Computation Of Zeros Research Articles

A Numerical Computation of Zeros of q-Generalized Tangent-Appell Polynomials

The intended objective of this study is to define and investigate a new class of q-generalized tangent-based Appell polynomials by combining the families of 2-variable q-generalized tangent polynomials and q-Appell polynomials. The investigation includes derivations of generating functions, series definitions, and several important properties and identities of the hybrid q-special polynomials. Further, the analogous study for the members of this q-hybrid family are illustrated. The graphical representation of its members is shown, and the distributions of zeros are displayed.

Improved explicit bounds for some functions of prime numbers

Using recent explicit asymptotic zero--free region and computations of zeros of the Riemann zeta function, obtained by Mossinghoff & Trudgian and Gourdon, respectively, we give an improvement for estimates of some functions related to distribution of primes, such as prime counting function, intervals containing at least one prime, Chebyshev's $\psi$ and $\vartheta$ functions.

Spectral parameter power series for fourth-order Sturm–Liouville problems

A general solution of the fourth-order Sturm–Liouville equation is presented in the form of a spectral parameter power series (SPPS). The uniform convergence of the series is proved and the coefficients of the series are calculated explicitly through a recursive intergration procedure. Based on the SPPS representation characteristic equations for spectral problems arising in mechanics and elasticity theory are obtained and it is shown that the spectral problems reduce to computation of zeros of corresponding analytic functions of the spectral parameter given by their Taylor series expansions. This leads to a simple and efficient numerical method for solving the spectral problems for fourth-order Sturm–Liouville equations. Several examples of application are discussed.

Computation of the Rankin–Selberg L-functions of Maass wave forms

This note outlines a method for numerically computing the Rankin–Selberg convolutions of Maass wave forms L-functions and reports on the computation of zeros of some of them. Bibliography: 11 titles.

Sorting-based localization and stable computation of zeros of a polynomial. I

Sorting-based localization and stable computation of zeros of a polynomial. I

Existence and computation of zeros of perturbed mappings

Sufficient conditions are given to assert that a perturbed mapping has only one zero in a specific open ball. This zero is estimated by means of a sequence of Newton's iterations. The proof of the result is constructive and is based upon continuation methods.

Automatic Computation of Zeros of Bessel Functions and Other Special Functions

An efficient algorithm is developed for the automatic calculation of the zeros of Bessel functions of arbitrary order, Coulomb wave functions, and other special functions. Remarkably, this can be done without any actual functional evaluations. Limitations and generalizations are discussed.

Efficient Computation of Zeros of the Moving Average Operator with Real Matricial Coefficients

This paper proposes a computationally efficient procedure for evaluating the zeros of the vector moving average operator whose coefficients are matrices. A feature of the proposed method is that it is not necessary to explicitly construct the determinantal equation from a given moving average operator in order to evaluate its zeros. Application of the procedure is illustrated and numerical examples are provided to demonstrate the relative computational accuracy of the suggested method.

Determination of eigenmodes by using Cauchy's integral formula

A deterministic method is proposed for the computation of zeros within a complex plane by using Cauchy's theorem. The integrals have been calculated for different numbers of integration points by the Gaussian integration formula. As an example, several eigenmodes of a TE/TM mode splitter have been determined by the proposed method.

Remarks on the generalized Newton method

We give some convergence results for the generalized Newton method for the computation of zeros of nondifferentiable functions which we proposed in an earlier work. Our results show that the generalized method can converge quadratically when used to compute the zeros of the sum of a differentiable function and the (multivalued) subgradient of a lower semicontinuous proper convex function. The method is therefore effective for variational inequalities and can be used to find the minimum of a function which is the sum of a twice-differentiable convex function and a lower semicontinuous proper convex function. A numerical example is given.

Computation of Zeros of Generalized State-Space Systems

Abstract The paper presents an efficient and numerically stable algorithm for the computation of finite zeros of generalized state-space systems. The proposed algorithm reduces the system matrix pencil to a regular pencil whose finite generalized eigenvalues are the finite zeros of the system. The reduction is performed using unitary (in complex case) or orthogonal (in real case) pencil similarity transformations. The method is general, it can handle both square and non-square systems, degene-rate systems as well as singular systems. It exploits efficiently the particular structure of the system matrix pencil. The proposed algorithm can be viewed as an extension of a similar algorithm for stan-dard state-space systems. Numerical properties and algorithmic details are discussed. Numerical examples are given for illustrating the performances of the algorithm.

The quantum-size effect in wedge-shaped films

The authors investigate the distribution of electronic states of thin metal or semiconductor films that satisfy size quantisation conditions. Hitherto the most widely studied case has been the parallel-sided slab. In the present work the quantum-size effect is calculated for wedge-shaped films. The calculation of the electronic spectrum involves the computation of zeros of Bessel functions when the inner radius of the wedge is zero (ideal wedge), and of their cross products when the inner radius is not zero (truncated wedge). The electronic spectrum splits into two-dimensional sub-bands which are similar to the well known sub-bands in a slab. The density of states tends to a constant value at high energy, just as for a slab. The singularity at the bottom of the sub-band is however of square-root form, in contrast to the step form found for a slab. In the case of the truncated wedge a second square-root singularity appears, provided the ratio of the inner to the outer radii is less than a critical value. When a truncated wedge tends to the slab shape, the density of states tends to the known slab form, as it should.

A simple quadrature-type method for the computation of real zeros of analytic functions in finite intervals

A new direct approximate method for the computation of zeros of analytic functions is proposed. For such a function possessing one real zero in a finite part of the real axis, this method permits the determination of this zero with a satisfactory accuracy by using a quite elementary algorithm. The present method is based on the Gauss- and Lobatto-Chebyshev quadrature rules for Cauchy type principal value integrals and is always convergent. The simplicity, accuracy and unrestricted convergence of the proposed method make it competitive to the analogous classical elementary methods. Numerical results are also presented.

A comparison between the NAM and related techniques with a recent method for the computation of zeros and zero directions

The annihilators N, M of the input and output maps, B,C, respectively of the linear state space model S(A, B, C) led to the NAM algorithm for the calculation of system zeros. A recent paper proposed an alternative algorithm and the present paper undertakes a comparison between the two. It is shown that the new approach resembles the NAM algorithm and indeed, in all but the general case the two approaches are shown to be conceptually identical ; in the general case the new method fails to identify the precise set of zeros and this obviates the need for a comparison. An examination of the computational aspects however demonstrates the advantages of the NAM algorithm in both the calculation of zeros and zero directions.

Computation of the abscissa of stability by repeated use of the Routh test

The numerical computation of the abscissa of stability of a polynomial, defined as the largest of the real parts of the zeros of the polynomial, is a problem that occurs repeatedly in computer-aided design of dynamical and control systems. The approach developed in this paper, by making repeated use of the Routh test, avoids the computation of zeros and provides a useful method.