Solving a system of multivariate polynomials (or finding extrema)

Specific Domains Numerics
1

I have a multivariate (in practice, 2–5 dimensions) function approximated with Chebyshev polynomials. I would like to find all the local extrema of the approximation. This is equivalent to solving the system obtained from partial derivatives, but there may be a more direct method (also, dealing with extrema at endpoints, these are bounded on [-1,1]^n.

Specifically, I have the approximation available as either weighted sums of a Chebyshev basis, or if preferable vectors of

(i j k ...) => c

where c would be the coefficient assigned to x_1^i x_2^j x_3^k \dots.

Is this a tractable problem? I only know a method for the unviariate case (Boyd 2013).

If yes, what would be the recommended way in Julia?

1 Like
2

maybe https://www.juliahomotopycontinuation.org

1 Like
3

I’m pretty sure Chebfun can do this in 2D.

1 Like
4

Yes, it can. But I am not sure what the algorithm is and if there is a Julia equivalent.

5

I asked Alex Townsend and he said it uses subdivision + Bezout resultants. I’m not sure it’s useful beyond 2D.

2 Likes
6

Assuming the number of solutions is finite, if it is possible to write out the equations in a standard monomial basis, you can maybe try Algebraic Systems Solving · AlgebraicSolving.jl (algebraic-solving.github.io)

2 Likes
7

The best software for solving polynomial systems is Bertini, PCHPack, Magma, and Singular. Both Mathematica and Maple have commands for solving polynomial systems. juliahomotopycontinuation appears to be a Julia implementation of Bertini, and AlgebraicSolving.jl appears to be a Julia version of Singular and what is in Mathematica and Maple. Before I start using the Julia packages, I want to be sure that they work. Many years ago I had an RA computing Groebner bases with SymPy. That was a total waste of time because SymPy was not reliable. It was also an example of getting what I paid for. Are there systematic efforts to test these Julia packages against alternatives?