Ahmet Yaşar Özban

Some new variants of Newton's method

Some new variants of Newtons method based on harmonic mean and midpoint integration rule have been developed and their convergence properties have been discussed. The order of convergence of the proposed methods are three. In addition to numerical tests verifying the theory, a comparison of the results for the proposed methods and some of the existing ones have also been given.

On the -Bernstein Polynomials of Unbounded Functions with

The aim of this paper is to present new results related to the -Bernstein polynomials of unbounded functions in the case and to illustrate those results using numerical examples. As a model, the behavior of polynomials is examined both theoretically and numerically in detail for functions on satisfying as , where and are real numbers.

On the Sets of Convergence for Sequences of the -Bernstein Polynomials with >1

The aim of this paper is to present new results related to the convergence of the sequence of the 𝑞-Bernstein polynomials {𝐵𝑛,𝑞(𝑓;𝑥)} in the case 𝑞g1, where 𝑓 is a continuous function on [0,1]. It is shown that the polynomials converge to 𝑓 uniformly on the time scale 𝕁𝑞={𝑞−𝑗}∞𝑗=0∪{0}, and that this result is sharp in the sense that the sequence {𝐵𝑛,𝑞(𝑓;𝑥)}∞𝑛=1 may be divergent for all 𝑥∈𝑅⧵𝕁𝑞. Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.

New methods for approximating square roots

Some new higher order iterative methods are obtained to approximate the positive square root of a positive real number. Moreover some numerical tests are performed to demonstrate the performances and accuracies of the new methods. The numerical results show that the methods we obtain are competitive with the existing ones.

Improved convergence criteria for Jacobi and Gauss-Seidel iterations

Some simple criteria for the convergence of the Jacobi, Gauss-Seidel and SOR iterations have been proposed in the work of Huang [ZAMM 76-1 (1996) 57-58]. In this study we present some modified forms of the criteria introduced in Huangs work. The new criteria also allow for the norms of the Jacobi iteration matrices to be greater than unity. Numerical examples are also given which show the effectiveness of the criteria.

Polynomial logistic distribution associated with a cubic polynomial

ABSTRACT Let P(x) be a polynomial monotone increasing on ( − ∞, +∞). The probability distribution possessing the distribution function is called the polynomial logistic distribution with associated polynomial P. This has recently been introduced by Koutras et al., who have also demonstrated its importance for modeling financial data. In this article, the properties of the polynomial logistic distribution with an associated polynomial of degree 3 have been investigated in detail. An example of polynomial logistic distribution describing daily exchange rate fluctuations for the US dollar versus the Turkish lira is provided.

Approximation of Discontinuous Functions by q -Bernstein Polynomials

This chapter presents an overview of the results related to the q-Bernstein polynomials with q > 1 attached to discontinuous functions on [0, 1]. It is emphasized that the singularities of such functions located on the set

On the system of rational difference equations xn = a/yn−3, yn = byn−3/xn−qyn−q

\displaystyle{\mathbb{J}_{q}:=\{ 0\} \cup \{ q^{-l}\}_{ l=0}^{\infty },\;\;q> 1,}