随机多项式与二分法的期望复杂度
Random polynomials and expected complexity of bisection methods for real solving
摘要：
我们的概率分析揭示了以下问题：为什么随机多项式似乎有几个，以及分离的真正的根，平均？为什么实根隔离的精确算法可能比数值算法表现得更好？本文利用Kac和Edelman与Kostlan的结果来估计具有i.i.d.的次数$d$多项式的实根分离性。遵循两个零均值正态分布的系数：对于SO（2）多项式，第$i$个系数的方差为${d \choose i}$，而对于Weyl多项式，其方差为${1/i!}$。通过应用统计物理学的结果，我们得到了\func{sturm}求解器的预期（位）复杂度，$\sOB(r d^2 τ)$，其中$r$是实根的数量，$τ$是最大系数位大小。我们的边界比最坏情况下的记录严格两个数量级。我们还推导出最坏情况下的输出敏感界限。本文第二部分证明了当系数为i.i.d.时，Bernstein基上$d$次多项式的期望实根数为$\sqrt{2d}\pm\OO(1)$。具有中等标准差的变量。我们的论文的结论与实验结果证实了我们的分析。
英文摘要：
Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or even better than numerical ones? We exploit results by Kac, and by Edelman and Kostlan in order to estimate the real root separation of degree $d$ polynomials with i.i.d.\ coefficients that follow two zero-mean normal distributions: for SO(2) polynomials, the $i$-th coefficient has variance ${d \choose i}$, whereas for Weyl polynomials its variance is ${1/i!}$. By applying results from statistical physics, we obtain the expected (bit) complexity of \func{sturm} solver, $\sOB(r d^2 τ)$, where $r$ is the number of real roots and $τ$ the maximum coefficient bitsize. Our bounds are two orders of magnitude tighter than the record worst case ones. We also derive an output-sensitive bound in the worst case. The second part of the paper shows that the expected number of real roots of a degree $d$ polynomial in the Bernstein basis is $\sqrt{2d}\pm\OO(1)$, when the coefficients are i.i.d.\ variables with moderate standard deviation. Our paper concludes with experimental results which corroborate our analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
