ON COMPLEX DYNAMICS OF SOME EIGHTH ORDER …€¦ · SUNIL KUMAR and JANAK RAJ SHARMA Advances and Applications in Mathematical Sciences, Volume 18, Issue 1, November 2018 158 5314

Advances and Applications in Mathematical Sciences Volume 18, Issue 1, November 2018, Pages 153-167 © 2018 Mili Publications

2010 Mathematics Subject Classification: 41A21, 65B99, 65H05.

Keywords: Rational Map, Basin of attraction, Extraneous fixed points.

Received January 5, 2018; Accepted April 26, 2018

ON COMPLEX DYNAMICS OF SOME EIGHTH ORDER

TECHNIQUES FOR NONLINEAR EQUATIONS

SUNIL KUMAR and JANAK RAJ SHARMA

Department of Mathematics

Sant Longowal Institute of Engineering and Technology

Longowal 148106, Punjab, India

E-mail: [email protected]

[email protected]

Abstract

There are many techniques to solve nonlinear equations. These techniques are categorized

by the order, informational efficiency and efficiency index. In this work we have taken the

criteria, namely basins of attraction for checking the convergence domain of the techniques.

This study is also called complex dynamics of iterative methods. We consider several techniques

of order eighth and present the basin of attraction for respective examples. We measured that

Kung-Traub and Sharma-Kumar techniques are consistently better than others.

1. Introduction

There are a number of different techniques for the numerical solution of

nonlinear equations [13]. These techniques are categorized by their order of

convergence (say, p), and the number of function and derivative evalutions

(say, n) per step. To check the effectiveness of such techniques, there are two

efficiency measures (see [13]) defined as n

pI (informational efficiency) and

npE

1

(efficiency index). Another measure, introduced recently, is the

complex dynamics of iterative techniques. For example, see, (Amat et al. [1],

Chicharro et al. [3], Chun et al. [4], Cordero et al. [5], Gutierrez et al. [7],

Neta et al. [10], Scott et al. [11]). In 1974, Kung and Traub [8] introduced the

concept of optimality. According to their hypothesis multipoint techniques


Advances and Applications in Mathematical Sciences, Volume 18, Issue 1, November 2018

154

without memory requiring 1n function-evaluations have order of

convergence at most .2n Such techniques are usually called optimal (see, for

example, [8]). An optimal technique of order 2p is the well known

Newton’s technique. Optimal techniques of order four were discussed in [1, 4,

10].

In this paper, we consider some eighth order optimal techniques and

study their complex dynamics. Moreover, we will make the relation between

conjugacy maps [5], extraneous fixed points [14] and the basins of attraction

in our numerical trial. Rest of the paper is organised as follows. Section 2, the

eighth order techniques that we have considered. In Section 3, we will check

the conjugacy maps for each technique and find the extraneous fixed points

[14]. Basins of attraction of the various optimal eighth order methods are

shown in section 4. To study the complex dynamics, we choose the eighth

order techniques proposed by Bi et al. [2], Cordero et al. [6], Kung-Traub [8],

Liu-Wang [9] and Sharma-Kumar [12].

2. Techniques for the Relative Examination

In this section we tabulate the eighth-order techniques that we have

taken here. To study the complex dynamics, we have taken eighth order

techniques proposed by Bi et al. [2], Cordero et al. [6], Kung-Traub [8], Liu-

Wang [9] and Sharma-Kumar [12].

Bi-Wu-Ren Technique (BWRT8):

,

k

kkk lg

lglw

,

52

2

k

k

kk

kkkk lg

wg

wglg

wglgwt

,

,,,

21

kkkkkkk

k

kk

kkkk wtlltgwtg

tg

tglg

tglgtl

(1)

where and

.,

,,kk

kkkkkk lt

lgltglltg

Cordero-Torregrosa-Vassileva Technique (CTVT8):



155

,

k

kkk lg

lglw

,

2 k

k

kk

kkkk lg

lg

wglg

wglglt

,

3

321

321

k

k

kkkkkk

kkkk lg

tg

ltlwt

tl

(2)

where 0,3,2,1 32 ii and

.

22

1

2

2

kk

k

kk

kk

k

kkk tgwg

tg

wglg

wglg

lg

tgt

Kung-Traub Technique (KTT8):

,

k

kkk lg

lglw

,

2kk

kk

k

kkk

wglg

lgwg

lg

lgwt

.

2

2

21

kkkk

kkkk

kk

kkk

k

kkk

tgwgtglg

tgwgwglg

wglg

tgwglg

lg

lgtl

(3)

Liu-Wang Technique (LWT8):

,

k

kkk lg

lglw

,

2 kk

k

k

kkk wglg

lg

lg

lgwt

,

4

2 21

2

1

kk

k

kk

k

kk

kk

k

kkk tglg

tg

tgwg

tg

wglg

wglg

lg

tgtl (4)

where .,1

Sharma-Kumar Technique (SKT8):

,

k

kkk lg

lglw



156

,

,2

kk

kkkk

lwg

wglgwt

,

,

12

3211

kk

kkkk

ltg

tglg

hhhtl

(5)

where 21, hh and 3h are defined as:

,1kk

kk

lw

twh

,

,

2

2kkkkkk

kkk

lwglwtw

lglth

.

,3kkkk

kkk

ltgtw

lglth

3. Compatible Conjugacy Maps For Quadratic Polynomials

Theorem 3.1 (Bi-Wu-Ren Technique, BWRT8). Rational map zRp

emerging from the technique (1) implemented on ,21 azazzp

zRaa p,21 is conjugate via the Mobius transformation given by

2

1

az

azzM

to

,9

z

z

D

NzzS

where

109843222045707045220823 zzzzzzzzNz

,1310869569514 6751 zzzz (6)

109873222024570704520432 zzzzzzzzDz

131086956958 56411 zzzz (7)

and for 1



157

6543221211017045220823 zzzzzzzNz

,4204570101 1110987 zzzzz (8)

44322121101704520432 zzzzzzDz

.82024570101 11109876 zzzzzz (9)

Theorem 3.2 (Cordero-Torregrosa-Vassileva Technique, CTVT8).

Rational map zRp emerging from the technique (2) implemented on

zRaaazazzp p,, 2121 is conjugate via the Mobius

transformation given by 2

1

az

azzM

to

,8

z

z

D

NzzS

where

323229

3230

32 136205416169244 zzzNz

32126

32127

3228 9567416443216171316 zzz

13

32124

32125 35430320725161791241216 zzz

3212

3214

32 84448448763457904371201 zz

122

3218

3215 294419511215281811336421344 zzz

2121

32110

32 2526438431231969462827691851 zz

32114

32112

3 4747305172184191266963683793 zz

32120

3219 499533056144477428806954 zz

3218

32119 7371482510244616340418114 zz

117

3216

32123 116847804478096075525120740 zzz



158

32113

32 883055361339482215314 z

32116 9071588313144 z

32111

3217 141328700205721052861761363 zz

,37024235845415 32115 z

312

3129

3231 17131613620541616 zzzDz

3314

3130

3313 95674169244443216 zzz

127

3216

3215 35430320725161791241216 zzz

32128

32126

32 84448448763457904371201 zz

32122

32125 19511215281811336421344 zz

32120

3218 312319694628275918512944 zz

32118

3219 419126696368379325264384 zz

32121

32116 477428806954474730517218 zz

32111

32110 166340418114499533056144 zz

3217

32112 755251207407371482510244 zz

32124 78044780960 z

32117

32113 88305536133948221531411684 zz

32123

32114 10528617613639071588313144 zz

32115

32119 3702423584541514132870020572 zz

and for 0,1,0 321

8765432 519461402270226141993011 zzzzzzzzNz



159

1514131211109 96168256344416471494 zzzzzzz

,41644 181716 zzz (10)

8765432 4714163442561689644164 zzzzzzzzDz

1431211109 226270402461519494 zzzzzz

.113099141 18171615 zzzz (11)

Theorem 3.3 (Kung-Traub Technique, KTT8). Rational map zRp



2

1

az

azzM

to

,8

z

z

D

NzzS

where

765432 34323116238215067602897410 zzzzzzzNz

12111098 4791006174925683214 zzzzz

.1052182 16151413 zzzz (12)

765432 25681749100647918252101 zzzzzzzDz

1312111098 76015062382311634323214 zzzzzz

.1074289 161514 zzz (13)

Theorem 3.4 (Liu-Wang Technique, LWT8). Rational map zRp

emerging from the technique (4) implemented on

zRaaazazzp p,, 2121 is conjugate via the Mobius

transformation given by 2

1

az

azzM

to

,8

z

z

D

NzzS



160

where

2111

111121413 43210213413 zzzzzNz

21215

212

2110 514581884167321163 zzz

2123

2127 245961427962 zz

2124

2129 22521417124186 zz

,665182152315202 2126

2218 zz

2112

113

114

12 41673102131341 zzzzzDz

2127

213

214 142796243221163 zzz

2125

21211 2418624596 zz

21210

2216 450141712315202 zz

2219

3128 794255866518125 zz

and for 0,0 21

,4916212013 65432 zxzzzzNz (14)

.13202116941 65432 zzzzzzDz (15)

Theorem 3.5 (Sharma-Kumar Technique, SKT8). Rational map zRp



2

1

az

azzM

to

,8

z

z

D

NzzS

where

765432 118412381067742400159426 zzzzzzzNz

,943138330619942 141312111098 zzzzzzz (16)



161

8765432 123811849426193301384391 zzzzzzzzDz

.6421594007421067 14131211109 zzzzzz (17)

3.1. Extraneous Fixed Points

It can be seen that all these techniques can be written as

,,,1 kkkgkkkk twlHlvll

where

.

k

kkk lg

lglv

Obviously the root 1 is a fixed point of the

technique, since .01 kv The points 1 at which 0gH are also

fixed points of the technique, whereas the second part on the right vanishes.

These points are termed extraneous fixed points (see [14]). In this section, we

will discuss the extraneous fixed points of each technique for the polynomial

.12 z Now the point is attractive if pR is less than one, indifferent if

pR is equal to one or repulsive if pR is greater than one, where

ztzwzHzvzzR gp ,, is the iteration function.

Theorem 3.6. For extraneous fixed points of Bi-Wu-Ren Technique (1) are

at 2770.0,1209.03040.0,3794.13521.0,0093.15 zizizz

,2064.0,1934.0,0062.01955.0,2870.02630.0,9331.0 zzizizi

,244.1,3799.0,3405.0,2238.02954.0,5800.02322.0 izziziz

.8228.26021.0 iz All extraneous points are repulsive.

Theorem 3.7 For extraneous fixed points of Cordero-Torregrosa-Vassileva

Technique (2) are at ,8680.03773.0,3187.14583.0 iziz

1110.0,4495.01227.0,5986.01254.0,1236.02525.0 ziziiz

,5986.01254.0,4495.01227.0,3441.21110.0,3441.2 izizizi

.3187.14583.0,8680.03773.0,1236.02525.0 iziziz All extraneous

points are repulsive.

Theorem 3.8 For extraneous fixed points of Kung-Traub Technique (3)

are at ,4196.02137.0,0204.13193.0,3908.03233.0 iziziz

,2450.3.0,0404.1.0,2937.0.0,3502.01694.0 iziziziz



162

3233.0,0204.13193.0,4196.02137.0,3502.01694.0 ziziziz

.3908.0 i All extraneous points are repulsive, except iz 2450.3.0 is

attractive.

Theorem 3.9. For extraneous fixed points of Liu-Wang Technique (4) are

at ,1453.03055.0,6291.03343.0,1826.15144.0 iziziz

.1826.15144.0,6291.03343.0,1453.03055.0 iziziz All extraneous

points are repulsive.

Theorem 3.10. For extraneous fixed points of Sharma-Kumar Technique

(5) are at iziziz 2529.02536.0,4848.02696.0,6029.03299.0

2261.0,3851.02156.0,3851.02156.0,3769.12261.0 ziziziz

.6029.03299.0,4848.02696.0,2529.02536.0,3769.1 izizizi All

extraneous points are repulsive.

4. Basins of Attraction

Example 1. In the first trial, we have test every one of the techniques to

get the simple zeros of the quadratic polynomial .12 z The result for the

basins of attraction are given in Figures (a)-(e).

Example 2. In our second trial, we have run every one of the techniques

to get the simple zeros for the cubic polynomial .13 z The result of the

basins of attraction are given in Figures (f)-(j).

Example 3. In our third trial, we have run every one of the techniques to

get the simple zeros for the biquadratic polynomial .14 z The result of the

basins of attraction are given in Figures (k)-(o).



163

Figures (a)-(e). Basins of attraction for polynomial .12 zzf



164

Figures (f)-(j). Basins of attraction for polynomial .13 zzf



165

Figures (k)-(o). Basins of attraction for polynomial .14 zzf



166

In order to draw the above figures the strategy taken into account is the

following. A color is assigned to each basin of attraction of a zero. We mark

with black the points of the figure for which the corresponding iterations

starting at them do not reach any root.

Notice from figures (a)-(e) that all methods perform well since there is no

black region in any figure. This means corresponding to quadratic polynomial

(example 1) every method possesses similar convergence behavior. In case of

cubic polynomial (example 2) the Kung-Traub and Sharma-Kumar methods

perform well since other figures have black spot in their basins. In the case of

bi-quadratic polynomial (example 3) again Kung-Traub and Sharma-Kumar

methods have better convergence behavior.

5. Conclusion

We have considered some eighth order techniques and checked their

performance by comparing the complex dynamics in terms of basins of

attraction. We have found that Kung-Traub and Sharma-Kumar techniques

are better than the other ones. Bi-Wu-Ren Technique performance poorly in

example 2 and in the example 3 the Bi-Wu-Ren Technique, Cordero-

Torregrosa-Vassileva Technique and Liu-Wang Technique perform very weak

(divergence area is more) than Kung-Traub and Sharma-Kumar techniques.

6. Acknowledgments

Authors would like to thank to Department of Mathematics, SLIET, Punjab

for the excellent lab facility and for continuous support.

References

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ON COMPLEX DYNAMICS OF SOME EIGHTH ORDER …€¦ · SUNIL KUMAR and JANAK RAJ SHARMA Advances and Applications in Mathematical Sciences, Volume 18, Issue 1, November 2018 158 5314