Enclosing all zeros of an analytic function — A rigorous approach

We present a method to find all zeros of an analytic function in a rectangular domain. The approach is based on finding guaranteed enclosures rather than approximations of the zeros. Well-isolated simple zeros are determined fast and with high accuracy. Clusters of zeros can in many cases be distinguished from multiple zeros by applying the argument principle to sufficiently high-order derivatives of the function. We illustrate the proposed method through five examples of varying levels of complexity.     

Asymptotic Distributions of Zeros of Quadratic Hermite–Pade Polynomials Associated with the Exponential Function

The asymptotic distributions of zeros of the quadratic Hermite--Pad\'{e} polynomials $p_{n},q_{n},r_{n}\in{\cal P}_{n}$ associated with the exponential function are studied for $n\rightarrow\infty$. The polynomials are defined by the relation $$(*)\qquad p_{n}(z)+q_{n}(z)e^{z}+r_{n}(z)e^{2z}=O(z^{3n+2})\qquad\mbox{as} \quad z\rightarrow0,$$ and they form the basis for quadratic Hermite--Pad\'{e} approximants to $e^{z}$. In order to achieve a differentiated picture of the asymptotic behavior of the zeros, the independent variable $z$ is rescaled in such a way that all zeros of the polynomials $p_{n},q_{n},r_{n}$ have finite cluster points as $n\rightarrow\infty$. The asymptotic relations, which are proved, have a precision that is high enough to distinguish the positions of individual zeros. In addition to the zeros of the polynomials $p_{n},q_{n},r_{n}$, also the zeros of the remainder term of (*) are studied. The investigations complement asymptotic results obtained in [17].     

On the zeros of the spectrogram of white noise

In a recent paper, Flandrin [16] proposed filtering based on the zeros of a spectrogram with Gaussian window. His results are based on empirical observations on the distribution of the zeros of the spectrogram of white Gaussian noise. These zeros tend to be uniformly spread over the time–frequency plane, and not to clutter. Our contributions are threefold: we rigorously define the zeros of the spectrogram of continuous white Gaussian noise, we explicitly characterize their statistical distribution, and we investigate the computational and statistical underpinnings of the practical implementation of signal detection based on the statistics of the zeros of the spectrogram. The crux of our analysis is that the zeros of the spectrogram of white Gaussian noise correspond to the zeros of a Gaussian analytic function, a topic of recent independent mathematical interest [24].     

Zeros of univariate interval polynomials

Polynomials with perturbed coefficients, which can be regarded as interval polynomials, are very common in the area of scientific computing due to floating point operations in a computer environment. In this paper, the zeros of interval polynomials are investigated. We show that, for a degree n interval polynomial, the number of interval zeros is at most n and the number of complex block zeros is exactly n if multiplicities are counted. The boundaries of complex block zeros on a complex plane are analyzed. Numeric algorithms to bound interval zeros and complex block zeros are presented.     

Enclosing clusters of zeros of polynomials

Lagrange interpolation and partial fraction expansion can be used to derive a Gerschgorin-type theorem that gives simple and powerful a posteriori error bounds for the zeros of a polynomial if approximations to all zeros are available. Compared to bounds from a corresponding eigenvalue problem, a factor of at least two is gained.The accuracy of the bounds is analyzed, and special attention is given to ensure that the bounds work well not only for single zeros but also for multiple zeros and clusters of close zeros.A Rouché-type theorem is also given, that in many cases reduces the bound even further.     

Convexity of the zeros of some orthogonal polynomials and related functions

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.     

New higher-order methods for the simultaneous inclusion of polynomial zeros

Higher-order methods for the simultaneous inclusion of complex zeros of algebraic polynomials are presented in parallel (total-step) and serial (single-step) versions. If the multiplicities of each zeros are given in advance, the proposed methods can be extended for multiple zeros using appropriate corrections. These methods are constructed on the basis of the zero-relation of Gargantini’s type, the inclusion isotonicity property and suitable corrections that appear in two-point methods of the fourth order for solving nonlinear equations. It is proved that the order of convergence of the proposed methods is at least six. The computational efficiency of the new methods is very high since the acceleration of convergence order from 3 (basic methods) to 6 (new methods) is attained using only n polynomial evaluations per iteration. Computational efficiency of the considered methods is studied in detail and two numerical examples are given to demonstrate the convergence behavior of the proposed methods.     

The Haar condition and multiplicity of zeros

Summary In this paper the problem is investigated of how to take the (possibly noninteger) multiplicity of zeros into account in the Haar condition for a linear function space on a given interval. Therefore, a distinction is made between regular and singular points of the interval, and a notion of geometric multiplicity, which always is a positive integer, is introduced. It is pointed out that, for regular zeros (i.e., zeros situate at regular points), aq-fold zero (in the sense that its geometric multiplicity equalsq), counts forq distinct zeros in the Haar condition. For singular zeros (i.e., zeros situated at singular points), this geometric multiplicity has to be diminished by some well-determinable integer.     

A class of Iteration functions for improving,simultaneously, approximations to the zeros of a polynomial

Approximations to all the zeros of a polynomial may be used to find a better approximation to one of the zeros. Using this fact, we present a simple approach to the derivation of a class of Iteration Functions for simultaneously improving approximations to zeros of a polynomial. Convergence properties are studied and computational results are included.     

The Use of Interval Arithmetic to Bound the Zeros of Real Polynomials

This paper discusses the problem of bounding the zeros of areal polynomial. Champagne's results in this field are given,and then Moore's extension of the Newton-Raphson Process forinterval arithmetic. Methods are then presented to deal withthe cases of complex zeros and multiple zeros, for which theNewton-Raphson-Moore Method is not applicable.     

Quadrature formula and zeros of para-orthogonal polynomials on the unit circle

Given a probability measure μ on the unit circle T, we study para-orthogonal polynomials B_n(^.,w) (with fixed w ∈ T) and their zeros which are known to lie on the unit circle. We focus on the properties of zeros akin to the well known properties of zeros of orthogonal polynomials on the real line, such as alternation, separation and asymptotic distribution. We also estimate the distance between the consecutive zeros and examine the property of the support of μ to attract zeros of para-orthogonal polynomials. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the zeros of basic finite Hankel transforms

In this article we prove that the basic finite Hankel transform whose kernel is the third-type Jackson q-Bessel function has only infinitely many real and simple zeros, provided that q satisfies a condition additional to the standard one. We also study the asymptotic behavior of the zeros. The obtained results are applied to investigate the zeros of q-Bessel functions as well as the zeros of q-trigonometric functions. A basic analog of a theorem of G. Pólya (1918) on the zeros of sine and cosine transformations is also given.     

Asymptotics of the zeros of degenerate hypergeometric functions

We find the asymptotics of the zeros of the degenerate hypergeometric function (the Kummer function) Φ(a, c; z) and indicate a method for numbering all of its zeros consistent with the asymptotics. This is done for the whole class of parameters a and c such that the set of zeros is infinite. As a corollary, we obtain the class of sine-type functions with unfamiliar asymptotics of their zeros. Also we prove a number of nonasymptotic properties of the zeros of the function Φ.     

On the existence of zeros of monotone operators in reflexive Banach spaces

This paper is devoted to the investigation on the existence of zeros of monotone operators in reflexive Banach spaces. We first present a sufficient condition under which single-valued monotone operators have zeros. The obtained theorem includes a previous result as a special case. A necessary and sufficient condition for the existence of zeros of maximal monotone operators is presented.     

On zeros of polynomials orthogonal with respect to a quasi-definite inner product on the unit circle

In this paper we present some results concerning the zeros of sequences of polynomials orthogonal with respect to a quasi-definite inner product on the unit circle. We study zero general properties, the existence of sequences with prefixed zeros and some situations concerning the polynomials with multiple zeros.     

Limit Curves for Zeros of Sections of Exponential Integrals

We are interested in studying the asymptotic behavior of the zeros of partial sums of power series for a family of entire functions defined by exponential integrals. The zeros grow on the order of \(O(n)\) , and after rescaling, we explicitly calculate their limit curve. We find that the rate at which the zeros approach the curve depends on the order of the singularities/zeros of the integrand in the exponential integrals. As an application of our findings, we derive results concerning the zeros of partial sums of power series for Bessel functions of the first kind.     

To solving multiparameter problems of algebra. 10. Computing zeros of a scalar polynomial

The paper considers the problem of computing zeros of scalar polynomials in several variables. The zeros of a polynomial are subdivided into the regular (eigen-and mixed) zeros and the singular ones. An algorithm for computing regular zeros, based on a decomposition of a given polynomial into a product of primitive polynomials, is suggested. The algorithm is applied to solving systems of nonlinear algebraic equations. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 119–130.     

On Isolation of Real and Nearly Real Zeros of a Univariate Polynomial and Its Splitting into Factors

We show two simple algorithms for isolation of the real and nearly real zeros of a univariate polynomial, as well as of those zeros that lie on or near a fixed circle on the complex plane. We also simplify slightly approximation of complex zeros of a polynomial with real coefficients.     

Applications of the monotonicity of extremal zeros of orthogonal polynomials in interlacing and optimization problems

We investigate monotonicity properties of extremal zeros of orthogonal polynomials depending on a parameter. Using a functional analysis method we prove the monotonicity of extreme zeros of associated Jacobi, associated Gegenbauer and q-Meixner-Pollaczek polynomials. We show how these results can be applied to prove interlacing of zeros of orthogonal polynomials with shifted parameters and to determine optimally localized polynomials on the unit ball.     

An algorithm for decomposing a polynomial system into normal ascending sets

We present an algorithm to decompose a polynomial system into a finite set of normal ascending sets such that the set of the zeros of the polynomial system is the union of the sets of the regular zeros of the normal ascending sets.If the polynomial system is zero dimensional,the set of the zeros of the polynomials is the union of the sets of the zeros of the normal ascending sets.