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A.7.1 Solving systems of polynomial equationsHere we turn our attention to the probably most popular aspect of the solving problem: given a system of complex polynomial equations with only finitely many solutions, compute floating point approximations for these solutions. This is widely considered as a task for numerical analysis. However, due to rounding errors, purely numerical methods are often unstable in an unpredictable way. Therefore, in many cases, it is worth investing more computing power to derive additional knowledge on the geometric structure of the set of solutions (not to mention the question of how to decide whether the set of solutions is finite or not). The symbolic-numerical approach to the solving problem combines numerical methods with a symbolic preprocessing. Depending on whether we want to preserve the multiplicities of the solutions or not, possible goals for a symbolic preprocessing are
interred command (see interred). Another goal might be
Moreover, the equational modelling of a problem frequently causes unwanted solutions, for instance, zero as a multiple solution. Not only for stability reasons, one is frequently interested to get rid of those. This can be done by computing the saturation of I with respect to an ideal having the excess components as set of solutions (see sat).
The SINGULAR libraries
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