Comments (9)
ChrisRackauckas
commented on June 11, 2024
1
Eigenvalues would need a polynomial equation solver. The generic fallbacks likely won't work
from symbolicutils.jl.
dlfivefifty
commented on June 11, 2024
- Eigenvalues
julia> @syms x::Real
(x,)
julia> [0 x; x 0]
2×2 Array{Any,2}:
0 x
x 0
julia> [0 x; x 0] |> eigvals
ERROR: MethodError: no method matching zero(::Type{Any})
Closest candidates are:
zero(::Type{Union{Missing, T}}) where T at missing.jl:105
zero(::Type{Missing}) at missing.jl:103
zero(::Type{LibGit2.GitHash}) at /Users/solver/Projects/julia-1.4/usr/share/julia/stdlib/v1.4/LibGit2/src/oid.jl:220
...
Stacktrace:
[1] zero(::Type{Any}) at ./missing.jl:105
[2] eigtype(::Type{T} where T) at /Users/solver/Projects/julia-1.4/usr/share/julia/stdlib/v1.4/LinearAlgebra/src/eigen.jl:302
[3] eigvals(::Array{Any,2}; kws::Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}}) at /Users/solver/Projects/julia-1.4/usr/share/julia/stdlib/v1.4/LinearAlgebra/src/eigen.jl:326
[4] eigvals(::Array{Any,2}) at /Users/solver/Projects/julia-1.4/usr/share/julia/stdlib/v1.4/LinearAlgebra/src/eigen.jl:326
[5] |>(::Array{Any,2}, ::typeof(eigvals)) at ./operators.jl:823
[6] top-level scope at REPL[8]:1
[7] eval(::Module, ::Any) at ./boot.jl:331
[8] eval_user_input(::Any, ::REPL.REPLBackend) at /Users/solver/Projects/julia-1.4/usr/share/julia/stdlib/v1.4/REPL/src/REPL.jl:86
[9] run_backend(::REPL.REPLBackend) at /Users/solver/.julia/packages/Revise/MgvIv/src/Revise.jl:1023
[10] top-level scope at none:0from symbolicutils.jl.
dlfivefifty
commented on June 11, 2024
Mathematica's solution is a Root type to represent polynomial roots:
In[20]:= ( {
{0, x, y, x},
{x, 0, x, x},
{0, x, x, x},
{x, x, x, x}
} ) // Eigenvalues
Out[20]= {Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 - 2 x #1^3 + #1^4 &,
1], Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 2],
Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 3],
Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 4]}from symbolicutils.jl.
owiecc
commented on June 11, 2024
Is there a workaround to solve a system of nonlinear equations in a different engine and get the solution back to Julia? I already have a big set of equations modelled with Symbolics.jl and I just saw systems of nonlinear equations are not implemented yet. I'd prefer not to reimplement everything from scratch.
from symbolicutils.jl.
anandijain
commented on June 11, 2024
Is there a workaround to solve a system of nonlinear equations in a different engine and get the solution back to Julia? I already have a big set of equations modelled with Symbolics.jl and I just saw systems of nonlinear equations are not implemented yet. I'd prefer not to reimplement everything from scratch.
@owiecc
Symbolics.symbolics_to_sympy might be of help, not sure though
from symbolicutils.jl.
YingboMa
commented on June 11, 2024
We could construct the characteristic polynomial and introduce the Root object
julia> using LinearAlgebra, Symbolics
julia> @variables x y λ
3-element Vector{Num}:
x
y
λ
julia> A = [0 x y x; x 0 x x; 0 x x x; x x x x]
4×4 Matrix{Num}:
0 x y x
x 0 x x
0 x x x
x x x x
julia> simplify(det(A - I*λ), expand=true)
x^4 + λ^4 + λ*(x^3) - (2x*(λ^3)) - (y*(x^3)) - (4(x^2)*(λ^2)) - (2y*λ*(x^2))Mathematica's root object is pretty smart. D is also defined on them.
In[10]:= ({{0, x, y, x}, {x, 0, x, x}, {0, x, x, x}, {x, x, x, x}}) //
Eigenvalues;
D[%, x]
Out[11]= {(-4 x^3 +
3 x^2 y - (3 x^2 - 4 x y) Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 1] +
8 x Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 1]^2 +
2 Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 1]^3)/(x^3 - 2 x^2 y -
8 x^2 Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 1] -
6 x Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 1]^2 +
4 Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 1]^3), (-4 x^3 +
3 x^2 y - (3 x^2 - 4 x y) Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 2] +
8 x Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 2]^2 +
2 Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 2]^3)/(x^3 - 2 x^2 y -
8 x^2 Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 2] -
6 x Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 2]^2 +
4 Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 2]^3), (-4 x^3 +
3 x^2 y - (3 x^2 - 4 x y) Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 3] +
8 x Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 3]^2 +
2 Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 3]^3)/(x^3 - 2 x^2 y -
8 x^2 Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 3] -
6 x Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 3]^2 +
4 Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 3]^3), (-4 x^3 +
3 x^2 y - (3 x^2 - 4 x y) Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 4] +
8 x Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 4]^2 +
2 Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 4]^3)/(x^3 - 2 x^2 y -
8 x^2 Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 4] -
6 x Root[
x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 4]^2 +
4 Root[x^4 - x^3 y + (x^3 - 2 x^2 y) #1 - 4 x^2 #1^2 -
2 x #1^3 + #1^4 &, 4]^3)}from symbolicutils.jl.
blegat
commented on June 11, 2024
For systems of polynomials equations, there is an interface in SemialgebraicSets.jl.
Two algorithms currently implement this interface:
- Compute Groebner basis, get the multiplication matrices and simultaneously diagonalize them with Schur factorization of a random linear combination of them. This is implemented in SemialgebraicSets.jl as well as the Groebner basis computation using Buchberger's algorithm. In the future, I would like to have other Groebner basis computation algorithm, e.g., by having https://github.com/ederc/GroebnerBasis.jl implement the interface.
- Homotopy continuation: https://www.juliahomotopycontinuation.org/
SymbolicUtils already has some code to transform symbolic expressions into multivariate polynomials using MultivariatePolynomials.jl so it should be too hard to link to SemialgebraicSets.jl.
from symbolicutils.jl.
ChrisRackauckas
commented on June 11, 2024
GroebnerBasis.jl is a problematic dependency. It doesn't tag for ages:
has issues with updating compats on time, and is GPL. That should not be deep in the dependency tree, so at most support via an addon package or Requires.
from symbolicutils.jl.
blegat
commented on June 11, 2024
Yes, the current approach is for other packages to have SemialgebraicSets/MultivariatePolynomials in their dependency, e.g., HomotopyContinuation has SemialgebraicSets it its dependency to implement its interface and DynamicPolynomials/TypedPolynomials have MultivariatePolynomials in their dependency. However SemialgebraicSets and MultivariatePolynomials have a lightweight dependency tree.
from symbolicutils.jl.
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from symbolicutils.jl.