On roots of random polynomials

We study the distribution of the complex roots of random polynomials of degree with i.i.d. coefficients. Using techniques related to Rice's treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.

     

Stochastic localization of roots of random polynomials

A procedure is proposed for stochastic localization of the roots of polynomials whose coefficients are random variables with a joint distribution density. The results are applied to problems of solvability of polynomial formulas.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 131–134, 1988.     

Expected number of real roots of random trigonometric polynomials

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials

$X_n\left(t\right)=u+\frac{1}{\sqrt{n}}\sum _{k=1}^n(A_{k}cos\left(kt\right)+B_{k}sin\left(kt\right))\text{,}t\in [0,2\pi ]\text{,}u\in \mathbb{R}$

whose coefficients $A_k,B_k$, $k\in \mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a,b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]?[0,2\pi ]$, we prove that

$\underset{n\to \infty }{lim}\frac{\mathbb{E}{N}_n[a,b]}{n}=\frac{b?a}{\pi \sqrt{3}}exp(?\frac{u^2}{2})\text{.}$

     

On the variance of the number of real roots of random algebraic polynomials

In this paper we obtain an estimate of the variance of the number of real roots of a random algebraic polynomial whose coefficients are dependent standard Gaussian random variables.     

Asymptotics of the variance of the number of real roots of random trigonometric polynomials

Let an,n 1 be a sequence of independent standard normal random variables.Consider the random trigonometric polynomial Tn(θ)=∑nj=1 aj cos(jθ),0≤θ≤2π and let Nn be the number of real roots of Tn(θ) in(0,2π).In this paper it is proved that limn →∞ Var(Nn)/n=c0,where 0相似文献   

Choosing roots of polynomials smoothly

We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given.     

On roots of quaternion polynomials

The topological structure of the zero-sets of quaternion polynomials is discussed. As was earlier proved by the author, such a zero-set consists of several points and two-dimensional spheres with centers on the real line. We also show that one can define multiplicities of components of each type in such way that their sum is equal to the algebraic degree of the polynomial considered. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

On random algebraic polynomials

This paper provides asymptotic estimates for the expected number of real zeros and -level crossings of a random algebraic polynomial of the form , where are independent standard normal random variables and is a constant independent of . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form .

     

On random interpolation polynomials

Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 148–150, July, 1992.     

Estimations of positive roots of polynomials

We obtain new estimates for positive roots of univariate polynomials. We discuss their efficiency and study their numerical and computational aspects. Bibliography: 12 titles.     

Zero distribution of random polynomials

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures and to quantitative results on the expected number of zeros in various sets. In the simplest case of Kac polynomials, given by the linear combinations of monomials with i.i.d. random coefficients, it is well known that under mild assumptions on the coefficients, their zeros are asymptotically uniformly distributed near the unit circumference. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by different bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures for associated sets in the plane and quantify this convergence. In our results, random coefficients may be dependent and need not have identical distributions.     

Bounds for positive roots of polynomials

A class of upper bounds for the positive roots of a polynomial is discussed, and it is shown that the bound (15) is nearly optimal in this class.     

Computing multiple roots of inexact polynomials

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials.

Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and it calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations. To fulfill the input requirement of Algorithm I, we develop a numerical GCD-finder containing a successive singular value updating and an iterative GCD refinement as the main engine of Algorithm II that calculates the multiplicity structure and the initial root approximation. While limitations exist in certain situations, the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic, even if the coefficients are inexact. This is perhaps the first blackbox-type root-finder with such capabilities.

To measure the sensitivity of the multiple roots, a structure-preserving condition number is proposed and error bounds are established. According to our computational experiments and error analysis, a polynomial being ill-conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved, and its multiple roots can be computed with high accuracy.

     

On the roots of cauchy polynomials

We discuss the butterfly-shaped region M_n in the complex plane which is defined as the set of all the roots of all normalized Cauchy polynomials of degree n. Besides the geometric structure, e.g. that the set M_n \sb {1} is star-shaped with respect to the origin, some results concerning the boundary of M_n are presented.