Computing the real roots of a polynomial by the exclusion algorithm

We describe a new algorithm for localizing the real roots of a polynomialP(x). This algorithm determines intervals on whichP(x) does not possess any root. The remainder set contains the real roots ofP(x) and can be arbitrarily small.     

On one representation of analytic functions by harmonic functions

Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ ⁿ . We study the representation of this function in the form of a series u(x) = u ₀(x) + |x|² u ₁(x) + |x|⁴ u ₂(x) + …, where u _k (x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula. Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162.     

Estimates of the number of zeros of some functions with algebraic Taylor coefficients

We prove two theorems about the number of zeros of analytic functions from certain classes that include the SiegelE-andG-functions. By using these theorems, we arrive at a new proof of the Gel'fond-Schneider theorem and improve the result that the numerical determinant does not vanish in the proof of the Shidlovskii theorem. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 817–824, June, 1997. Translated by M. A. Shishkova     

On Eneström-Kakeya theorem and related analytic functions

We prove some extensions of the classical results concerning the Eneström-Kakeya theorem and related analytic functions. Besides several consequences, our results considerably improve the bounds by relaxing and weakening the hypothesis in some cases.     

A verified method for bounding clusters of zeros of analytic functions

In this paper, we propose a verified method for bounding clusters of zeros of analytic functions. Our method gives a disk that contains a cluster of m   zeros of an analytic function f(z)$f(z)$. Complex circular arithmetic is used to perform a validated computation of n  -degree Taylor polynomial p(z)$p(z)$ of f(z)$f(z)$. Some well known formulae for bounding zeros of a polynomial are used to compute a disk containing a cluster of zeros of p(z)$p(z)$. A validated computation of an upper bound for Taylor remainder series of f(z)$f(z)$ and a lower bound of p(z)$p(z)$ on a circle are performed. Based on these results, Rouché's theorem is used to verify that the disk contains the cluster of zeros of f(z)$f(z)$. This method is efficient in computation of the initial disk of a method for finding validated polynomial factor of an analytic function. Numerical examples are presented to illustrate the efficiency of the proposed method.     

Orthonormal basis of the octonionic analytic functions

By confirming a conjecture proposed in Li and Peng (2001) [1], we obtain the orthonormal basis for the octonionic analytic functions.     

On the coefficients of a polynomial with restricted zeros

LetP(Z)=αn Zn + αn-1Zn-1 +…+α0 be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P(z). In this paper we prove some sharp inequalities concerning the coeffi-cients of the polynomial P(z) with restricted zeros. We also establish a sufficient condition for the separation of zeros of P(z).     

Numerical computation of polynomial zeros by means of Aberth's method

An algorithm for computing polynomial zeros, based on Aberth's method, is presented. The starting approximations are chosen by means of a suitable application of Rouché's theorem. More precisely, an integerq 1 and a set of annuliA _i,i=1,...,q, in the complex plane, are determined together with the numberk _i of zeros of the polynomial contained in each annulusA _i. As starting approximations we choosek _i complex numbers lying on a suitable circle contained in the annulusA _i, fori=1,...,q. The computation of Newton's correction is performed in such a way that overflow situations are removed. A suitable stop condition, based on a rigorous backward rounding error analysis, guarantees that the computed approximations are the exact zeros of a nearby polynomial. This implies the backward stability of our algorithm. We provide a Fortran 77 implementation of the algorithm which is robust against overflow and allows us to deal with polynomials of any degree, not necessarily monic, whose zeros and coefficients are representable as floating point numbers. In all the tests performed with more than 1000 polynomials having degrees from 10 up to 25,600 and randomly generated coefficients, the Fortran 77 implementation of our algorithm computed approximations to all the zeros within the relative precision allowed by the classical conditioning theorems with 11.1 average iterations. In the worst case the number of iterations needed has been at most 17. Comparisons with available public domain software and with the algorithm PA16AD of Harwell are performed and show the effectiveness of our approach. A multiprecision implementation in MATHEMATICA is presented together with the results of the numerical tests performed.Work performed under the support of the ESPRIT BRA project 6846 POSSO (POlynomial System SOlving).     

A note on the interlacing of zeros and orthogonality

Let be a sequence of monic polynomials with deg(t_n)=n such that, for each n∈N, the zeros of t_n are real and simple and t_n and t_n+1 have no common zeros. We discuss the connection between the orthogonality of the sequence, the positivity of a certain ratio, and the interlacing of the zeros of t_n and t_n+1 for n≥1, n∈N.     

Real meromorphic functions and linear differential polynomials

We determine all real meromorphic functions f in the plane such that f has finitely many zeros, the poles of f have bounded multiplicities, and f and F have finitely many non-real zeros, where F is a linear differential polynomial given by F = f (k) +Σk-1j=0ajf(j) , in which k≥2 and the coefficients aj are real numbers with a0≠0.     

带有临界点的解析函数的圆模式逼近

一个圆模式是指复平面C上具有特定交角的一种圆格局.给定有界单连通区域ΩC内一个具有有限多个临界点的解析函数F,首先利用有分枝圆模式枝术构造了F的离散近似解,然后证明了这个近似解序列在Ω的紧子集上一致收敛于该解析函数F.这为带有临界点的解析函数的数值计算提供一种新的方法.     

Starlike approximations of analytic functions

When an analytic function is not univalent, it is often of interest to approximate it by univalent functions. In this paper we introduce a measure of the non-univalency of a function and we derive a method for constructing the best starlike univalent approximations of analytic functions with respect to it, suitable for both practical problems and numerical implementation.     

Fourier coefficients of Zygmund functions and analytic functions with quasiconformal deformation extensions

An open problem is to characterize the Fourier coefficients of Zygmund functions.This problem was also explicitly suggested by Nag and later by Teo and Takhtajan-Teo in the course of study of the universal Teichmu¨ller space.By a complex analysis approach,we give a characterization for the Fourier coefficients of a Zygmund function by a quadratic form.Some related topics are also discussed,including those analytic functions with quasiconformal deformation extensions.     

Normality of meromorphic functions with multiple zeros and shared values

In this paper, we study the normality of a family of meromorphic functions and general criteria for normality of families of meromorphic functions with multiple zeros concerning shared values are obtained.     

The zeros of Euler's Psi function and its derivatives

We investigate the locations of the points of inflexion of Euler's Psi function, and the positions of the stationary points of its derivative. We also establish some trigonometric approximations to Psi which lead to improved estimates for the positions of its zeros. Finally we consider the behaviour of the horizontal separation between the branches.     

Lower bounds for the zeros of analytic functions

Summary In the present work the problem of finding lower bounds for the zeros of an analytic function is reduced by a Hilbert space technique to the well-known problem of finding upper bounds for the zeros of a polynomial. Several lower bounds for all the zeros of analytic functions are thus found, which are always better than the well-known Carmichael-Mason inequality. Several numerical examples are also given and a comparison of our bounds with well-known bounds in literature and/or the exact solution is made.     

关于f~((k))-af~n的零点

设 f (z) 为平面内超越亚纯函数 ,a≠ 0 为常数 ,证明了当 n≥ k+3 时 ,f( k) -afn有无穷多个零点 .     

A note on some improvements of the simultaneous methods for determination of polynomial zeros

Applying Gauss-Seidel approach to the improvements of two simultaneous methods for finding polynomial zeros, presented in [9], two iterative methods with faster convergence are obtained. The lower bounds of the R-order of convergence for the accelerated methods are given. The improved methods and their accelerated modifications are discussed in view of the convergence order and the number of numerical operations. The considered methods are illustrated numerically in the example of an algebraic equation.