Elementary approximations for zeros of Bessel functions

Let j_vk, y_vk and c_vk denote the kth positive zeros of the Bessel functions J_v(x), Y_v(x) and of the general cylinder function C_v(x) = cos αJ_v(x)?sin αY_v(x), 0 ? α < π, respectively. In this paper we extend to c_vk, k = 2, 3,..., some linear inequalities presently known only for j_vk. In the case of the zeros y_vk we are able to extend these inequalities also to k = 1. Finally in the case of the first positive zero j_v1 we compare the linear enequalities given in [9] with some other known inequalities.     

A new semilocal convergence theorem for the Weierstrass method for finding zeros of a polynomial simultaneously

In this paper we study the convergence of the famous Weierstrass method for simultaneous approximation of polynomial zeros over a complete normed field. We present a new semilocal convergence theorem for the Weierstrass method under a new type of initial conditions. Our result is obtained by combining ideas of Weierstrass (1891) and Proinov (2010). A priori and a posteriori error estimates are also provided under the new initial conditions.     

A unified semilocal convergence analysis of a family of iterative algorithms for computing all zeros of a polynomial simultaneously

In this paper, we first present a family of iterative algorithms for simultaneous determination of all zeros of a polynomial. This family contains two well-known algorithms: Dochev-Byrnev’s method and Ehrlich’s method. Second, using Proinov’s approach to studying convergence of iterative methods for polynomial zeros, we provide a semilocal convergence theorem that unifies the results of Proinov (Appl. Math. Comput. 284: 102–114, 2016) for Dochev-Byrnev’s and Ehrlich’s methods.     

A posteriori error bounds for the zeros of a polynomial

Summary An algorithm for the computation of error bounds for the zeros of a polynomial is described. This algorithm is derived by applying Rouché's theorem to a Newton-like interpolation formula for the polynomial, and so it is suitable in the case where the approximations to the zeros of the polynomial are computed successively using deflation. Confluent and clustered approximations are handled easily. However bounds for the local rouding errors in deflation, e.g. in Horner's scheme, must be known. In practical application the method can, especially in some ill-conditioned cases, compete with other known estimates.     

Iteration of a class of hyperbolic meromorphic functions

We look at the class which contains those transcendental meromorphic functions for which the finite singularities of are in a bounded set and prove that, if belongs to , then there are no components of the set of normality in which as . We then consider the class which contains those functions in for which the forward orbits of the singularities of stay away from the Julia set and show (a) that there is a bounded set containing the finite singularities of all the functions and (b) that, for points in the Julia set of , the derivatives have exponential-type growth. This justifies the assertion that is a class of hyperbolic functions.

     

A single oval of Cassini for the zeros of a polynomial

We derive two ovals of Cassini, each containing all the zeros of a polynomial. The computational cost to obtain these ovals is similar to that of the Brauer set for the companion matrix of a polynomial, although they are frequently smaller. Their derivation is based on the Gershgorin set for an appropriate polynomial of the companion matrix.     

On the zeros of a polynomial

For a polynomial of degree n, we have obtained an upper bound involving coefficients of the polynomial, for moduli of its p zeros of smallest moduli, and then a refinement of the well-known Eneström-Kakeya theorem (under certain conditions).     

A globally convergent algorithm for determining approximate real zeros of a class of functions

BIT Numerical Mathematics - It is shown that Newton's method can be used to define a globally convergent algorithm for approximating real zeros of a certain class of functions. Included in this...     

A parallel wilf algorithm for complex zeros of a polynomial

A global recursive bisection algorithm is described for computing the complex zeros of a polynomial. It has complexityO(n ³ p) wheren is the degree of the polynomial andp the bit precision requirement. Ifn processors are available, it can be realized in parallel with complexityO(n ² p); also it can be implemented using exact arithmetic. A combined Wilf-Hansen algorithm is suggested for reduction in complexity.     

On some iteration functions for the simultaneous computation of multiple complex polynomial zeros

Second order methods for simultaneous approximation of multiple complex zeros of a polynomial are presented. Convergence analysis of new iteration formulas and an efficient criterion for the choice of the appropriate value of a root are discussed. A numerical example is given which demonstrates the effectiveness of the presented methods.     

On Cauchy's bound for zeros of a polynomial

In this paper we have improved Cauchy's bound for zeros of a polynomialp(z)=z ⁿ+a _n+1 z ^n-1+a _n-2 z ^n-2+......+a ₁ z+a ₀. Our result is best possible and sharpens some other results also.     

A method for computing all the zeros of a polynomial with real coefficients

The problem of finding all the zeros of a polynomialP _n (x)=x ⁿ +a _n–1 x ^n–1+...+a ₁ x+a ₀, where the coefficientsa _i are real, can be posed as a system ofn nonlinear equations. The structure of this system allows an efficient numerical solution using a damped Newton method; in particular it is possible to generate the triangular factors of the associated Jacobian matrix directly. This approach provides a natural generalisation of the well-known method of Durand and Kerner.     

Simultaneous inclusion of the zeros of a polynomial

Summary A family of inclusion sets for the zeros of a complex polynomial is derived from the Lagrangean interpolation formulas. The optimization of the inclusion leads to a special type of matrix eigenvalue problem previously considered by several authors in connection with minimal Gershgorin discs.     

Uniformn-analytic polynomial approximations of functions on rectifiable contours in ℂ

We study approximations of functions byn-analytic polynomials in the uniform norm on closed rectifiable Jordan curves in the complex plane. It is shown that, in contrast to the case of uniform approximations by complex polynomials, there are no topological criteria for the existence of such approximations. We obtain a criterion for the existence ofn-analytic polynomial approximations in terms of analytic properties of these curves. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 603–609, April, 1996. The author is extremely grateful to A. G. Vitushkin and P. V. Paramonov for the statement of the problem and their attention to the work. The work was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00225.