ABSTRACT

In this thesis, we have developed two new iterative schemes for solving nonlinear equations.
The two schemes have been constructed from Taylor’s series expansion and Adomian
decomposition method. The two schemes have been compared with other existing iterative
methods using one way analysis of variance (ANOVA). They are found to be efficient and
better than some of the existing schemes. The results show that Newton-Raphson method and
New scheme 1 have more advantage with a maximum of seven iterations each, while new
scheme 2 has nine. Basto et al. and Abbasbany have equal number of thirteen iterations each.
The Adomian has sixteen iterations. Thirty numerical examples are given and solved to justify
the efficiency of the new iterative schemes.

 

 

TABLE OF CONTENTS

COVER PAGE
FLY PAGE
TITLE PAGE
DECLARATION…………………………………………………………………………………………………….. i
CERTIFICATION ………………………………………………………………………………………………….. ii
DEDICATION ……………………………………………………………………………………………………… iii
ACKNOWLEDGEMENT ………………………………………………………………………………………. iv
ABSTRACT ……………………………………………………………………………………………………………v
TABLE OF CONTENTS……………………………………………………………………………………….. vi
CHAPTER ONE : GENERAL INTRODUCTION …………………………………………………….1
1.1 Introduction ……………………………………………………………………………………………………….1
1.2 Motivation for this study ………………………………………………………………………………………5
1.3 Problem studied in the thesis …………………………………………………………………………………6
1.4 Aim and objectives ……………………………………………………………………………………………..6
1.5 Limitation of the study …………………………………………………………………………………………7
1.6 Definitions of term ………………………………………………………………………………………………7
1.7 Theorems used in the study …………………………………………………………………………………..8
1.8 Outline of the thesis …………………………………………………………………………………………….9
CHAPTER TWO : LITERATURE REVIEW ………………………………………………………… 10
2.1 Introduction …………………………………………………………………………………………………….. 10
2.2 Generalizations of Newton’s method …………………………………………………………………… 11
2.3 The Adomian decomposition method …………………………………………………………………… 14
2.4 Studies based on Adomian decomposition method …………………………………………………. 16
2.5 Studies based on Homotopy perturbation method…………………………………………………… 19
CHAPTER THREE : CONSTRUCTION OF THE NEW SCHEMES ……………………… 22
3.1 Introduction …………………………………………………………………………………………………….. 22
3.2 The present work ……………………………………………………………………………………………… 23
3.3 Construction of the new schemes ………………………………………………………………………… 26
3.3.1 New iterative scheme 1 …………………………………………………………………………………… 26
3.3.1.1 Convergence analysis for new the scheme 1 …………………………………………………….. 29
3.3.2 New iterative scheme 2 …………………………………………………………………………………… 34
vii
3.3.2.1 Convergence analysis for new scheme 2 …………………………………………………………. 36
CHAPTER FOUR : ANALYSIS OF RESULTS ……………………………………………………… 41
4.1Introduction ……………………………………………………………………………………………………… 41
4.2 Thirty Examples of Different Nature ……………………………………………………………………. 42
Table 4.1Comparison between Number of Iterations for Thirty Different Examples …………. 43
4.3 Summary of Results Obtained for Some Solved Examples ………………………………………. 44
4.4 Results obtained from ANOVA ………………………………………………………………………….. 47
CHAPTER FIVE : ……………………………………………………………………………………………….. 51
SUMMARY, CONCLUSION AND RECOMMENDATIONS ………………………………….. 51
5.1 Summary ………………………………………………………………………………………………………… 51
5.2 Conclusion………………………………………………………………………………………………………. 52
5.3 Recommendations ……………………………………………………………………………………………. 52
References ……………………………………………………………………………………………………………. 54
Appendix ……………………………………………………………………………………………………………… 57
1

 

 

CHAPTER ONE

GENERAL INTRODUCTION
1.1 Introduction
Solving nonlinear equations is an important part of numerical analysis. In recent years,
interests have grown considerably in developing effective iterative methods (IM) for
computing solutions for large systems in science and engineering. The development of faster
and more robust IM and preconditions which can be efficiently mapped to a variety of
problems is of fundamental importance in that it will be of great assistance to scientists and
engineers throughout many disciplines. In numerical analysis this approach is in contrast
to direct methods which attempt to solve the problem by a finite sequence of operations.
Numerical analysis assumes this task, and with it the limitations of practical calculations.
Numerical answers are usually tentative and, at best, known to be accurate only to within
certain bounds. IM are often useful even for linear problems involving a large number of
variables, where direct methods would be prohibitively expensive. Intuitively iterative
methods keep on improving upon subsequent iterations. With iteration methods, the
operational cost can often be reduced.
In this thesis, we study iterative methods for solving nonlinear equations, f x  0, where
f x is any continuously differentiable real valued function. The iterative methods we try to
develop for this class of equations will require knowledge of initial guess for desired roots of
the equation.
2
Traub (1964), Numerical methods for solving nonlinear equations are divided into two
categories; the interval methods and the initial point methods.. Bisection method is an
example of interval methods. The initial point methods use one or more initial points as the
starting values to find the approximate solution using recurrence relation. In this study, we
concentrate mainly on one point initial methods. The major disadvantage of these methods
however, is that their convergences are not guaranteed and the choice of initial guess requires
some insight. These methods are however, usually faster than the interval methods. Secant
method and Newton method are examples of initial point methods. There are several methods
for solving nonlinear equations and here we introduce a few of them.
Newton or Newton-Raphson method is the most widely used method for finding roots of an
equation. According to Traub (1964), it begins with an initial approximation, 0 x and
generates a sequence of successive iterates  
k k0 x converging quadratically to simple roots.
In Secant Method, which is a variant of Newton-Raphson’s method it use finite difference to
approximate the derivative of the function y = f (x) close to the root by the line (secant) and
requires two initial points   1 1 , n n x f and   n n x , f , where   n n f  f x .Taking the point of
intersection of this line with the x-axis as the subsequent iterate. We get
, 1,2
1
1
1 





 f n
f f
x x x n
n n
n n
n where xn1 and xn are two consecutive iterates. Since a secant
line is defined using two points on the graph of f (x), as opposed to a tangent line that requires
information at only one point on the graph, we need two initial approximations 0 x and 1 x .
Traub (1964), the method has a super linear convergence.
3
The Bisection Method tries to decrease the size of the interval in which a solution exist. If
the function f xsatisfies     < 0, 0 0 f a f b then the equation starts with one sign at 0 a and
ends with the opposite sign at 0 b , and if  0   0  = 0 f a f b , then either 0 a or 0 b or both are roots
of f(x) = 0. This method consists of finding midpoint 0 a and 0 b . If   1 2 0 0
m  1 a  b is the
midpoint of this interval, then the root will lie either in the interval   0 1 a ,m or in the interval
  1 0 m ,b provided that   0 1 f m  . If   0 1 f m  , then 1 m is the required root. Repeating this
procedure, we obtain the bisection method
 , 0,1,
2
1
1      m a b a n n n n n .
Where     1 , 1  an an mn if f an f mn1  < 0 (1.1)
and
    bn1 mn1,bn  if f mn 1 f bn  < 0 
We take the midpoint of the last interval as an approximation to the root. Traub (1964), if f (x)
is continuous in the interval [a, b] which contains the root, the method converges.
Hafiz and Bahgat (2012), several iterative methods have been developed to solve nonlinear
algebraic equations and the system of nonlinear equations. These methods have been
improved using Taylor polynomials, homotopy perturbation method and Adomian
decomposition methods.
4
The homotopy perturbation method (HPM) was developed for solving nonlinear systems, He
(1999). HPM linearizes any given problem (converting it to a series of linear equations). The
method gives a rapid convergence of the solution and only a few iterations lead to accurate
result. In contrast to the traditional perturbation methods, this method does not require a small
parameter in an equation. In this method, a homotopy with an imbedding parameter p ∈ [0, 1]
is constructed, and the imbedding parameter is considered as a “small parameter”. Li (2009),
when p=0, the system of equation usually reduces to a sufficiently simplified form, which
normally admits a rather simple solution. As p increases to 1, the system under goes a
sequence of deformations, the solution of each is close to that at the previous stage of
deformation. When p=1, the system takes the original form of the equation and the final stage
of deformation gives the desired solution.
Adomian (1984), developed a new method known as the Adomian decomposition method
(ADM) for solving functional equations of any kind: ordinary differential equations (ODEs),
algebraic, partial differential equations (PDEs), integral equations, etc. The ADM breaks any
given problem into linear and nonlinear parts. The linear operator representing the linear
portion of the equation is inverted and the inverse operator is then applied to both sides of the
equation, before applying the initial or boundary conditions. The term that contains the
independent variable alone is taken as the initial approximation. The unknown function is
then decomposed into a series whose components are to be determined. The components are
determined in terms of polynomials called Adomian polynomials whose successive terms are
determined using a recurrent relation.
5
Traub (1964), classified one-point iterative methods into,
(i) One-point methods without memory
(ii) One-point methods with memory
If the value of the root is ascertained by using new data only at one point and no previous data
is used, then it is called one-point iterative method without memory. An example of one-point
iterative methods without memory is the Newton-Raphson method. Hence, if n1 x is estimated
by new data at n x and no previous data is used, we have   n n x  x 1 , then  is called onepoint
iteration function without memory.
If the value of the root is ascertained by using new data at one point and by using the previous
data at either one or more than one points, then the iterative method is called one-point
method with memory. Secant method is an example of one point iterative method with
memory. If n1 x is estimated by new data at n x and the previous data at n n m x x   , , 1  , we have
  n n n n m x x x x     , , , 1  1  and is called one-point iteration function with memory.
1.2 Motivation for this study
Many methods and algorithms have been developed to solve problems of nonlinear algebraic
equations over the years. Despite these efforts, no single algorithm is capable of solving any
and all nonlinear problems. Depending on the system and the degree of nonlinearity, one
solution scheme may be preferred over another. To keep up with recent computational
challenges in the field of numerical analysis, it is imperative to develop new schemes that are
capable of taking advantage of the latest advances in numerical analysis. This is what
6
motivates us to undertake this study. Advances such as Adomian decomposition method,
Basto et al. method.
1.3 Problem studied in the thesis
In our present work, we have tried to develop two new iterative schemes which are based on
Taylor’s series and Adomian method. The first scheme truncates the Taylor’s series after the
third term while the second scheme truncates the series after the fourth term. Moreover in
both schemes, it is assumed that  
   1.


f x
f x
1.4 Aim and objectives
The aim of the study is to develop and analyse new iterative schemes for solving nonlinear
equations.
The objectives of the study are
(i) To review iterative schemes between 1998 and 2012 which have been
developed from Adomian decomposition method, Homotopy perturbation method
and variants of Newton-Raphson’s method for solving nonlinear equations.
(ii) To develop new schemes that could compete with previous schemes and
probably have further advantages.
(iii)To compare the new schemes with the existing known iterative schemes.
7
1.5 Limitation of the study
This work is limited to initial point methods and specifically, only to one point iterative
methods. Moreover, for those problems whose second derivatives are far away from the first
derivatives, the assumption  
   1


f x
f x is a limitation.
1.6 Definitions of term
(i) Let n x  x be the truncation error in the nth iterate where x is the required
root. If there exists a number p 1 and a constant c  0 such that
c
x x
x x
p
n
n
n







1 lim ,
then p is called the order of convergence of the method, Burden and Faires (2011) .
Note that;
If p 1, we say that   n x is linearly convergent.
If p >1, we have super linear convergence.
If p  2 , we have quadratic convergence.
8
1.7 Theorems used in the study
We now state the following theorems without proofs, which will be used in this study. The
proofs can be found in any standard analysis text book such as Burden and Faires (2011).
Theorem 1.7.1: Mean value Theorem
If f C[a,b] (whereC[a,b] is the space of continuous functions on[a,b]) and f xis
differentiable ina,b, then there exists a point ca,bsuch that f b f a  b  af c.
Theorem 1.7.2: Fixed Point Theorem
If g C[a,b]and gx[a,b]for all x[a,b],then g has a fixed point in [a,b]. If in addition,
gxexists on a,b and a positive constant k < 1exists with gx  k , for all xa,b, then
the fixed point in [a,b] is unique.
Theorem 1.7.3: Rolle’s Theorem
Suppose f C[a,b] and f is differentiable on a,b. If f a f b, then a number c in
a,b exists with f c  0.
Theorem 1.7.4: Extreme Value Theorem
Suppose f C[a,b] and , [ , ] 1 2 c c  a b with       1 2 f c  f x  f c , for all x[a,b]. In
addition, if f is differentiable on a,b, then the numbers 1 c and 2 c occur either at the
endpoints of [a,b] or where f   0 .
9
Theorem1.7.5: Taylor’s Theorem
Suppose f Cn[a,b], (where Cn[a,b]is the space of n-times continuously differentiable
functions on[a,b]) and that f n1 exists on[a,b], with [ , ] 0 x  a b . Then for every x[a,b] ,
there exists a number  x between xo and x with
            x x  R x
n
f x f x f x x x f x x x f x n
n
n
    

     0
2 0
0
0
0 0 0 2! !

Where    
    1
0
1
1 !




 n
n
n x x
n
R x f x

(called the remainder term or truncation error).
1.8 Outline of the thesis
The present thesis is structured as follows:
Chapter 1, which is the present chapter constitutes the general introduction, the aims and
objectives of the study and limitations of the study. It also contains some basic definitions and
theorems that are important to the study being done.
Chapter 2 is a review of previous work that is related to the work under study.
Chapter 3 is a description of the Adomian method and how it is used to develop the two new
iterative methods.
Chapter 4 contains analysis of numerical results to illustrate the effectiveness of the new
methods.
Chapter 5 contains summary, conclusion and recommendations. It also contains discussion of
potential areas for further work.

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