Warning: this post is the "numerical" analog of
Are there workable algebraic geometry approaches for the pentagon equation?
I've replaced "algebraic geometry" by "numerical" in its content, otherwise it's almost the same.
This question is mainly intended for experts in numerical analysis.
A competence in the pentagon equation is not necessary.
A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring.
A fusion ring is given by a finite set of integer matrices checking some axioms (see here p 22).
A solution of its pentagon equation is the main condition for having a structure of fusion category,
in which it encodes associativity (see here). For example, the irreducible complex representations
of a finite group, equipped with $\oplus$ and $\otimes$, generate a fusion ring and a fusion category.
The problem is that, in practice the pentagon equations we meet are huge, for example, a system of $2000000$ polynomial equations with $50000$ variables (of degree $3$ with integer coefficients), so that proving the existence of a solution (and a fortiori finding one) is very hard.
Note that there are pentagon equations without solution.
Question: Are there workable numerical approaches ?
(for highlighting or proving the existence of a solution, or for finding a numerical one)
In fact, the pentagon equation is more structured than just a system of scalar equations, it's a system of several invertible (unitary) matrix equations of the form $$A_1 A_2 A_3 = A_4 A_5$$
(and $A_i^* A_i = I$), such that $A_i = \tau_i B_i \tau'_i$, with $\tau_i$,$\tau'_i$ fixed permutation matrices and $B_i$ a block diagonal matrix (each block is a matrix variable), so that there are many holes (see here p 29-30).
Note that in general $\nu_1\nu_2\nu_3 \neq \nu_4\nu_5$ (with $\nu_i = \tau_i\tau'_i$), so that $B_i =I$ can't be a solution.
I'm interested in the equation given by this fusion ring. There are $16227$ matrix equations of dimensions from $1$ to $91$, and $2097$ invertible (unitary) matrix variables of dimensions from $1$ to $17$ (see here p 31).