The kissing polynomials and their Hankel determinants

In this paper, we investigate algebraic, differential and asymptotic properties of polynomials $p_n(x)$ that are orthogonal with respect to the complex oscillatory weight $w(x)=\mathrm {e}^{\mathrm {i}\omega x}$ on the interval $[-1,1]$, where $\omega>0$. We also investigate related quantities such as Hankel determinants and recurrence coefficients. We prove existence of the polynomials $p_{2n}(x)$ for all values of $\omega \in \mathbb {R}$, as well as degeneracy of $p_{2n+1}(x)$ at certain values of $\omega $ (called kissing points). We obtain detailed asymptotic information as $\omega \to \infty $, using recent theory of multivariate highly oscillatory integrals, and we complete the analysis with the study of complex zeros of Hankel determinants, using the large $\omega $ asymptotics obtained before.

Innovative Genetic Algorithmic Approach to Select Potential Patches Enclosing Real and Complex Zeros of Nonlinear Equation

In this article, an innovative Genetic Algorithm is proposed to find potential patches enclosing roots of real valued function f:R→R. As roots of f can be real as well as complex, the function is reframed on to complex plane by writing it as f(z). Thus, the problem now is transformed to finding potential patches (rectangles in C) enclosing z such that f(z)=0, which is resolved into two components as real and imaginary parts. The proposed GA generates two random populations of real numbers for the real and imaginary parts in the given regions of interest and no other initial guesses are needed. This is the prominent advantage of the method in contrast to various other methods. Additionally, the proposed ‘Refinement technique' aids in the exhaustive coverage of potential patches enclosing roots and reinforces the selected potential rectangles to be narrow, resulting in significant search space reduction. The method works efficiently even when the roots are closely packed. A set of benchmark functions are presented and the results show the effectiveness and robustness of the new method.

Computer tools for the construction and analysis of some efficient root-finding simultaneous methods

Using the tools provided by computer algebra system Mathematica, we consider two iterative methods of high efficiency for the simultaneous approximation of simple or multiple (real or complex) zeros of algebraic polynomials. The proposed methods are based on the fourth-order Schr?der-like methods of the first and second kind. We prove that the order of convergence of both basic total-step simultaneous methods is equal to five. Using corrective approximations produced by methods of order two, three and four for finding a single multiple zero, the convergence order is increased from five to six, seven, and eight, respectively. The increased convergence speed is attained with negligible number of additional arithmetic operations, which significantly increases the computational efficiency of the accelerated methods. Convergence properties of the proposed methods are demonstrated by numerical examples and graphics visualization by plotting trajectories of zero approximations. Flows of iterative processes, presented by these trajectories, point to the stability and robustness of the proposed methods.