Chapter 4 Geometrically constructed families of iterative ...shodhganga.inflibnet.ac.in/bitstream/10603/5708/11/11_chapter 4.pdf · Chapter 4 Geometrically constructed families of

Chapter 4

Geometrically constructed families ofiterative methods

4.1 Introduction

The aim of this CHAPTER3 is to derive one-parameter families of Newton’s method [7, 11–

16], Chebyshev’s method [7, 30–32, 59–61], Halley’s method [7, 30–32, 60–73], super-

Halley method [30–32, 61, 71–73], Chebyshev-Halley type methods [74], C-methods [30],

Euler’s method [7, 30, 64], osculating circle method [33] and ellipse method [31, 32, 61]

for finding simple roots of nonlinear equations, permitting f ′(xn) = 0 at some points in

the vicinity of required root. Halley’s method, Chebyshev’s method, super-Halley method,

Chebyshev-Halley type methods, Euler’s method, osculating circle method and as an excep-

tional case, Newton’s method are seen as the special cases of these families. All methods

of various families are cubically convergent to simple roots except Newton’s or a family of

Newton’s method. Further, by making some semi-discrete modifications to above proposed

cubically convergent families, a number of interesting new classes of third-order as well as

fourth-order multipoint iterative methods can be derived which are free from second-order

derivative. Furthermore, numerical examples show that all proposed one-point as well as

multipoint iterative families are effective and comparable to the well-known existing classi-

cal methods.

As discussed previously, one of the most basic problems in computational mathemat-

ics is to find solutions of nonlinear equations (1.1). To solve equation (1.1), one can use

3The main contents of this CHAPTER have been published in [206].

93

iterative methods such as Newton’s method [7, 11–16] and its variants namely, Halley’s

method [7, 30–32, 60–73], Chebyshev’s method [7, 30–32, 59–61] and super-Halley method

[30–32, 61, 71–73] respectively. Among all these iterative methods, Newton’s method is

probably the well-known and mostly used algorithm. Its geometric construction consists in

considering a straight line

y = ax+b, (4.1)

and then determining unknowns ‘a’ and ‘b’ by imposing the tangency conditions :

y(xn) = f (xn), y′(xn) = f ′(xn), (4.2)

thereby obtaining a tangent line

y(x)− f (xn) = f ′(xn)(x− xn), (4.3)

to the graph of f (x) at {xn, f (xn)}. The point of intersection of this line with x−axis, gives

the celebrated Newton’s method

xn+1 = xn− f (xn)f ′(xn)

. (4.4)

The tangent line in Newton’s method can be seen as first degree polynomial of f (x) at x = xn.

Though, Newton’s method is simple and fast with quadratic convergence, but its major draw-

back is that, it may fail miserably if at any stage of computation, the derivative of the function

f (x) is either zero or very small in the vicinity of the required root. Therefore, it has poor

convergence and stability problems as it is very sensitive to initial guess.

The well-known third-order methods which entail an evaluation of f ′′(x) are close rel-

atives of Newton’s method and can be obtained by admitting geometric derivation from the

quadratic curves, e.g. parabola, hyperbola, circle or ellipse.

In this category, the Euler’s method [7, 30, 64] can be constructed by considering a

parabola

x2 +ax+by+ c = 0, (4.5)

and imposing the tangency conditions :

y(xn) = f (xn), y′(xn) = f ′(xn) and y′′(xn) = f ′′(xn), (4.6)

94

one gets

y(x)− f (xn) = f ′(xn)(x− xn)+f ′′(xn)

2(x− xn)2. (4.7)

The point of intersection of (4.7) with x−axis gives an iteration scheme

xn+1 = xn−{

21±√

1−2L f (xn)

}f (xn)f ′(xn)

, (4.8)

where

L f (xn) =f (xn) f ′′(xn){ f ′(xn)}2 . (4.9)

The well-known Halley’s method [7, 30–32, 60–73] admits its geometric derivation from

a hyperbola in the form

axy+ y+bx+ c = 0. (4.10)

After the imposition of conditions (4.6) on (4.10) and taking the intersection of (4.10) with

x−axis as next iterate, the Halley’s iteration process reads as

xn+1 = xn−{

22−L f (xn)

}f (xn)f ′(xn)

. (4.11)

The classical Chebyshev’s method [7, 30–32, 59–61] admits its geometric derivation

from parabola in a following form :

ay2 + y+bx+ c = 0. (4.12)

Similarly after imposing the tangency conditions (4.6) and taking the intersection of (4.12)

with x−axis as next iterate, Chebyshev’s iteration process reads as

xn+1 = xn−{

1− L f (xn)2

}f (xn)f ′(xn)

. (4.13)

Amat et al. [30] studied another type of third-order methods by considering a hyperbola

in a following form :

any2 +bnxy+ y+ cnx+dn = 0, (4.14)

and by the similar conditions as before, one can get an iteration process reads as

xn+1 = xn−1+

12

L f (xn)

1+bn

(f (xn)f ′(xn)

)

f (xn)f ′(xn)

, (4.15)

where ‘bn’ is a parameter depending on ‘n’.

For particular values of ‘bn’, some interesting particular cases of this family are

95

(i) For bn = 0, formula (4.15) corresponds to classical Chebyshev’s method [7, 30–32, 59].

(ii) For bn =− f ′′(xn)2 f ′(xn)

, formula (4.15) corresponds to famous Halley’s method [60, 30–32].

(iii) For bn =− f ′′(xn)f ′(xn)

, formula (4.15) corresponds to super-Halley method [71–73].

(iv) For bn →±∞, formula (4.15) corresponds to classical Newton’s method [7, 11–16].

Osculating circle method [33] and ellipse method [31, 32, 61] are more cumbersome from

computing point of view as compared to Halley’s, Chebyshev’s or super-Halley methods.

For more detailed survey of these most important techniques, some excellent text books

[1, 7, 14, 21] are available in literature. The problem with existing methods is that these

techniques may fail to converge in some cases if an initial guess is far from zero point or if

the derivative of the function is small or even zero in the vicinity of required root. Recently,

Kanwar and Tomar [33, 231, 232] using different approach derived new classes of iterative

techniques having quadratic and cubic convergence respectively. These techniques provide

an alternative to the failure situation of existing classical techniques.

The purpose of this study is to eliminate the defects of existing classical methods by

simple modifications of iteration processes.

4.2 Development of iteration schemes

Based on the two theorems [5], namely Theorem 1.2.2 and Theorem 1.2.3 (discussed in

Chapter 1), proposed methods can be derived by admitting geometrical derivation from the

hyperbolic curve, cubic curve and general quadratic curve.

4.2.1 Derivation from a hyperbola

Let us now, develop an iteration scheme by fitting the function ‘ f (x)’ of equation (1.1) in the

following form of a hyperbola :

an

{ye−α(x−xn)− f (xn)

}2+bn(x− xn)


}

+{

ye−α(x−xn)− f (xn)}

+ cn(x− xn)+dn = 0, (4.16)

96

where α ∈ R. The unknowns an,cn and dn are now determined in terms of ‘bn’ by using

tangency conditions (4.6) on equation (4.16). This gives

an =− f ′′(xn)+α2 f (xn)−2α f ′(xn)

2{ f ′(xn)−α f (xn)}2 − bn

f ′(xn)−α f (xn),

cn =−{ f ′(xn)−α f (xn)} and dn = 0.

. (4.17)

At a root

y(xn+1) = 0, (4.18)

and it follows from (4.16) that

xn+1 = xn− {an{ f (xn)}2− f (xn)}cn−bn f (xn)

. (4.19)

If (4.16) is an exponentially fitted straight line, then an = 0 = bn and from (4.19), one can

have

xn+1 = xn− f (xn)f ′(xn)−α f (xn)

. (4.20)

This is a one-parameter family of Newton’s method [205, 231] already discussed in Chapter

2 and do not fail even if f ′(xn) = 0 or very small in the vicinity of required root, unlike

Newton’s method. Ignoring the term in ‘α’, one gets Newton’s method [7, 11–16].

Further, making use of ‘an’ and ‘cn’ in (4.19), one can obtain

xn+1 = xn−1+

12

L∗f (xn)

1+bn

{f (xn)

{ f ′(xn)−α f (xn)}}

f (xn)f ′(xn)−α f (xn)

, (4.21)

where ‘bn’ is a parameter and

L∗f (xn) =f (xn)

{f ′′(xn)+α2 f (xn)−2α f ′(xn)

}{

f ′(xn)−α f (xn)}2 . (4.22)

This is a modification over the formulas (4.15) of Amat et al. [30] and do not fail even if

f ′(xn) is very small or zero in the vicinity of required root.

For particular values of ‘bn’, some interesting particular cases of (4.21) are as under :

(i) For bn = 0, one can obtain

xn+1 = xn−{

1+12

L∗f (xn)}


. (4.23)

This is a modification over the well-known cubically convergent Chebyshev’s method.

97

(ii) Putting bn =−L∗f (xn)2u , one gets

xn+1 = xn− 22−L∗f (xn)


. (4.24)

This is a modification over the well-known cubically convergent Halley’s method.

(iii) Substituting bn =−L∗f (xn)u , one can get

xn+1 = xn−{

1+12

L∗f (xn)

1−L∗f (xn)

}f (xn)

f ′(xn)−α f (xn). (4.25)

This is a modification over the well-known cubically convergent super-Halley method.

(iv) Letting bn =−α∗L∗f (xn)

u ,α∗ ∈ R, one can have

xn+1 = xn−{

1+12

L∗f (xn)

1−α∗L∗f (xn)

}f (xn)

f ′(xn)−α f (xn). (4.26)

This is a modification over the well-known cubically convergent Chebyshev-Halley

type methods [74].

(v) If bn →±∞, one obtains


. (4.27)

This is a modification over the well-known quadratically convergent Newton’s method.

It can be verified that in the absence of the term in parameter ‘α’, method (4.23),

method (4.24), method (4.25), method (4.26) and method (4.27) reduce to Chebyshev’s

method [7, 30–32, 59–61], Halley’s method [60, 30–32, 61–73], super-Halley method [30–

32, 61, 71–73], Chebyshev-Halley type methods [74] and Newton’s method [7, 11–16] re-

spectively. The sign of parameter ‘α’ in all these methods is chosen so that the denominator

is largest in magnitude.

Following theorem gives an order of convergence of the proposed iterative scheme (4.21) :

98

Theorem 4.2.1. Let f : I ⊆ R→ R be a sufficiently differentiable function defined on an

open interval I, enclosing a simple zero of f (x) (say x = r ∈ I). Assume that initial guess

x = x0 is sufficiently close to ‘r’ and f ′(xn)−α f (xn) 6= 0 in I. Then an iteration scheme

defined by formula (4.21) is cubically convergent and satisfies the following error equation:

en+1 ={

2C22 −C3−3αC2 +

12

α2 +bn (C2−α)}

e3n +O

(e4

n), (4.28)

where en = xn− r is an error at nth iteration and Ck =( 1

k!

) f k(r)f ′(r) ,k = 2,3, . . .

Proof. Using Taylor’s series expansion for f (xn) about x = r and taking into account that

f (r) = 0 and f ′(r) 6= 0, one can have

f (xn) = f ′(r)ben +C2e2n +C3e3

n +O(e4n)c, (4.29)

f ′(xn) = f ′(r)b1+2C2en +3C3e2n +O(e3

n)c, (4.30)

and

f ′′(xn) = f ′(r)b2C2 +6C3en +12C4e2n +O(e3

n)c. (4.31)

Using equations (4.29)-(4.31), one gets


= en− (C2−α)e2n−

(2C3−2C2

2 +2αC2−α2)e3n +O

(e4

n), (4.32)

and

L∗f (xn) =f (xn)

{f ′′(xn)+α2 f (xn)−2α f ′(xn)

}{

f ′(xn)−α f (xn)}2 =2(C2−α)en− (6C2

2 −6C3−6C2α+3α2)e2n

+O(e3n). (4.33)

Using equations (4.32) and (4.33), one can obtain

12

L∗f (xn)

1+bn

{f (xn)

{ f ′(xn)−α f (xn)}} =(C2−α)en−

(3C2

2−3C3−3αC2+32

α2+bn (C2−α))e2

n+O(e3n).

(4.34)

Substituting equations (4.32) and (4.34) in formula (4.21), one obtains

en+1 ={

2C22 −C3−3αC2 +

12

α2 +bn (C2−α)}

e3n +O

(e4

n). (4.35)

This completes proof of the theorem.

99

4.2.2 Derivation from a cubic curve

Another iterative processes can also be derived by considering different exponentially fitted

osculating curves. For instance, if one takes a cubic equation namely

an


}3+bn


}2+


}+dn(x−xn)= 0,

(4.36)

and then using tangency conditions (4.6) on (4.36), one gets

an =Cf ′′(xn)+α2 f (xn)−2α f ′(xn)

{ f ′(xn)−α f (xn)}2 ,

bn =− f ′′(xn)+α2 f (xn)−2α f ′(xn)2{ f ′(xn)−α f (xn)}2 ,

dn =−{ f ′(xn)−α f (xn)}.

. (4.37)

At a root

y(xn+1) = 0, (4.38)

and it follows from (4.36) that

xn+1 = xn−{

1+12

L∗f (xn)+C{

L∗f (xn)}2

}f (xn)

f ′(xn)−α f (xn). (4.39)

This is a modification over the Amat et al. C-methods [30, 71] and do not fail even if f ′(xn)

is very small or zero in the vicinity of required root. It can be verified that in the absence of

the term in parameter ‘α’, method (4.39) reduces to Amat et al. C-methods [30, 71].

A mathematical proof for an order of convergence of the proposed iterative scheme (4.39)

is given below :




defined by formula (4.39) is cubically convergent and satisfies the following error equation:

en+1 ={

(2−4C)C22 −C3 +(−3+8C)

(C2α− α2

2

)}e3

n +O(e4n). (4.40)

100

Proof. Using equations (4.29)-(4.31), one gets


= en− (C2−α)e2n−

(2C3−2C2

2 +2αC2−α2)e3n +O

(e4

n), (4.41)

and

L∗f (xn) =f (xn)

{f ′′(xn)+α2 f (xn)−2α f ′(xn)

}{

f ′(xn)−α f (xn)}2 =2(C2−α)en− (6C2

2 −6C3−6C2α+3α2)e2n

+O(e3n). (4.42)

Substituting equations (4.41) and (4.42) in formula (4.39), one can obtain

en+1 ={

(2−4C)C22 −C3 +(−3+8C)

(C2α− α2

2

)}e3

n +O(e4n). (4.43)


4.2.3 Derivation from a general quadratic curve

Let us consider a following general quadratic equation :

(x− xn)2 +an

(ye−α(x−xn)− f (xn)

)2+bn(x− xn)+ cn

(ye−α(x−xn)− f (xn)

)+dn = 0,

(4.44)

and determine the unknowns bn, cn and dn in the terms of parameter ‘an’ by tangency condi-

tions (4.6), one can have

bn =2[1+an{ f ′(xn)−α f (xn)}2]{ f ′(xn)−α f (xn)}

f ′′(xn)+α2 f (xn)−2α f ′(xn),

cn =− 2[1+an{ f ′(xn)−α f (xn)}2]

f ′′(xn)+α2 f (xn)−2α f ′(xn),

dn =0.

. (4.45)

Taking next iterate ‘xn+1’ as a point of intersection of (4.44) with x−axis, one gets another

two-parameter family of iteration scheme as

xn+1 = xn−{


}× D1

D2, (4.46)

101

where

D1 = 2+an{

f ′(xn)−α f (xn)}2 (

2+L∗f (xn)), (4.47)

and

D2 = 1+an{

f ′(xn)−α f (xn)}2±

√{1+an { f ′(xn)−α f (xn)}2

}2−L∗f (xn) D1. (4.48)

This is a modification over formula (20) of Sharma [32] and do not fail even if f ′(xn) is very

small or zero in the vicinity of required root. Iterative family (4.46) requires computation of

square root at each step and sometimes is not convenient for practical usage. Therefore, it

can be further reduced to two parameter rational iterative family(similar to Chun [31]

), or

by using the well-known approximation(similar to Jiang and Han [61]

)

√1+ x≈ 1+

12

x, |x|< 1,

and is given by

xn+1 = xn−2[1+an{ f ′(xn)−α f (xn)}2]D1

4[1+an{ f ′(xn)−α f (xn)}2

]2−L∗f (xn)D1


. (4.49)

Special cases:

For particular values of ‘an’, some interesting particular cases of (4.46) are

(i) For an = 0, one can obtain

xn+1 = xn− 2

1±√

1−2L∗f (xn)


. (4.50)

This is a modification over the well-known cubically convergent Euler’s method [7, 30].

(ii) Taking an = 1, one gets


D∗1

D∗2, (4.51)

where

D∗1 = 2+

{f ′(xn)−α f (xn)

}2 (2+L∗f (xn)

), (4.52)

and

D∗2 = 1+

{f ′(xn)−α f (xn)

}2±√{

1+{ f ′(xn)−α f (xn)}2}2−L∗f (xn) D∗

1. (4.53)

This is a modification over the cubically convergent osculating circle method derived

by Kanwar et al. [33].

102

(iii) Letting an = 2{ f ′(xn)}2

(2−L∗f (xn)

) , one can have

xn+1 = xn− 22−L∗f (xn)


. (4.54)

This is a modification over the well-known cubically convergent Halley’s method.

(iv) Substituting an =L∗f (xn)

4{ f ′(xn)}2(

1−L∗f (xn)) , one gets

xn+1 = xn−{

1+12

L∗f (xn)

1−L∗f (xn)

}f (xn)

f ′(xn)−α f (xn). (4.55)

This is a modification over the well-known cubically convergent super-Halley method.

(v) If an →±∞, one gets

xn+1 = xn−{

1+12

L∗f (xn)}


. (4.56)

This is a modification over the well-known cubically convergent Chebyshev’s method.

(vi) For an =−1/{ f ′(xn)}2, one can get


. (4.57)

This is a modification over the well-known quadratically convergent Newton’s method.

It can be verified that in the absence of the term in parameter ‘α’, method (4.50), method

(4.51), method (4.54), method (4.55), method (4.56) and method (4.57) reduce to Euler’s

method [7, 30, 64], osculating circle method [33], Halley’s method [60, 30–32, 61–73],

super-Halley method [30–32, 61, 71–73], Chebyshev’s method [7, 30–32, 59–61] and New-

ton’s method [7, 11–16] respectively. The sign of parameter ‘α’ in all these methods is

chosen so that the denominator is largest in magnitude.

Next, Theorem 4.2.3 presents a mathematical proof for an order of convergence of the

proposed iterative scheme (4.46).

103


open interval I, enclosing a simple zero of f (x) (say x = r ∈ I). Assume that initial guess x =

x0 is sufficiently close to ‘r’ and f ′(xn)−α f (xn) 6= 0 in I. Then an iteration scheme defined

by formula (4.46) is cubically convergent for any finite ‘an’ except an = −1/{ f ′(xn)}2 (in

that case it has quadratic convergence) and satisfies the following error equation :

en+1 =

{(2C2−α

)α+an{ f ′(r)}2(4C2

2 −6C2α+3α2)

2(1+an{ f ′(r)}2

) −C3

}e3

n +O(e4

n). (4.58)

Proof. Using equations (4.29)-(4.31), one can get


= en− (C2−α)e2n−

(2C3−2C2

2 +2αC2−α2)e3n +O

(e4

n), (4.59)

and

L∗f (xn) =f (xn)

{f ′′(xn)+α2 f (xn)−2α f ′(xn)

}{

f ′(xn)−α f (xn)}2 =2(C2−α)en−

(6C2

2 −6C3−6C2α+3α2)e2n

+O(e3n). (4.60)

Using equations (4.29), (4.30) and (4.60), one can have

D1 = 2(1+an{ f ′(r)}2)+2an{ f ′(r)}2(5C2−3α

)en +O(e2

n), (4.61)

and

D2 = 2(1+an{ f ′(r)}2)+

(4an{ f ′(r)}2(2C2−α

)−2(C2−α))en +O(e2

n). (4.62)

Substituting equations (4.59), (4.61) and (4.62) in formula (4.46), one gets

en+1 =

{(2C2−α

)α+an{ f ′(r)}2(4C2

2 −6C2α+3α2)

2(1+an{ f ′(r)}2

) −C3

}e3

n +O(e4

n). (4.63)


Alternatively, all these modified techniques((4.21), (4.39) and (4.46)

)can also be ob-

tained by using the idea of Ben-Israel [211]. By applying the existing classical techniques to

a modified function, say f (x) := e−∫

a(x)dx f (x), for some suitable function f (x), one can get

the corresponding modified iterative schemes.

104

4.2.4 Worked examples

Now, let us make an attempt to present some numerical results obtained by employing ex-

isting classical methods namely, Chebyshev’s method (CM), Halley’s method (HM), super-

Halley method (SHM), Euler’s method (EM), osculating circle method (OCM) and newly

developed methods namely, modified Chebyshev’s method (MCM) (4.23), modified Hal-

ley’s method (MHM) (4.24), modified super-Halley method (MSHM) (4.25), modified Eu-

ler’s method (MEM) (4.50) and modified osculating circle method (MOCM) (4.51) respec-

tively to solve nonlinear equations given in Table 4.1. Here, for simplicity, the formulae are

tested for |α|= 1. The results are summarized in Table 4.2 (Number of iterations), Table 4.3

(Computational order of convergence) and Table 4.4 (Total number of function evaluations)

respectively. Computations have been performed using MAT LABr version 7.5 (R2007b) in

double precision arithmetic. We use ε = 10−15 as a tolerance error. The following stopping

criteria are used for computer programs :

(i)|xn+1− xn|< ε, (ii)| f (xn+1)|< ε.

105

Table 4.1

Test examples, interval (I) containing the root, initial guesses (x0) and exact root (r)

Example No. Examples I Initial guesses Exact root (r)

0.44.2.1 exp(1− x)−1 = 0. [0,2]

1.51.000000000000000

4.2.2 x exp(−x) = 0. [1,3] 2 0.000000000000000

4.2.3 sin(x) = 0. [0,2] 1.5 0.000000000000000

4.2.4 exp(−x)− sin(x) = 0. [5,7] 5 6.285049273382587

4.2.5 arctan(x) = 0. [−1,2] 2 0.000000000000000

04.2.6 x3 +4x2−10 = 0. [0,2]

0.11.3652300134140907

-1

4.2.7 x exp(−x)−0.1 = 0. [−2,2] 0.8 0.11832559158630

1

2.94.2.8 exp(x2 +7x−30)−1 = 0. [2,4]

3.153.000000000000000

-14.2.9 cos(x)− x = 0. [−2,2]

10.7390851332151607

106

Table 4.2

Number of iterations (D and F below stand for divergent and failure respectively)

Example No. CM MCM HM MHM SHM MSHM EM MEM OCM MOCM

(4.23) (4.24) (4.25) (4.50) (4.51)

4 4 3 3 3 3 4 3 3 34.2.1

4 3 3 3 3 3 3 3 3 3

4.2.2 D 1 D 1 D 1 D 1 D 1

4.2.3 5∗ 5 6 5 4∗ 4 4 5 4 5

4.2.4 11∗ 5 5 4 4 4 4 4 4 4

4.2.5 D 5 4 5 5 D 5 6 4 5

F 4 F 4 F 4 F 4 F 44.2.6

46 4 6 4 5 4 4 4 21 4

5 3 4 3 5 3 6 3 5 3

4.2.7 D 3 5 3 5 3 4 3 4 3

F 3 F 3 F 3 F 3 F 3

D 5 4 4 4 4 4 4 D 54.2.8

5 5 4 4 6 5 6 6 5 5

D 5 5 4 8 4 4 5 4 54.2.9

3 4 3 3 3 3 3 3 3 4

∗ Converges to undesired root.

107

Table 4.3

Computational order of convergence (COC)

Example No. CM MCM HM MHM SHM MSHM EM MEM OCM MOCM

(4.23) (4.24) (4.25) (4.50) (4.51)

3.00 3.02 2.94 2.99 3.01 3.01 2.99 2.89 3.19 3.144.2.1

3.01 2.97 2.82 2.99 3.00 3.00 2.92 3.25 2.57 2.91

4.2.2 . . . ≡≡ . . . ≡≡ . . . ≡≡ . . . ≡≡ . . . ≡≡4.2.3 — 2.96 3.01 2.99 — 2.44 3.00 3.00 3.06 2.99

4.2.4 — 2.98 3.00 2.85 2.92 2.76 3.00 3.01 3.03 2.88

4.2.5 . . . 2.89 2.97 2.99 2.91 . . . 3.00 3.00 2.81 2.99

== 2.90 == 2.96 == 2.86 == 2.96 == 2.904.2.6

3.02 2.92 3.01 2.97 2.81 2.91 3.00 2.97 2.88 2.92

2.98 2.76 2.92 2.78 2.99 2.81 3.03 2.81 3.01 2.80

4.2.7 — 3.28 3.01 3.19 3.05 3.10 3.04 3.11 3.06 3.11

== 3.06 == 3.31 == 3.13 == 3.16 == 3.15

. . . 3.01 3.00 3.01 3.00 3.01 3.04 3.03 . . . 3.014.2.8

2.99 2.98 3.00 2.97 3.00 3.02 3.00 3.09 2.99 2.97

. . . 2.98 3.05 2.89 3.01 3.13 3.04 3.00 3.02 3.004.2.9

2.41 3.01 2.52 2.89 3.03 2.86 3.03 3.17 2.96 2.99

Here, the symbols namely

(i) ‘—’ represents COC for a case of undesired root.

(ii) ‘. . . ’ represents COC for a case of divergence.

(iii) ‘==’ represents COC for a case of failure.

(iv) ‘≡≡’ represents COC for a case when number of iterations are not sufficient.

108

Table 4.4

Total number of function evaluations (TNOFE)

Example CM MCM HM MHM SHM MSHM EM MEM OCM MOCM

No. (4.23) (4.24) (4.25) (4.50) (4.51)

12 12 9 9 9 9 12 9 9 94.2.1

12 9 9 9 9 9 9 9 9 9

4.2.2 ∼ 3 ∼ 3 ∼ 3 ∼ 3 ∼ 3

4.2.3 _ 15 18 15 _ 12 12 15 12 15

4.2.4 — 15 15 12 12 12 12 12 12 12

4.2.5 ∼ 15 12 15 15 ∼ 15 18 12 15

= 12 = 12 = 12 = 12 = 124.2.6

138 12 18 12 15 12 12 12 63 12

15 9 12 9 15 9 18 9 15 9

4.2.7 ∼ 9 15 9 15 9 12 9 12 9

= 9 = 9 = 9 = 9 = 9

∼ 15 12 12 12 12 12 12 ∼ 154.2.8

15 15 12 12 18 15 18 18 15 15

∼ 15 15 12 24 12 12 15 12 154.2.9

9 12 9 9 9 9 9 9 9 12

Here, the symbols namely

(i) ‘ _’ represents TNOFE for a case of undesired root.

(ii) ‘∼’ represents TNOFE for a case of divergence.

(iii) ‘ =’ represents TNOFE for a case of failure.

109

4.3 Generalized families of multipoint iterative methods

without memory

The practical difficulty associated with third-order methods given by formula (4.21) may be

an evaluation of second-order derivative. In past and recent years, many new modifications

of Newton’s method free from second-order derivative have been proposed by researchers

[7, 27, 34, 35, 82, 9, 47–55, 57, 58, 75–81, 87–109] by discretization of second-order deriva-

tive or by considering different quadrature formulae for computation of an integral arising

from Newton’s theorem [8, 52–55] or by considering Adomian decomposition method [39–

44]. All these modifications are targeted at increasing a local order of convergence with the

view of increasing their efficiency index [13]. Some of these multipoint iterative meth-

ods are of third-order [7, 52–55, 57, 58, 75–81, 87–95], while others are of order four

[7, 34, 35, 82, 9, 47–51, 96–109]. Recently, fourth-order methods proposed by Nedzhi-

bov et al. [101], Basu [102], Maheswari [103], Cordero et al. [104], Petkovic and Petkovic

(a survey) [105], Ghanbari [106], Soleymani [47, 107], Cordero and Torregrosa [108], Chun

et al. [109] etc. have optimal order of convergence [9]. The efficiency index [13] of these

optimal multipoint iterative methods is equal to 3√

4≈ 1.587. These multipoint methods cal-

culate new approximations to a zero of f (x) by sampling f (x) and possibly its derivatives for

a number of values of independent variable, per step. Nedzhibov et al. [101] have modified

Chebyshev-Halley type methods [74] to derive several cubic and quartic order convergent

multipoint iterative methods free from second-order derivative.

Since all these techniques are variants of Newton’s method and main practical difficulty

associated with these multipoint iterative methods is that an iteration can be aborted due to

the overflow or leads to divergence if the derivative of the function at any iterative point is

singular or almost singular (namely f ′(xn)≈ 0).

Here, it is intended to develop and unify the class of third-order and fourth-order

multipoint iterative methods free from second-order derivative in which f ′(xn) = 0 is

permitted. The main idea of proposed methods lies in discretization of second-order deriva-

tive involved in family (4.21) (similar to Nedzhibov et al. [101]). Therefore, again this work

can be viewed as a generalization over Nedzhibov et al. [101] families of multipoint iter-

110

ative methods. An additional feature of these processes is that they may possess a number

of disposable parameters which can be used to ensure the convergence is to a certain order

for simple roots. This section, deals with the derivation of three different families free from

second-order derivative involved in the family (4.21).

4.3.1 First Family

Let us take u ≡ u(xn) = f (xn)f ′(xn)−α f (xn)

, and expanding the function f (xn− u) about a point

x = xn with f (xn) 6= 0 by Taylor’s series expansion, one can have

f (xn−u) = f (xn)−u f ′(xn)+u2

2f ′′(xn)+O(u3). (4.64)

Inserting the value of ‘u’ defined above in the coefficient of f ′(xn), one can obtain

f (xn−u)( f ′(xn)−α f (xn))+α{ f (xn)}2

f ′(xn)−α f (xn)=

u2

2f ′′(xn)+O(u3). (4.65)

Further, L∗f (xn) from equation (4.22) may be written as follows :

L∗f (xn) =f (xn) f ′′(xn)

{ f ′(xn)−α f (xn)}2 −α f (xn)

f ′(xn)−α f (xn)− α f (xn) f ′(xn){ f ′(xn)−α f (xn)}2 . (4.66)

Taking the approximate value of f ′′(xn) from formula (4.65), one can simplify formula of

L∗f (xn) given in (4.66) as

L∗f (xn)≈ 2 f (xn−u)f (xn)

−α2{


}2

. (4.67)

If |u| is sufficiently small, then the second term in above equation can be neglected and one

obtains

L∗f (xn)∼= 2 f (xn−u)f (xn)

. (4.68)

Substituting this approximate value of L∗f (xn) in (4.21), a following corresponding multipoint

family is obtained

xn+1 = xn−

1+f (xn−u)

f (xn)[

1+bn

(f (xn)

f ′(xn)−α f (xn)

)]


. (4.69)

111

This is a modification over the formula (2.1) of Nedzhibov et al. [101] and do not fail

even if f ′(x) is very small or zero in the vicinity of required root. For different specific

values of ‘bn’, the following various families of multipoint iterative methods can be derived

from (4.69), e.g.

(i) For bn = 0, one gets formula


− f (xn−u)f ′(xn)−α f (xn)

. (4.70)

This is a modification over the well-known cubically convergent Traub’s method [7,

pp. 168].

(ii) Letting bn =−L∗f (xn)2u , one can obtain

xn+1 = xn− { f (xn)}2

{ f ′(xn)−α f (xn)}{ f (xn)− f (xn−u)} . (4.71)

This is a modification over the well-known cubically convergent Newton-Secant method

[7, pp. 180].

(iii) Substituting bn =−L∗f (xn)u , one obtains


[f (xn)− f (xn−u)

f (xn)−2 f (xn−u)

]. (4.72)

This is a modification over the well-known quartically convergent Traub-Ostrowski’s

method [7, 82, 101].

It can be seen that in absence of the term in parameter ‘α’, method (4.70), method (4.71)

and method (4.72) reduce to Traub’s method [7, pp. 168], Newton-Secant method [7, pp.

180] and Traub-Ostrowski’s method [7, 82, 101] respectively. The sign of parameter ‘α’ in

all these families is chosen so that the denominator is largest in magnitude.

A mathematical proof for an order of convergence of the proposed iterative family (4.69)

is given in following theorem :

112




defined by formula (4.69), is cubically convergent for bn ∈ R and satisfies the following

error equation :

en+1 = (C2−α)(bn +2C2−α)e3n +O(e4

n). (4.73)




n +O(e4n)c, (4.74)

and

f ′(xn) = f ′(r)b1+2C2en +3C3e2n +O(e3

n)c. (4.75)

From equations (4.74) and (4.75), one gets


= en +(−C2 +α)e2n +O(e3

n). (4.76)

Further, one can get

f (xn−u) = f ′(r)b(C2−α)e2n +O(e3

n)c. (4.77)

Using equations (4.74)-(4.77), one can have

f (xn−u)

f (xn)[

1+bn

(f (xn)


)] = (C2−α)en +((α−C2)(bn +3C2)+2C3−α2)e2

n +O(e3n).

(4.78)

Substituting equations (4.76) and (4.78) in formula (4.69), one gets

en+1 = (C2−α)(bn +2C2−α)e3n +O(e4

n). (4.79)


113

4.3.2 Second family

Replacing second-order derivative in (4.22) by the finite difference between first-order deriva-

tives as

f ′′(xn)≈ f ′(xn)− f ′(xn−θu)θu

,θ ∈ R−{0}, (4.80)

where u is as defined earlier.

One obtains

L∗f (xn)≈ f ′(xn)− f ′(xn−θu)θ{ f ′(xn)−α f (xn)} +α2

{f (xn)


}2

− 2α f ′(xn) f (xn){ f ′(xn)−α f (xn)}2 . (4.81)

If |u| is sufficiently small, then the second term in above equation can be neglected and then

one can get

L∗f (xn)∼= f ′(xn)− f ′(xn−θu)θ{ f ′(xn)−α f (xn)} −

2α f ′(xn) f (xn){ f ′(xn)−α f (xn)}2 . (4.82)

Substituting this approximate value of L∗f (xn) in (4.21), a following multipoint family is

obtained

xn+1 = xn−{

1+12

(f ′(xn)− f ′(xn−θu)

)−2αθu f ′(xn)θ(

f ′(xn)+(bn−α) f (xn))

}f (xn)

f ′(xn)−α f (xn). (4.83)

The formula (4.83) is a modification over formula (2.5) given by Nedzhibov et al. [101] and

do not fail even if f ′(xn) = 0 or very small in the vicinity of required root. For particular

values of ‘bn’ and ‘θ’, some particular cases of formula (4.83) are

(i) For bn =−L∗f (xn)2u and θ = 1, one gets

xn+1 = xn− 2 f (xn)

f ′(xn)+ f ′(

xn− f (xn)f ′(xn)−α f (xn)

)+2α2 f (xn) f ′(xn)


. (4.84)

This is a modification over the well-known cubically convergent Weerakoon and Fer-

nando method [8].

(ii) Putting bn = −L∗f (xn)2u and θ = −1, one can get a following new cubically convergent

family of multipoint iterative methods

xn+1 = xn− 2 f (xn)

3 f ′(xn)− f ′(

xn + f (xn)f ′(xn)−α f (xn)

)+2α2 f (xn) f ′(xn)


. (4.85)

114

(iii) Substituting bn =−L∗f (xn)2u and θ =−1

2 , one can get an another new cubically convergent

family of multipoint iterative methods

xn+1 = xn− f (xn)

2 f ′(xn)− f ′(

xn + f (xn)2{ f ′(xn)−α f (xn)}

)+α2 f (xn) f ′(xn)


. (4.86)

(iv) Letting bn =−L∗f (xn)2u and θ = 1

2 , one gets

xn+1 = xn− f (xn)

f ′(

xn− f (xn)2{ f ′(xn)−α f (xn)}

)+α2 f (xn) f ′(xn)


. (4.87)

This is a modification over the cubically convergent method derived by Traub [7, pp.

182].

(v) Taking bn =−L∗f (xn)u and θ = 2

3 , one can obtain


f ′(

xn− 23u

)− f ′(xn)+ 4

3{ f ′(xn)}2

f ′(xn)−α f (xn)− 4

3α f (xn)

2 f ′(

xn− 23u

)−2 f ′(xn)+ 4

3{ f ′(xn)}2+α2{ f (xn)}2


.

(4.88)

This is a modification over the well-known quartically convergent Jarratt’s method [34].

It is easy to observe that in absence of the term in parameter ‘α’, method (4.84), method

(4.85), method (4.86), method (4.87) and method (4.88) reduce to Weerakoon and Fernando

method [8], third-order multipoint iterative method derived by Traub [7, pp. 182] and Jar-

ratt’s method [34, 35] respectively. Here, the parameter ‘α’ should be chosen so that the

denominator is largest in magnitude.


is presented in the next theorem.

115



x = x0 is sufficiently close to ‘r’ and f ′(xn)− f (xn) 6= 0 in I. Then an iteration scheme defined

by formula (4.83) is cubically convergent and satisfies the following error equation :

en+1 =(

2C22 −

12

C3(2−3θ)+bn(C2−α)−3C2α+2α2)

e3n +O(e4

n). (4.89)




n +O(e4n)c, (4.90)

and

f ′(xn) = f ′(r)b1+2C2en +3C3e2n +O(e3

n)c. (4.91)



= en +(−C2 +α)e2n +O(e3

n). (4.92)

Further, one can get

f ′(xn−θu) = f ′(r)b1+2C2(1−θ)en +(3C3(1−θ)2 +2C2(C2−α)θ

)e2

n +O(e3n)c. (4.93)

Using equations (4.90)-(4.93), one can obtain

(1−2αθu) f ′(xn)− f ′(xn−θu)2θ( f ′(xn)+(bn−α) f (xn))

=(−α+

C2

θ

)en +

12θ

(3C3 +2(αθ−C2)(bn +C2−2α)−4C2

2)e2

n

+O(e3n). (4.94)

Substituting equations (4.92) and (4.93) in formula (4.83), one gets

en+1 =(

2C22 −

12

C3(2−3θ)+bn(C2−α)−3C2α+2α2)

e3n +O(e4

n). (4.95)

Proof of the theorem is now complete.

116

4.3.3 Third family

Replacing second-order derivative in (4.22) by the finite difference as

f ′′(xn)≈5 f ′(xn)−4 f ′

(xn− u

2

)− f ′(xn−u)3u

, (4.96)

where u is as defined earlier. One obtains

L∗f (xn)≈5 f ′(xn)−4 f ′

(xn− u

2

)− f ′(xn−u)3{ f ′(xn)−α f (xn)} +α2

{f (xn)


}2

− 2α f ′(xn) f (xn){ f ′(xn)−α f (xn)}2 .

(4.97)

If |u| is sufficiently small, then the second term in above equation can be neglected and one

can obtain

L∗f (xn)∼=5 f ′(xn)−4 f ′

(xn− u

2

)− f ′(xn−u)3{ f ′(xn)−α f (xn)} − 2α f ′(xn) f (xn)

{ f ′(xn)−α f (xn)}2 . (4.98)

Substituting this approximate value of L∗f (xn) in (4.21), the following multipoint family is

obtained

xn+1 = xn−{

1+

(5 f ′(xn)−4 f ′

(xn− u

2

)− f ′(xn−u))−6αu f ′(xn)

6(

f ′(xn)+(bn−α) f (xn))

}f (xn)

f ′(xn)−α f (xn).

(4.99)

Then for bn =−L∗f (xn)2u in (4.99), one gets

xn+1 = xn− 6 f (xn)

f ′(xn)+4 f ′(

xn− f (xn)2{ f ′(xn)−α f (xn)}

)+ f ′

(xn− f (xn)


)+6αu f ′(xn)

.

(4.100)

This is a modification over the cubically convergent formula explored by Hasanov et al.

[101, 230], which do not fail even if f ′(xn) = 0 in the vicinity of required root. Here, the

parameter ‘α’ should be chosen such that the denominator is largest in magnitude. It can be

observed that in absence of the term in parameter ‘α’, the family (4.100) reduces to Hasanov

et al. method [101, 230].

Next, Theorem 4.3.3 presents a mathematical proof for an order of convergence of the

proposed iterative family (4.99).

117



x = x0 is sufficiently close to ‘r’ and f ′(xn)−α f (xn) 6= in I. Then an iteration scheme defined

by formula (4.99) is cubically convergent and satisfies the following error equation :

en+1 =(2C2

2 +bn(C2−α)−3C2α+2α2)e3n +O(e4

n). (4.101)




n +O(e4n)c, (4.102)

and

f ′(xn) = f ′(r)b1+2C2en +3C3e2n +O(e3

n)c. (4.103)



= en +(−C2 +α)e2n +O(e3

n). (4.104)

Further, one can have

f ′(xn−u) = f ′(r)b1+2C2(C2−α)e2n +O(e3

n)c, (4.105)

and

f ′(

xn− u2

)= f ′(r)

⌊1+C2en +

(3C3

4+C2(C2−α)

)e2

n +O(e3n)

⌋. (4.106)

Using equations (4.102)-(4.106), one can obtain

(5−6αu) f ′(xn)−4 f ′(xn− u

2

)− f ′(xn−u)6( f ′(xn)+(bn−α) f (xn))

=(C2−α)en +(2C3−2α2− (C2−α)(bn +3C2)

)e2

n

+O(e3n). (4.107)

Substituting equations (4.104) and (4.107) in the formula (4.99), one can get

en+1 =(2C2

2 +bn(C2−α)−3C2α+2α2)e3n +O(e4

n). (4.108)


On the similar lines, one can further derive some more new cubically convergent families

of multipoint iterative methods free from second-order derivative by discretizing the second-

order derivative involved in families (4.39) and (4.46).

118

4.3.4 Further exploration of family (4.69)

One can also construct a different family of fourth-order convergent multipoint iterative

methods by suitably choosing parameter ‘bn’. Again consider the family (4.69)

xn+1 = xn−

1+f (xn−u)

f (xn)[

1+bn

(f (xn)


)]


, (4.109)

where α ∈ R and u = f (xn)f ′(xn)−α f (xn)

.

Taking bn = −2C2 + α ≈ − f ′′(xn)f ′(xn)

+ α, in an error equation (4.79), the family (4.109) has at

least fourth-order convergence, i.e. after simplifications, one gets new fourth-order multi-

point iterative family given by


− f (xn−u) f ′(xn){ f ′(xn)}2− f (xn) f ′′(xn)

. (4.110)

Further, one can derive another fourth-order multipoint iterative family free from second-

order derivative by using (4.65) and is given by


[1+

f (xn−u) f ′(xn)u{ f ′(xn)}2−2α{ f (xn)}2−2 f (xn−u)( f ′(xn)−α f (xn))

].

(4.111)

This is a modification over the well-known Traub-Ostrowski’s method [7, 82] and is also

an optimal fourth-order method. This family requires same evaluations of the function and

its derivative as that of Traub-Ostrowski’s method per iteration. Thus, an efficiency index

[13] of this method is equal to 3√

4∼= 1.587, which is better than the one of Newton’s method2√

2∼= 1.414 and method (4.72) (derived in this chapter) 3√

3∼= 1.442. More importantly, this

method will not fail even if the derivative of the function is very small or even zero in the

vicinity of required root, unlike Traub-Ostrowski’s method [1, 43]. Here, the parameter ‘α’

should be chosen so that denominator is largest in magnitude.

One can see in absence of the term in parameter ‘α’, the family (4.111) reduces to well-

known Traub-Ostrowski’s method [7, 82].


is given by next theorem.

119



x = x0 is sufficiently close to ‘r’ and f ′(xn)−α f (xn) 6= in I. Then an iteration scheme defined

by formula (4.111) has optimal order of convergence four and satisfies the following error

equation :

en+1 = (C2−α)(C22 −C3−αC2 +2α4)e4

n +O(e5n). (4.112)




n +O(e4n)c, (4.113)

and

f ′(xn) = f ′(r)b1+2C2en +3C3e2n +4C4e3

n +O(e4n)c. (4.114)



= en +(−C2 +α)e2n +(2C2

2 −2C3−2αC2 +α2)e3n +O(e4

n). (4.115)

Further, one can obtain

f (xn−u) = f ′(r)b(C2−α)e2n +(−2C2

2 +2C3 +2αC2−α2)e3n +O(e4

n)c. (4.116)

Using equations (4.113), (4.114) and (4.116), one gets

f (xn−u) f ′(xn)u{ f ′(xn)}2−2α{ f (xn)}2−2 f (xn−u)( f ′(xn)−α f (xn))

=(C2−α)en +(−C22 +C3)e2

n

+(−2C2C3 +3C4 +2C2

2α−C3α

−3C2α2 +2α3)e3n +O(e4

n).

(4.117)

Substituting equations (4.115) and (4.117) in formula (4.111) and simplifying, one obtains

en+1 = (C2−α)(C22 −C3−αC2 +2α4)e4

n +O(e5n). (4.118)


120

4.3.5 Worked examples

Here, an attempt has been made to present some numerical results obtained by employ-

ing existing classical methods namely, Steffensen’s method (SM), Traub’s method (T M),

Newton-Secant method (NSM), Weerankoon and Fernando method (WFM), Hasanov et al.

method (HAM), Traub-Ostrowski’s method (TOM), Jarratt’s method (JM) and newly devel-

oped methods namely, modified Traub’s method (MT M) (4.70), modified Newton-Secant

method (MNSM) (4.71), modified Weerankoon and Fernando method (MWFM) (4.84),

modified Hasanov et al. method (MHAM) (4.100), modified Traub-Ostrowski’s method

(MTOM) (4.111) and modified Jarratt’s method (MJM) (4.88) respectively to solve nonlin-

ear equations given in Table (4.5). Here, for simplicity, the formulae are tested for |α| = 1.

The results are summarized in Table 4.6 (Number of iterations), Table 4.7 (Computational

order of convergence) and Table 4.8 (Total number of function evaluations) respectively.

Computations have been performed using MAT LABr version 7.5 (R2007b) in double pre-

cision arithmetic. We use ε = 10−15 as a tolerance error. The following stopping criteria are

used for computer programs :

(i)|xn+1− xn|< ε, (ii)| f (xn+1)|< ε.

121

Table 4.5

Test examples, interval (I) containing the root, their initial guesses (x0) and exact

root (r)

Example No. Examples I Initial guesses Exact root

0.44.3.1 exp(1− x)−1 = 0. [0,2]

1.51.0000000000000000

1.54.3.2 sin(x) = 0. [0,2]

1.510.0000000000000000

4.3.3 exp(−x)− sin(x) = 0. [5,7] 5 6.285049273382587

-1

4.3.4 arctan(x) = 0. [−1,2] -2 0.000000000000000

2

0

4.3.5 x3 +4x2−10 = 0. [0,2] 0.1 1.365230013414097

2

14.3.6 (x−1)3−1 = 0. [0,3]

32.000000000000000

2.934.3.7 exp(x2 +7x−30)−1 = 0. [2,4]

3.33.000000000000000

-14.3.8 cos(x)− x = 0. [−1,2]

20.7390851332151607

4.3.9 log(x) = 0. [.5,3.5] 3 1.000000000000000

122

Tabl

e4.

6

Num

ber

ofite

ratio

ns(D

and

Fbe

low

stan

dsfo

rdi

verg

enta

ndfa

ilure

resp

ectiv

ely)

Exa

mpl

eSM

TM

MT

MN

SMM

NSM

WFM

MW

FMH

AM

MH

AM

TOM

MTO

MJM

MJM

No.

(4.7

0)(4

.71)

(4.8

4)(4

.100

)(4

.111

)(4

.88)

44

44

44

44

43

33

44.

3.1

34

34

44

34

43

33

4

44∗

43∗

44

43

43∗

54∗

54.

3.2

43

45∗

47

43

44∗

53∗

5

4.3.

33

54

44

9∗4

34

44

44

D4

43

34

43

33

43

4

4.3.

4D

D4

46

D5

36

55

D5

DD

44

6D

53

65

5D

5

13F

4F

5F

5F

5F

4F

5

4.3.

538

644

75

85

75

54

55

184

44

43

43

43

43

4

1F

1F

1F

5F

1F

1F

54.

3.6

165

54

54

54

53

43

4

DD

64

44

44

43

33

34.

3.7

D6

76

66

66

64

44

4

58

44

43

54

510

55

54.

3.8

53

53

43

43

43

53

4

4.3.

9D

D5

D6

D5

D6

D5

35

∗C

onve

rges

toun

desi

red

root

123

Tabl

e4.

7

Com

puta

tiona

lord

erof

conv

erge

nce

(CO

C)

Exa

mpl

eSM

TM

MT

MN

SMM

NSM

WFM

MW

FMH

AM

MH

AM

TOM

MTO

MJM

MJM

No.

(4.7

0)(4

.71)

(4.8

4)(4

.100

)(4

.111

)(4

.88)

3.00

2.99

2.85

2.99

2.98

3.00

2.98

2.99

2.98

3.96

3.78

3.95

2.88

4.3.

12.

983.

033.

922.

972.

993.

002.

423.

003.

004.

034.

114.

033.

00

5.05

—2.

97—

3.94

3.00

2.85

4.29

3.94

—3.

98—

3.96

4.3.

25.

042.

902.

97—

3.94

3.00

2.84

5.05

3.94

—3.

982.

803.

96

4.3.

35.

435.

862.

934.

313.

582.

972.

865.

383.

585.

353.

685.

463.

66

...

4.99

2.93

4.52

3.66

2.98

2.92

4.55

3.67

4.73

3.84

4.79

3.86

4.3.

4..

...

.2.

524.

913.

97..

.3.

123.

313.

974.

873.

78..

.3.

80

...

...

2.52

4.91

3.97

...

3.12

3.31

3.97

4.87

3.78

...

3.80

2.00

==2.

88==

2.98

==2.

98==

2.98

==3.

90==

3.02

4.3.

52.

002.

992.

923.

162.

993.

082.

993.

162.

993.

703.

923.

703.

01

2.00

2.99

2.84

2.94

2.98

2.78

2.99

2.80

2.98

3.39

3.94

3.83

2.96

≡≡==

≡≡==

≡≡==

2.99

==≡≡

==≡≡

==3.

004.

3.6

1.97

3.00

2.83

2.89

2.99

2.87

2.82

2.89

2.99

3.26

3.40

3.26

2.61

...

...

2.84

3.03

3.02

3.07

3.01

3.03

3.02

4.09

4.09

4.09

2.50

4.3.

7..

.2.

862.

992.

982.

972.

952.

992.

982.

973.

733.

743.

692.

91

2.00

3.11

2.92

3.12

2.20

2.75

2.99

2.44

3.00

2.53

3.96

3.48

3.14

4.3.

82.

002.

242.

982.

333.

023.

262.

992.

193.

003.

394.

003.

463.

10

4.3.

9..

...

.3.

77..

.3.

00..

.2.

98..

.3.

00..

.3.

983.

532.

95

Her

e,th

esy

mbo

lsus

ed,a

real

read

yde

fined

inth

isch

apte

rund

erth

eTa

ble

4.3.

124

Tabl

e4.

8

Tota

lnum

ber

offu

nctio

nev

alua

tions

(TN

OFE

)

Exa

mpl

eSM

TM

MT

MN

SMM

NSM

WFM

MW

FMH

AM

MH

AM

TOM

MTO

MJM

MJM

No.

(4.7

0)(4

.71)

(4.8

4)(4

.100

)(4

.111

)(4

.88)

1212

1212

1212

1212

1212

1212

164.

3.1

912

912

1212

912

129

99

16

8_

12_

1212

1212

16_

15_

154.

3.2

89

12_

1221

1212

16_

15_

15

4.3.

36

1512

1212

2712

1216

1212

1212

∼12

129

912

1212

129

129

12

4.3.

4∼

∼12

1218

∼15

1224

1515

∼15

∼∼

1212

18∼

1512

2415

15∼

15

26=

12=

15=

15=

20=

12=

15

4.3.

576

192

1221

1524

1528

2015

1215

15

3612

1212

1212

1212

169

129

12

2=

3=

3=

15=

4=

3=

154.

3.6

4815

1512

1512

1516

2012

129

12

∼∼

1812

1212

1212

1212

1212

124.

3.7

∼18

2118

1818

1818

1816

1616

16

1024

1212

129

1516

2030

1515

154.

3.8

109

159

129

1212

169

159

12

4.3.

9∼

∼15

∼18

∼15

∼24

∼15

915

Her

e,th

esy

mbo

lsus

ed,a

real

read

yde

fined

inth

isch

apte

rund

erth

eTa

ble

4.4.

125

4.4 Discussion and conclusions

This work proposes many new families of Chebyshev’s method, Halley’s method, super-

Halley method, Chebyshev-Halley type methods, C-methods, Euler’s method, osculating

circle method and ellipse method for finding simple roots of nonlinear equations, permitting

f ′(xn) = 0, at some points in the vicinity of required root. Further, an attempt is made to

propose some new families of third-order and fourth-order multipoint iterative methods free

from second-order derivative by discretization. The numerical examples considered in Ta-

ble 4.1 and Table 4.5 overwhelmingly support that proposed one-point as well as multipoint

iterative methods can compete with any of the existing methods including classical New-

ton’s method and Traub-Ostrowski’s method. A reasonably close initial guess is necessary

for these multipoint methods to converge. This condition, however, applies practically to

all iterative methods for solving equations. Fourth-order multipoint iterative method of first

family have optimal order with computational efficiency index [13] equal to 3√

4∼= 1.587 and

third-order multipoint iterative methods of first and second families have efficiency indices

[13] equal to 3√

3∼= 1.442, which are better than the one of Newton’s method√

2∼= 1.414. In

third family, a modified third-order multipoint family of Hasanov et al. method [101, 226]

has been obtained, permitting f ′(xn) = 0, at some points in the vicinity of required root.

Further, one can also construct many new families of iterative methods free from first and

second-order derivatives in computing process by the forward-difference approximations.

Furthermore, the behaviours of existing classical iterative methods (one-point as well

as multipoint) and the proposed modifications can be compared by their corresponding cor-

rection factors. The factor f (xn)f ′(xn)

, which appears in the classical iterative schemes, is now

modified to f (xn)f ′(xn)−α f (xn)

, where α ∈ R. Here, the parameter ‘α’ should be chosen such that

the denominator is largest in magnitude. As is well-known that, if an initial guess ‘xn’ is

close enough to the solution ‘r’ with f ′(r) 6= 0, the classical iterates are well defined and

converge to the required root. Now this modified factor has two major advantages over its

predecessor. The first one is that it is always well defined, even if f ′(xn) = 0. The second one

is that an absolute value of denominator in the formula is greater than | f ′(xn)| provided ‘xn’

is not accepted as an approximation to the required root ‘r’. This means that the numerical

126

stability of iterative schemes having this factor is better than the classical iterates. Moreover,

if ‘xn’ is sufficiently close to ‘r’ (simple root), then f (xn) will be sufficiently close to zero,

and in particular the considered denominator f ′(xn)−α f (xn) will also be close to zero if

f ′(xn) ≈ 0. In these problems where f ′(xn) −→ 0, a small value of the parameter ‘α’ may

lead to divergence or failure. Such situations can be handled by choosing ‘α’ in such a way

such that the correction factor is as small as possible, i.e. denominator f ′(xn)−α f (xn) is

sufficiently large in magnitude.

Finally, it is concluded that these techniques can be used as an alternative to existing tech-

niques or in some cases where existing techniques are not successful, therefore, have definite

practical utility.

127

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