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, $\sOB(r d^2 \tau)$, where $r$ is the number of real roots and $\tau$ 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.
Deep Dive into Random polynomials and expected complexity of bisection methods for real solving.
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, $\sOB(r d^2 \tau)$, where $r$ is the number of real roots and $\tau$ 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
One of the most important procedures in computer algebra and algebraic algorithms is root isolation of univariate polynomials. The goal is to compute intervals in the real case, or squares in the complex case, that isolate the roots of the polynomial and to compute one such interval, or square, for every root.
We restrict ourselves to exact algorithms, i.e. algorithms that perform arithmetic with rational numbers of arbitrary size. The best known algorithms are subdivision algorithms, based on Sturm sequences (sturm), or on Descartes’ rule of sign (descartes), or on Descartes’ rule and the Bernstein basis representation (bernstein). Subdivision algorithms mimic binary search and their complexity depends on separation bounds. They are given an initial interval, or compute one containing all real roots. Then, they repeatedly subdivide it until it is certified that zero or one real root is contained in the tested interval.
Thanks to important recent progress [7,8,10,11], the complexity of sturm, descartes and bernstein is, in the worst case, O B (d 4 τ 2 ), where d is the degree of the polynomial and τ the maximum coefficient bitsize. The bound holds even when the polynomial is non-squarefree, and we also compute (all) the multiplicities. This requires a preprocessing of complexity O B (d 2 τ), in order to compute the square-free factorization. The new polynomial has coefficients of size O(d + τ). The complexity of this stage, although significant in practice, is asymptotically dominated. In this paper we consider the behavior of sturm on random polynomials of various forms. Our results can be extended to descartes and bernstein.
Another important exact solver (cf) is based on the continued fractions expansion of the real roots e.g. [1,33,35]. Several variants of this solver exist, depending on the method used to compute the partial quotients of the real roots. Assuming the Gauss-Kuzmin distribution holds for the real algebraic numbers, it was proven [35], that the expected complexity is O B (d 4 τ 2 ). By spreading the roots, the expected complexity becomes O B (d 3 τ) [35]. The currently known worst-case bound is O B (d 4 τ 2 ) [25]. This paper reduces the gap between sturm cf.
Numerical algorithms compute an approximation, up to a desired accuracy, of all complex roots. They can be turned into isolation algorithms by requiring the accuracy to be equal to the theoretical worst-case separation bound. The current record is O B (d 3 τ) and is achieved by recursively splitting the polynomial until one obtains linear factors that approximate sufficiently the roots [32,27]. It seems that the bounds could be improved to O B (d 2 τ) with a more sophisticated splitting process. We should mention that optimal numerical algorithms are very difficult to implement.
Even though the complexity bounds of the exact algorithms are worse than those of the numerical ones, recent implementations of the former tend to be competitive, if not superior, in practice, e.g. [19,30,11,35]. Our work attempts to provide an explanation for this. There is a huge amount of work concerning root isolation and the references stated represent only the tip of the iceberg; we encourage the reader to refer to the references.
Most of the work on random polynomials, which typically concerns polynomials in the monomial basis, focuses on the number of real roots. Kac’s [20] celebrated result estimated the expected number of real roots of random polynomials (named after himself) as 2 π log d + O(1), when the coefficients are standard normals i.i.d. or uniformly distributed, and d is the degree of the polynomial. We refer the reader to e.g. [5,24,12] for a historical perspective and to [3] for various references. A geometric interpretation of this result and many generalizations appear in [9]. We mainly examine SO(2) polynomials, where the i-th coefficient is an i.i.d. Gaussian random variable of zero mean and variance d i . According to [9], they are “the most natural definition of random polynomials”, see also [34]. Their expected number of real roots is √ d. For Weyl polynomials, the i-th coefficient is an i.i.d. Gaussian random variable of zero mean and variance 1/i!, and the expected number of real roots is about
where higher-order terms are not known to date [31]. For results on complex roots we refer to e.g. [14,13].
Our first contribution concerns the expected bit complexity of sturm, when the input is random polynomials with i.i.d. coefficients; notice that their roots are not independently distributed! In other words, we have to go beyond the theory of Kac, and Edelman and Kostlan, in order to study the statistical behavior of root differences and, more precisely, the minimum absolute difference. We examine SO(2) and Weyl random polynomials, and exploit the relevant progress achieved in statistical physics. In fact, these polynomial classes are of particular interest in statistical physics because they model zero-crossings in diffusion
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