In our quest for the design, the analysis and the implementation of a subdivision algorithm for finding the complex roots of univariate polynomials given by oracles for their evaluation, we present sub-algorithms allowing substantial acceleration of subdivision for complex roots clustering for such polynomials. We rely on Cauchy sums which approximate power sums of the roots in a fixed complex disc and can be computed in a small number of evaluations --polylogarithmic in the degree. We describe root exclusion, root counting, root radius approximation and a procedure for contracting a disc towards the cluster of root it contains, called $\varepsilon$-compression. To demonstrate the efficiency of our algorithms, we combine them in a prototype root clustering algorithm. For computing clusters of roots of polynomials that can be evaluated fast, our implementation competes advantageously with user's choice for root finding, MPsolve.
翻译:在我们寻求设计、分析和实施一个子分类算法以找到由神器提供的单亚多元体的复杂根部,供其评估时,我们提出次分类法,允许大大加速为这种多面体的复杂根组群的子分类法。我们依靠Cauchy 的金额,这些金额近似根部在固定复杂盘中的功率总和,可以在少量的评估中计算出来 -- -- 程度的多面体。我们描述了根排斥、根计、根半径近似和将一张盘的磁盘承包到其中的根组的程序,称为$\varepsilon-Compression。为了展示我们的算法的效率,我们把它们结合到原型根组群算算法中。对于计算能够快速评估的多面组的根组,我们的实施与用户选择的根根查方法MPsolve竞争优势。