Abstract
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 approximation of the power sums of the roots in a fixed complex disc by Cauchy sums, each computed in a small number of evaluations of an input polynomial and its derivative, that is, in a polylogarithmic number 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.
Victor’s research has been supported by NSF Grant CCF 1563942 and PSC CUNY Award 63677 00 51.
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Notes
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- 2.
they are not publicly realeased yet.
- 3.
3.2.1 available here: https://numpi.dm.unipi.it/software/mpsolve.
- 4.
In [20, Sect. 2], called “The result”, we read: “The method is involved and many details still need to be worked out. In this report also many proofs will be omitted. A full account of the new results shall be given in a monograph” which has actually never appeared. [3] deduced a posteriori estimates, depending on root separation and Mahler’s measure, that is, on the roots themselves, not known a priori.
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Imbach, R., Pan, V.Y. (2022). Accelerated Subdivision for Clustering Roots of Polynomials Given by Evaluation Oracles. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2022. Lecture Notes in Computer Science, vol 13366. Springer, Cham. https://doi.org/10.1007/978-3-031-14788-3_9
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