Some iterative methods for solving non-linear equations

العناوين الأخرى

بعض الطرق التكرارية لحل المعادلات اللاخطية

مقدم أطروحة جامعية

Khidr, Fuad Wahid

مشرف أطروحة جامعية

Said, Rustum Karim

أعضاء اللجنة

Abd Allah, Herish Umar

الجامعة

جامعة صلاح الدين

الكلية

كلية العلوم

القسم الأكاديمي

قسم الرياضيات

دولة الجامعة

العراق

الدرجة العلمية

ماجستير

تاريخ الدرجة العلمية

2010

الملخص الإنجليزي

Indeed the calculation of roots of functions is one of the oldest of mathematical problems.

The solution of quadratics was known to the ancient Babylonians, and to the early middle Ages.

The solution of cubics was first solved in a closed form by G.

Cardano in the mid-16th century, and the quartic soon afterwards.

However N.H.

Abel in the early 19th century showed that polynomials of degree five or more could not be solved by a formula involving radicals of expressions in the coefficients, as those of degree up to four could be.

Since then (and for some time before in fact), researchers have concentrated on numerical (iterative) methods such as the famous Newton’s method of the 17th century.

Of course there have been a plethora of new methods in the 20th and early 21st century, especially since the advent of electronic computers [3].

Determining the roots of an equation have attracted the attention of pure and applied mathematicians for centuries.

Many problems may be formulated in terms of finding zeros.

These roots cannot in general be expressed in closed form.

Thus, in order to solve non-linear equations, we have to use approximate methods.

One of the most important techniques to study these equations is the use of iterative processes, starting with an initial approximation.

The Newton method is the most popular method for solving non-linear equations [1, 4].

Multipoint iterative methods for finding solutions of non-linear equations have been a constant interesting field of study in numerical analysis.

Recently, many researches for these methods were carried out (e.

g.

[17-25] and references therein) they attempted to develop higher order methods by adding finite evaluations of the function in the multipoint methods to obtain less iterations than the classical Newton method.

In this work, some higher order methods are presented with their convergence for finding simple root of non-linear equations.

Also, we developed some iterative methods and established their convergence.

All methods in this thesis are tested on the same numerical examples, and in most cases we get that the proposed method has at least equal performance as compared with the other known methods.

In the following we give a brief overview of the thesis by chapters.

The present work consists of six chapters as follows : Chapter one : deals with historical background of iterative methods for solving non-linear equations.

Also we give some definitions and necessary concepts.

Chapter two : contains the new iterative method for solving non-linear equations of order at least three.

A convergence analysis of the proposed method was given with some numerical results.

Chapter three : Three iterative methods of higher order are derived for solving non-linear equations ; the first method is of order at least three and the other two iterative methods of order four and six, also the algorithms for the methods are given.

A convergence analysis of the presented methods has been given with some numerical results.

Chapter four : Two iterative methods derived for finding simple roots Of non-linear equations which are based on an improvement of the Sebah and Gourdon method and the algorithms for the methods are given.

Also a detailed convergence analysis of the present methods has been given with some numerical results.

Chapter five : Another approach presented based on decomposition technique to construct new two iterative methods involving only the first derivative of the function of order at least six and seven respectively.

A detailed convergence analysis of the present methods is given with some numerical results as well.

Finally, Chapter six : deals with conclusions and recommendations for future works

التخصصات الرئيسية

الرياضيات

عدد الصفحات

68

قائمة المحتويات

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لغة النص

الإنجليزية

نوع البيانات

رسائل جامعية

رقم السجل

BIM-310389