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Logcf: An Efficient Tool for Real Root Isolation

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Abstract

Computing upper bounds of the positive real roots of some polynomials is a key step of those real root isolation algorithms based on continued fraction expansion and Vincent's theorem. The authors give a new algorithm for computing an upper bound of positive roots in this paper. The complexity of the algorithm is O(n log(u+1)) additions and multiplications where u is the optimal upper bound satisfying Theorem 3.1 of this paper and n is the degree of the polynomial. The method together with some tricks have been implemented as a software package logcf using C language. Experiments on many benchmarks show that logcf is competitive with RootIntervals of Mathematica and the function realroot of Maple averagely and it is much faster than existing open source real root solvers in many test cases.

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Authors and Affiliations

  1. RISE & School of Computer and Information Science, Southwest University, Chongqing, 400700, China

    Liyun Dai

  2. LMAM & School of Mathematical Science, Peking University, Beijing, 100871, China

    Zhe Fan, Bican Xia & Hanwen Zhang

Authors
  1. Liyun Dai
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  2. Zhe Fan
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  3. Bican Xia
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  4. Hanwen Zhang
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Corresponding author

Correspondence to Liyun Dai.

Additional information

The research was supported by the National Science Foundation of China under Grant Nos. 61802318, 61732001 and 61532019.

This paper was recommended for publication by Editor LI Hongbo.

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Dai, L., Fan, Z., Xia, B. et al. Logcf: An Efficient Tool for Real Root Isolation. J Syst Sci Complex 32, 1767–1782 (2019). https://doi.org/10.1007/s11424-019-7361-7

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  • DOI: https://doi.org/10.1007/s11424-019-7361-7

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