Symbolic vs Numeric Math - Performance

Question

Do symbolic math calculations (especially for solving nonlinear polynomial systems) cause huge performance (calculation speed) disadvantage compared to numeric calculations? Are there any benchmark/data about this?

Found a related question: Symbolic computation vs. numerical computation

Another one: Computational Efficiency of Forward Mode Automatic vs Numeric vs Symbolic Differentiation

Answer 1

I am the individual who answered the Scicomp question you reference in your question. I personally am not aware of any empirical metrics performed to compare run-time performance for symbolic versus numerical solutions to systems of polynomial equations.

However, it should be fairly intuitive that symbolic solutions will have a bit more overhead for most aspects of solving the problem due to things such as manipulation of terms in the equation symbolically, searching how to simplify/rearrange equations to make them easier to solve, searching through known closed form solutions, etc. One major issue with symbolic solvers is that you may not have a closed form solution you can find and use, so solving it numerically would have to happen either way.

The only way I can see symbolic solvers outperforming numerical solutions in terms of run-time is if the symbolic solver can quickly enough recognize your problem as one with a known analytical solution or if it arrives at the solution eventually while the numerical solver never does (aka it diverges).

Given you can find a numerical solver that converges, I think the numerical case will generally be much more efficient since there's just much less overhead to make progress in refining your solution. Since you mention solving systems of polynomial equations, I suspect there are also some tailored algorithms for your type of problem that may be superior to typical nonlinear equation solving schemes.