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Title: Some Efficient Iterative Schemes for Finding Roots of Nonlinear Equations
Authors: Akram, Saima
Keywords: Physical Sciences
Issue Date: 2019
Publisher: Bahauddin Zakariya University Multan
Abstract: Keywords: Nonlinear Equations, System of Nonlinear Equations, Iterative Methods, Weight Function, Optimal Iterative Methods, Rational Interpolation, Hermite Interpolation, Derivative-free Methods, With-memory Methods, E¢ ciency Index, Simple Zeros, Multiple Zeros, Basins of Attraction. 2000 Mathematics Subject ClassiÖcation: 34A34, 65H05, 65B99, 41A05, 41A25, 65A99. The root Önding problem has remained a very important problem in almost all the branches of engineering and science. There exist some complicated direct root Önding techniques for cubic or quartic equations but roots of higher degree nonlinear equations cannot be determined using these direct or noniterative techniques. Therefore, there is a natural need of some numerical root Önding algorithms for solving nonlinear equations and systems of nonlinear equations. Several numerical methods to Önd the roots of nonlinear equations have been presented in the last few decades. The root Önding iterative method starts with an initial guess of the required root of the nonlinear equation and improve this approximation iteratively until we obtain the approximate root of required accuracy. In earlier years, several single step root Önding methods were developed for the solution of nonlinear equations. For example, Newton, Laguerre, Grae§e, Baristow and Mueller proposed single step root Önding methods. Among them the i Newtonís method is most famous, which requires the evaluation of derivative of the function at each iterative step. The order of convergence and e¢ ciency of single step methods can not be increased after a certain bound. Traub [115] classiÖed the iterative methods as single step and Multistep root Önding methods. Multistep root Önding methods allow us not to discard information that had already been computed. These methods require evaluations of the nonlinear function and derivatives of nonlinear function at several values of the independent variable. Thus, higher order root-Önding methods are developed by using this approach that has motivated the researchers towards the study of multistep root-Önding methods. Multistep root Önding methods overcome the theoretical limits of any single step method regarding the order of convergence and informational and computational e¢ ciency. Thus, they are of great practical importance than single step methods. Therefore, the researchers have been contributing a lot in the development of these algorithms. Multi-step root Önding methods that use only information from the current iteration are called methods without-memory and the root Önding methods that use information from the current and previous iteration are termed as methods with-memory. Ostrowski [82] deÖned the e¢ ciency index of an iterative method as r 1 kf ; where r is the convergence order of the method and kf is the number of function evaluations required per iteration. Kung and Traub [61] conjectured that a without-memory multipoint method requiring k + 1 function evaluations per iteration have optimal order at most 2 k and it can attain the e¢ ciency index at most 2 k1 k : The methods satisfying above hypothesis of Kung and Traub are known as optimal. The main aim of this study is to investigate and introduce some new optimal and computationally e¢ cient multistep iterative methods using di§erent techniques such as rational interpolation, Hermite interpolation, using free parameters and weight functions for Önding simple and repeated roots of nonlinear equations as well as to solve systems of nonlinear equations. Some new i root Önding methods with-memory are presented in this thesis that are highly e¢ cient to obtain approximations of better accuracy. Moreover, we have proposed a new family of optimal eighth order methods to Önd multiple roots of nonlinear equations. In addition, we study the comparison of new and existing root Önding methods in terms of numerical experiments and dynamical planes. Numerical results are given by taking several real world problems including all kinds of nonlinear functions. The comparison of the dynamical behavior of di§erent root Önding methods is done by using the idea of basins of attraction. In this thesis, we have proposed a new family of optimal fourth order iterative methods to Önd simple roots of nonlinear equations along with its extension to solve systems of nonlinear equations. Convergence analysis for both cases shows that the order of convergence of the new methods is at least four. Numerical experiments and dynamical planes show that the new methods are better alternates to the existing methods of similar kind. We have also developed two new families of eighth order convergent derivativefree methods without-memory to solve nonlinear equations. The worth of these methods lies in the fact that they are optimal in the sense of Kung and Traubís hypothesis and are extendable to highly e¢ cient methods (withmemory methods). We have deÖned a procedure of the construction of optimal derivative free methods without memory that are extendable to with-memory. The convergence analysis for the new methods is also are presented. The comparison of the dynamical planes of di§erent methods is done by drawing their regions of convergence using polynomials of di§erent degree. In this thesis, we have presented two new general families of derivative free nstep optimal iterative schemes without-memory based on rational and Hermite interpolation that satisfy the conjecture of Kung and Traub [61]. Some particular members of the family and their analysis of convergence are also studied. The comparison of the proposed root Önding methods without ii memory with the existing iterative methods in terms of numerical results and basins of attraction is presented. With-memory multi-step iterative methods that use information from the current and previous iterations, increase the convergence order and computational e¢ ciency of the multistep iterative methods without-memory without using additional function evaluations. The increase in the order of convergence is based on one or more accelerating parameters which appear in the error equations of the without-memory methods. For this reason, several multi-step with- and without-memory iterative methods have been developed in recent years. For a background study, regarding the acceleration of convergence order via with-memorization, one may see e.g. [69, 70, 90]. In this thesis, we have proposed some new e¢ cient root Önding methods with-memory based on newly developed optimal eighth order derivative free methods without memory using four accelerating parameters. For this, we approximate the involved free parameters by using Newtonís interpolating polynomials that pass through already saved iterative points. The R-order of convergence [78] of the new root Önding methods with-memory is 15:5156 by using only four function evaluations and thus, their e¢ ciency index is 1:9847. We have also presented a general class of with-memory methods as an extension of newly developed family of nstep derivative-free optimal methods without-memory by using a self-accelerating parameter. The convergence order of the methods without memory is increased from 2 n to 2 n + 2n1 + 2n2 without using additional function evaluations. An extensive comparison of our with-memory methods is done with the existing with-memory methods using some nonlinear equations. In this thesis, we have also constructed a new family of optimal eighth order convergent methods to Önd multiple roots of nonlinear equations. This family is based on weight function approach and the modiÖed Newtonís method for multiple roots. Analysis of convergence is presented for the presented scheme with the help of symbolic computations on programming package Mathematica 8. In addition, we have also demonstrated the applicability of the presented schemes on some real world problems and illustrated that the proposed methods are more e¢ cient among the available multiple root Önding schemes. The numerical tests of all the problems considered in this thesis have been carried out by using the programming package Maple 16 based on highprecision calculations on few initial estimations. Comparison of the performance of proposed and existing methods has also been carried out by drawing their dynamical planes in the complex plane, that allows us to know how wide is the set of initial guesses that converges to the required roots. Both of the comparisons give us complementary information that helps us to fully understand the numerical performance of the root Önding methods.
Gov't Doc #: 20229
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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