A Computer Algebra System

Systems of polynomial equations appear throughout mathematics, science, and engineering, but solving them is widely considered to be part of numerical analysis, not computer algebra or algebraic geometry. Due to rounding errors, though, purely numerical methods often are unstable in unpredictable ways. They usually don’t find all solutions and frequently have problems with overdetermined systems. Moreover, they can barely treat underdetermined systems, and they certainly get into trouble near singularities—points where the solution set (which could be a curve or a surface) does not behave regularly. Therefore, in many situations, it is worth investing more computing power to derive additional knowledge about the solution set’s structure or sometimes even to solve the system symbolically. If the system depends on parameters, which could vary for different computations, a symbolic solution is particularly ideal: it provides a simultaneous solution for all parameter values and gives perfect insight into how the solution behaves when those parameters vary. Unfortunately, complexity makes symbolic methods slow and applicable only to relatively small systems. However, the continuous progress of computing facilities and algorithms for symbolic computations makes more systems accessible for symbolic methods. Singular, which is free software under GNU public license, provides such access. This article should encourage you to test Singular’s abilities with symbolic-numerical solving of a system of polynomial equations such as