符号数学与数值数学-性能 [英] Symbolic vs Numeric Math - Performance

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问题描述

与数值计算相比,符号数学计算(尤其是用于求解非线性多项式系统)是否会导致巨大的性能(计算速度)劣势?是否有与此相关的基准/数据?

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?

找到了一个相关问题: https://scicomp.stackexchange.com/questions/21754/symbolic-computation-vs-numerical-computation

Found a related question: https://scicomp.stackexchange.com/questions/21754/symbolic-computation-vs-numerical-computation

另一种:正向模式自动计算效率vs数值vs符号差异

推荐答案

我是回答您在问题中引用的Scicomp问题的个人。我个人不知道用于比较多项式方程组的符号解和数值解的运行时性能的经验指标。

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.

但是,应该很直观由于诸如符号化方程中的项的操作,寻找如何简化/重新排列方程以使其更易于求解,搜索已知的封闭形式解等之类的问题,符号化解决方案在解决问题的大多数方面将有更多的开销符号求解器的一个主要问题是您可能没有一个可以找到和使用的封闭式解决方案,因此无论从数字上解决它都必​​须以任何一种方式进行。

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.

我可以看到符号求解器在运行时方面优于数字求解器的唯一方法是,如果符号求解器可以足够快地将您的问题识别为已知解析解,或者最终解决方案而数值求解器从不解决(aka它

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).

鉴于您可以找到一个收敛的数值求解器,我认为数值情况通常会更有效,因为精炼方面的开销要少得多您的解决方案。由于您提到了多项式方程组的求解系统,因此我怀疑还有一些针对您的问题类型的量身定制的算法可能会比典型的非线性方程组求解方案优越。

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.

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