A New Family of High-Order Ehrlich-Type Iterative Methods

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

A Family of Optimal Eighth Order Iteration Functions for Multiple Roots and Its Dynamics

In this manuscript, we present a new general family of optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity using weight functions. An extensive convergence analysis is presented to verify the optimal eighth order convergence of the new family. Some special cases of the family are also presented which require only three functions and one derivative evaluation at each iteration to reach optimal eighth order convergence. A variety of numerical test functions along with some real-world problems such as beam designing model and Van der Waals’ equation of state are presented to ensure that the newly developed family efficiently competes with the other existing methods. The dynamical analysis of the proposed methods is also presented to validate the theoretical results by using graphical tools, termed as the basins of attraction.

Two Classes of Iteration Functions and Q-Convergence of Two Iterative Methods for Polynomial Zeros

In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard-type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions.

Element Iteration Respecification Unstressed Word-Final [u] in Portuguese

This study assumes that the internal structure of vowels consists of the combination of elements which can be iterated. According to previous research, it is also assumed that element iteration in Portuguese provides vowels with inherent weight; word-finally, such vowels are stress-attractors. Notwithstanding, Portuguese has a great amount of words with unstressed final [u] (i.e., a vowel consisting of the iteration of {U}), as it is the case of inflected forms of nouns and adjectives. After analysing diachronic and morphological data, it is proposed that element iteration can have different representations at the lexical and the post-lexical levels. On the basis of this observation, it is proposed that element iteration functions as a weight- and stress-assigner only when lexically specified

An Iteration Function Having Optimal Eighth-Order of Convergence for Multiple Roots and Local Convergence

In the study of dynamics of physical systems an important role is played by symmetry principles. As an example in classical physics, symmetry plays a role in quantum physics, turbulence and similar theoretical models. We end up having to deal with an equation whose solution we desire to be in a closed form. But obtaining a solution in such form is achieved only in special cases. Hence, we resort to iterative schemes. There is where the novelty of our study lies, as well as our motivation for writing it. We have a very limited literature with eighth-order convergent iteration functions that can handle multiple zeros m≥1. Therefore, we suggest an eighth-order scheme for multiple zeros having optimal convergence along with fast convergence and uncomplicated structure. We develop an extensive convergence study in the main theorem that illustrates eighth-order convergence of our scheme. Finally, the applicability and comparison was illustrated on real life problems, e.g., Van der Waal’s equation of state, Chemical reactor with fractional conversion, continuous stirred reactor and multi-factor problems, etc., with existing schemes. These examples further show the superiority of our schemes over the earlier ones.

Structural Stability of a Family of Exponential Polynomial Maps

We perturbed a family of exponential polynomial maps in order to show both analytically and numerically their unpredictable orbit behavior. Due to the analytical form of the iteration functions the family has numerically different behavior than its correspondent analytical one, which is a topic of paramount importance in computer mathematics. We discover an unexpected oscillatory parametrical behavior of the perturbed family.

Modeling and Analysis of Chaos-based Spread Spectrum Scheme using Irregular LDPC Code and Non-Coherent 16-DCSK under Fading and Jamming

In chaos-based spread spectrum systems, the use of spreading code and chaotic binary sequence expands the bandwidth of the information-bearing signal but this expansion results in SNR degradation under the constraint of constant channel capacity according to Hartley-Shannon law. To compensate for this drawback, our proposed model employs an irregular low-density parity-check (LDPC) code with its iterative decoding algorithm. Coupled with this forward error correction (FEC) coding, we used non-coherent (NC) 16-ary differential chaos shift keying (16-DCSK) that additionally provides the ability of data encryption due to its use of chaotic signals compared with the conventional modulation schemes. Analytical expressions of bit error probability (BEP) are derived under the assumption of the three-ray model along with partial band noise jamming (PBNJ) over a Rayleigh fading channel. Simulation results assert that the proposed system can mitigate the effect of PBNJ via lowering BEP by coding gain and processing gain under identical transmission power. It is also confirmed that a higher level of security can be provided by the use of proposed two iteration functions of Duffing Map-based chaotic binary sequence than the security level of one iteration function of Logistic Map, based on the balance and autocorrelation analysis.

Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application

The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the first derivative; the eighth order method uses three evaluations of the function and one evaluation of the first derivative; and sixteenth order method uses four evaluations of the function and one evaluation of the first derivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions are compared with the existing competitors on some standard academic problems. The conjugacy maps of the presented method and other existing eighth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. We apply the new scheme to find the optimal launch angle in a projectile motion problem and Planck’s radiation law problem as an application.