Abstract
We describe Ccluster, a software for computing natural \(\varepsilon \)-clusters of complex roots in a given box of the complex plane. This algorithm from Becker et al. (2016) is near-optimal when applied to the benchmark problem of isolating all complex roots of an integer polynomial. It is one of the first implementations of a near-optimal algorithm for complex roots. We describe some low level techniques for speeding up the algorithm. Its performance is compared with the well-known MPSolve library and Maple.
Rémi’s work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 676541.
Victor’s work is supported by NSF Grants # CCF-1116736 and # CCF-1563942 and by PSC CUNY Award 698130048.
Chee’s work is supported by NSF Grants # CCF-1423228 and # CCF-1564132.
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Notes
- 1.
Irina Voiculescu informed us that her student Dan-Andrei Gheorghe has independently implemented the same algorithm in a Masters Thesis Project (May 18, 2017) at Oxford University. Sewon Park and Martin Ziegler at KAIST, Korea, have implemented a modified version of Becker et al. (2016) for polynomials having only real roots being the eigenvalues of symmetric square matrices with real coefficients. See the technical report CS-TR-2018-415 at https://cs.kaist.ac.kr/research/techReport.
- 2.
https://julialang.org/. Download our code in https://github.com/rimbach/Ccluster.
- 3.
http://arblib.org/. Download our code in https://github.com/rimbach/Ccluster.jl.
- 4.
We treat two-valued predicates for simplicity; the discussion could be extended to predicates (like \(\widetilde{T}^{G}_{*}\)) which returns a finite set of values.
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Imbach, R., Pan, V.Y., Yap, C. (2018). Implementation of a Near-Optimal Complex Root Clustering Algorithm. In: Davenport, J., Kauers, M., Labahn, G., Urban, J. (eds) Mathematical Software – ICMS 2018. ICMS 2018. Lecture Notes in Computer Science(), vol 10931. Springer, Cham. https://doi.org/10.1007/978-3-319-96418-8_28
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