) ./ c)
+                update!(prog, length(x))
+            end
+            return true
+        end
+        total = Atomic(0)
+        failures = Atomic(0)
+        i = 1
+        while total.x < d[dim] + 1
+            num_threads_chunk = min(num_threads, d[dim] + 1 - total.x)
+            Threads.@threads for j in 0:num_threads_chunk-1
+                if populate([cur; ZZ(i + j)], dim + 1; num_threads=1, offset=max(offset, j + 1))
+                    @atomic total.x += 1
+                    @atomic failures.x = 0
+                else
+                    @atomic failures.x += 1
+                end
+            end
+            i += num_threads_chunk
+            if failures.x >= retry
+                return false
+            end
+        end
+        return true
+    end
+    res = populate([ZZ(1)], 2; num_threads=nr_thrds)
+    if !res
+        finish!(prog)
+        throw(CuytLeeError("Failed to collect enough data points for interpolation. This could happen if the black box function is singular at zero, or if the expected total degree is incorrect."))
+    end
+    perm = sortperm(x)
+    x = x[perm]
+    coeffs_den = coeffs_den[perm]
+    coeffs_num = coeffs_num[perm]
+    # We interpolate the denominator and numerator separately
+    den = zero(R)
+    num = zero(R)
+    for i in 0:d_den
+        y = [coeffs_den[j][i+1] for j in 1:length(x)]
+        den += _homogenize(newton(R, x, y, d), i)
+    end
+    for i in 0:d_num
+        y = [coeffs_num[j][i+1] for j in 1:length(x)]
+        num += _homogenize(newton(R, x, y, d), i)
+    end
+    finish!(prog)
+    return num // den
+end
+
+@doc Markdown.doc"""
+    cuyt_lee_with_shift(R::QQMPolyRing, bb::Function, shift::Vector{Int}; <keyword arguments>)
+
+Compute the multivariate rational function corresponding to the black box function `bb` using Cuyt and Lee's interpolation algorithm,
+with a given shift applied to the input of the black box function.
+
+**Note**: This is an internal function. For a user-facing function that automatically applies random shifts, see `cuyt_lee`.
+"""
+function cuyt_lee_with_shift(
+    R::QQMPolyRing,
+    bb::Function,
+    shift::Vector{Int};
+    retry::Int=10,
+    nr_thrds::Int=1,
+    show_progress::Bool=false,
+    desc::String="Multivariate rational interpolation"
+)::FracFieldElem{QQMPolyRingElem}
+    t = length(gens(R))
+    if t == 0
+        return R(bb(Vector{QQFieldElem}())) // one(R)
+    end
+    f_shifted = cuyt_lee_shifted(R, z -> bb(z .+ shift); retry=retry, nr_thrds=nr_thrds, show_progress=show_progress, desc=desc)
+    x = gens(R) .- shift
+    num = evaluate(numerator(f_shifted), x)
+    den = evaluate(denominator(f_shifted), x)
+    return num // den
+end
+
+@doc Markdown.doc"""
+    cuyt_lee(R::QQMPolyRing, bb::Function; <keyword arguments>)
+
+Compute the multivariate rational function corresponding to the black box function `bb` using Cuyt and Lee's interpolation algorithm.
+
+# Arguments
+- `R::QQMPolyRing`: the multivariate polynomial ring over the rationals.
+- `bb::Function`: a black box function that takes a vector of rational numbers as input and returns a rational number as output.
+- `initial_shift::Vector{Int}=_random_point(length(gens(R)))`: the initial shift to use for the interpolation.
+- `retry::Int=10`: the maximum number of consecutive failures allowed when evaluating the black box function or interpolating the points.
+- `nr_thrds::Int=1`: the number of threads to use when evaluating the black box function.
+- `show_progress::Bool=false`: whether to show a progress bar while collecting points.
+- `desc::String="Multivariate rational interpolation"`: the description to show in the progress bar.
+
+# Example
+```jldoctest
+julia> using AlgebraicSolving
+
+julia> using AlgebraicSolving.Interpolation
+
+julia> R, _ = polynomial_ring(QQ, ["a", "b"]);
+
+julia> f = t -> (t[1] + t[2])^2 / (t[1] - t[2])^2;
+
+julia> cuyt_lee(R, f)
+(a^2 + 2*a*b + b^2)//(a^2 - 2*a*b + b^2)
+```
+"""
+function cuyt_lee(
+    R::QQMPolyRing,
+    bb::Function;
+    initial_shift=_random_point(length(gens(R))),
+    retry::Int=10,
+    nr_thrds::Int=1,
+    show_progress::Bool=false,
+    desc::String="Multivariate rational interpolation"
+)::FracFieldElem{QQMPolyRingElem}
+    t = length(gens(R))
+    if t == 0
+        return R(bb(Vector{QQFieldElem}())) // one(R)
+    end
+    shift = initial_shift
+    for i in 1:retry
+        try
+            return cuyt_lee_with_shift(R, bb, shift; retry=retry, nr_thrds=nr_thrds, show_progress=show_progress, desc=desc)
+        catch e
+            if isa(e, CuytLeeError)
+                if show_progress
+                    @warn "Interpolation failed, retrying with a different shift... Retries left: $(retry - i)"
+                end
+                shift = _random_point(t)
+            else
+                rethrow(e)
+            end
+        end
+    end
+    throw(CuytLeeError("Interpolation failed after maximum number of retries."))
+end

src/interp/main.jl

@@ -0,0 +1,18 @@
+module Interpolation
+
+using Markdown
+using Nemo
+import Nemo.Generic: FracFieldElem
+using ..Progress
+
+export thiele, newton, cuyt_lee
+
+# Interpolation algorithms
+include("thiele.jl")
+include("newton.jl")
+include("cuyt_lee.jl")
+
+# Applications
+include("resultant.jl")
+
+end # module Interpolation