Applied Mathematical Sciences, Vol. 14, 2020, no. 10, 461 - 473HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2020.914186

On the Stability of Weierstrass Type Method

with King’s Correction for Finding all Roots of

Non-Linear Function with Engineering Application

Saima Akram 1, Mudassir Shams 2, Naila Rafiq 3, Nazir Ahmad Mir 2

1 Center for Advanced Studies in Pure and Applied MathematicsBahauddin Zakariya University, Multan, Pakistan

2 Department of Mathematics and StatisticsRiphah International University I-14, Islamabad 44000, Pakistan

3 Department of Mathematics, NUML, Islamabad, Pakistan

This article is distributed under the Creative Commons by-nc-nd Attribution License.

Copyright c© 2020 Hikari Ltd.

Abstract

In this article, we present a new family of simultaneous iterative method for deter-

mining all the roots of non-linear equation. Using King’s family of iterative methods

as corrections, we accelerate the convergence order of basic weierstrass method from

2 to 5. Convergence analysis, basins of attraction, computational efficiency, numer-

ical test examples and log of residual demonstrate the performance and efficiency of

the newly constructed simultaneous method as compared to other existing methods

in the literature.

Keywords: Distinct Roots; Non-Linear Equation; Simultaneous IterativeMethods; Basins of Attraction; Computational Efficiency

1. INTRODUCTION

The problem of solving a non-linear equation is one of the oldest problem ofscience in general and in mathematics in particular. These non-linear equationshave a diverse application in many areas of science and engineering. The meth-ods for the simultaneous determination of all roots of a non-linear equationare important roots solvers, as they overcome the difficulty of the successiveremoval of the linear factor and have a very fast convergence. Apart fromthese, such methods have a wider region of convergence, are more stable and

462 Saima Akram, Mudassir Shams, Naila Rafiq and Nazir Ahmad Mir

can be applied to parallel computing. For details on simultaneous methods,their convergence order, computational efficiency and parallel implementationcan be seen in [1, 2, 3, 4, 5, 6, 7, 9, 10, 14, 15, 16, 20, 24, 26, 27, 31].

The main aim of this paper is to develop family of simultaneous methodswith higher convergence order having more efficiency than the existing simulta-neous methods in the literature. Further, a very high computational efficiencyis achieved by using the suitable correction due to King’s family of iterative [18]method of low computational efficiency in weierstrass function. The correc-tion enable us to achieve a very fast convergence (equal to five) with minimalnumber of function evaluations in each step. Basins of attraction for differentvalues of parameter involve in our family of simultaneous methods shows thebetter convergence properties as compared to existing methods ZPH of sameconvergence order. Computational efficiency and local computational orderof convergence (�(k)) [30] shows the dominance behavior in efficiency of oursimultaneous method as compared to ZPH.

Consider non-linear equation, say:

f(x) = 0: (1)

2. CONSTRUCTIONS OF SIMULTANEOUS METHODS

Here, we construct a family of fifth order simultaneous method which ismore efficient than the similar methods existing in literature.

2.1. Construction of Simultaneous Method for Distinct Roots. Con-sidering fourth order King’s [18] family of iterative method:

z(k) = y(k) −(

f(x(k)) + �f(y(k))

f(x(k)) + (� − 2)f(y(k))

)f(y(k))

f ′(x(k)); k = 0; 1; 2; ::: (2)

where y(k) = x(k) − f(x(k))

f ′(x(k)):

Weierstrass [11] presents the following simultaneous method of convergenceorder 2 as:

y(k)i = x

(k)i −

f(x(k)i )

n

Πj 6=ij=1

(x(k)i − x

(k)j )

:(i; j = 1; 2; 3; :::; n) (3)

Replacing x(k)j by z

(k)j in (3), we have:

y(k)i = x

(k)i −

f(x(k)i )

n

Πj 6=ij=1

(x(k)i − z

(k)j )

; (i; j = 1; 2; 3; :::; n) (4)

where

z(k)j = y

(k)j −

(f(x

(k)j ) + �f(y

(k)j )

f(x(k)j ) + (� − 2)f(y

(k)j )

)f(y

(k)j )

f ′(x(k)j )

;

On the stability of Weierstrass type method with King’s correction 463

and y(k)j = x

(k)j −

f(x(k)j )

f ′(x(k)j )

.

Thus, we have constructed a new family of simultaneous method (4) forextracting all the distinct roots of non-linear equation (1).

2.2. Convergence Analysis. Here, we study the convergence analysis offamily of simultaneous methods (4) in form of the following theorem:

Theorem 1. [1] Let �1; �

2; :::; �n be n simple roots of non-linear equation

(1). If x(0)1 ; x

(0)2 ; :::; x

(0)n be the initial approximations of the roots respectively

and sufficiently close to actual roots, the convergence order of method (4) isfive.

Proof. Let

�i = xi − � i;�′i = yi − � i;

be the errors in xi and yi approximations respectively For simplification weomit iteration index. From (4), we have:

yi − � i = xi − � i − w(xi);

�′i = �i − �iw(xi)

�i;

�′i = �i(1−Gi); (5)

where w(xi) = f(xi)nΠj 6=ij=1

(xi−zj)and

Gi =w(xi)

�i=

f(xi)

�in

Πj 6=ij=1

(xi − zj)=

n

Πj=1

(xi − �j)

�in

Πj 6=ij=1

(xi − zj)=

(xi − � i)n

Πj 6=ij=1

(xi − �j)

�in

Πj 6=ij=1

(xi − zj)

=n∏

j 6=ij=1

(xi − �j)(xi − zj)

: (6)

Now, if � i is a simple root, then for a small enough �, |xi − zj| is bounded awayfrom zero, and so

xi − �jxi − zj

=xi − zj + zj − �j

xi − zj= 1 +

zj − �jxi − zj

= 1 + O (� 4);

464 Saima Akram, Mudassir Shams, Naila Rafiq and Nazir Ahmad Mir

where zj − �j =O (� 4) see [18]:

n∏j 6=ij=1

(xi − �j)(xi − zj)

= (1 + O (�4)) n−1 = 1 + (n− 1) O (�4) = 1 + O (�4):

Implies that:

Gi = 1 + O (� 4);

Gi − 1 = O (� 4):

Thus, (6) gives:

�′

i = O (�) 5: (7)

which shows that convergence order of method (4) is five. Hence the theorem.

2.3. Complex Dynamical Study of Families of Simultaneous IterativeMethods. Here, we discuss the dynamical study of iterative methods NMSand ZPH. Let us recall some basic concepts of this study in the backgroundcontexture of complex dynamics. For more details on the dynamical behav-ior of the iterative methods one can consult [12, 25, 28]. Taking a rationalfunction <f : C −→ C, where C denotes the complex plane, the orbit x0 ∈ Cis defines a set such as orb(x) = {x0;<f (x0);<2

f (x0); :::;<mf (x0); :::}.The con-

vergence orb(s) → x∗ is understood in the sense if limk→∞<(x) = x∗ exist. A

point is x0 ∈ C known as attracting if∣∣<k′(x)

∣∣ < 1. An attracting pointx∗ ∈ C defines basin of attraction <(x∗), as the set of starting points whoseorbit tends to x∗. To generate basins of attraction, we take grid 2000 × 2000of square [−2:5 × 2:5]2 ∈ C. To each root of (1), we assign a color to whichthe corresponding orbit of the iterative methods starts and converges to afixed point. Take color map as Jet. We use |f(xi)| < 10−5 as a stoppingcriteria and maximum number of iteration is taken as 5 due to golbal con-vergence properties of simultaneous iterative methods as compared to sin-gle root find method see [13, 17, 21, 22, 23, 25]. Dark black point are as-signs if the orbit of the simultaneous iterative methods does not convergesto root after 5 iterations. we obtaine basins of attractions for the followingthree test function f1(x) = x5 − x2 + x + 125; f2(x) = ex − x2 + 25 andf3(x) = sin

(s−1

2

)sin(s−2

2

)sin(s−2.5

2

): The exact root of f1(x) is -5.00072, ex-

act root of f2(x) are 2.1+1.5i,2.1-1.5i,-0.8+2.5i,-0.8-2.5i,-2.5 and root of f3(x)are 1,2,2.5. Brightness in color means less number of iterations steps. Fi-nally, in Table 1, we present Elapsed time of basins of attraction correspond toiterative map NMS and ZPH using tic toc command in MATLAB (R2011b).

Table 1: Elapsed Time in Seconds

Methods f1(x) f2(x) f3(x)ZPH 23.90345 Fig. 1(a) 2.110827 Fig. 1(c) 6.893102 Fig. 1(e)

NMS 2.884889 Fig. 1(b) 28.52985 Fig. 1(d) 9.757365 Fig. 1(f)

On the stability of Weierstrass type method with King’s correction 465

Fig. 1(a)

Fig. 1(b)

Fig. 1(c)

466 Saima Akram, Mudassir Shams, Naila Rafiq and Nazir Ahmad Mir

Fig. 1(d)

Fig. 1(e)

Fig. 1(f)

On the stability of Weierstrass type method with King’s correction 467

Fig. 1(a,b), presents the basins of attraction of Methods NMS and ZPH fornon-linear function f1(s); Fig. 1(c,d), presents the basins of attraction ofMethods NMS and ZPH for non-linear function f2(s), Fig. 1(e,f), presents thebasins of attraction of Methods NMS and ZPH for non-linear function f3(s)respectively. Table 1, clearly shows that the elapsed time of NMS is less thanZPH, showing dominance efficiency and convergence behavior of our newlyconstructed method (NMS) over ZPH.

3. COMPUTATIONAL ASPECT

Here, we compare the computational efficiency and convergence behaviorof the ZPH. [8] method and the new method(4). As presented in [21], theefficiency of an iterative method can be estimated using the efficiency indexgiven by

EF (m) =log r

D; (8)

where D is the computational cost and r is the order of convergence of theiterative method. Using arithmetic operation per iteration with certain weightdepending on the execution time of operation to evaluate the computationalcost D. The weights used for division,multiplication and addition plus sub-traction are wd; ws and w

ASrespectively. For a given polynomial of degree m

and n roots, the number of division, multiplication addition and subtractionper iteration for all m roots are denoted by Dm; Mm and ASm. The cost ofcomputation can be calculated as:

D = D(m) = wASASm + wmMm + wdDm; (9)

thus (8) becomes:

EF (m) =log r

wASASm + wmMm + wdDm

: (10)

Considering the number of operations of a complex polynomial with real andcomplex roots reduce to operation of real arithmetic, given in Table 2 as apolynomial of degree m taking the dominion term of order (m2): Applying(10) and data given in Table 2, we calculate the percentage ratio�((4); (X))[21] given by:

�((4); (ZPH)) =

(EF (4)

EF (ZPH)− 1

)× 100

�((ZPH); (4)) =

(EF (ZPH)

EF (4)− 1

)× 100

Fig. 2(a), shows computational efficiency of methods (NMS) w.r.t methodZPH, and Fig. 2(b), shows computational efficiency of method ZPH w.r.t

468 Saima Akram, Mudassir Shams, Naila Rafiq and Nazir Ahmad Mir

Fig. 2(a)

Fig. 2(b)

(NMS). Figure 2(a,b) clearly shows that the newly constructed simultaneousmethod (4) is more efficient as compared to ZPH method.

Table 2: The number of basic arithmetic operations

Methods r ASm Mm Dm

NMS 5 7m2 +O(m) 7m2 +O(m) 2m2 +O(m)ZPH 5 11m2 +O(m) 3m2 +O(m) 2m2 +O(m)

4. NUMERICAL RESULTS

Here, some numerical examples are considered in order to demonstrate theperformance of our newly constructed simultaneous methods, namely NMS.We compare our method with ZPH [8] method of convergence order five (ab-breviated as ZPH). All the computations are performed using Maple 18 with64 digits floating point arithmetic. We take ∈ = 10−30as tolerance and use thefollowing stopping criteria:

e(k)i =

∣∣∣f (x(k+1)i

)∣∣∣ <∈;where e

(k)i represents the absolute error of function values. Numerical tests

examples from [19, 20, 25] are provided in Tables 4:1-4:5 and 4:6. In all Tables,

On the stability of Weierstrass type method with King’s correction 469

CO represents the convergence order, n represents the number of iterations,�(k) local computational order of convegence and CPU represents executiontime in seconds for approximating roots of (1).The figures 3, show that residuefall of the methods NMS and ZPH for the examples 1−3, shows that methodsNMS is more efficient as compared to ZPH. We observe that numericalresults of the method NMS are better than ZPH method on same numberof iteration.

Example1: Fractional Conversion [25] (An Engineering Applica-tion)

As expression describe in [28, 29]

f4(x) = x4 − 7:79075x3 + 14:7445x2 + 2:511x− 1:674

is the fractional conversion of nitrogen, hydrogen feed at 250 atm. and 227k.Theexact roots of (1) is �1 = 3:9485 + 0:3161i; �2 = 3:9485 − 0:3161i; �3 =−0:3841; �4 = 0:2778: The initial estimates for f4(x)are taken as:

(0)x1 = 3:5 + 0:3i;

(0)x2 = 3:5− 0:3i;

(0)x 3 = −0:3 + 0:01i;

(0)x 4 = 1:8 + 0:01i:

Table.3

Method CO CPU n e(k)1 e

(k)2 e

(k)3 e

(k)4 �

(k)1 �

(k)2 �

(k)3 �

(k)4

1 3.1 3.2 0.1 18.5 - - - -

ZPH 5 0.015 2 0.5 0.5 4.5e-3 10.9 -0.59 -0.59 2.32 0.81

3 0.019 0.016 1.7e-6 1.3 5.71 5.92 2.45 0.10

4 5.5e-6 3.5e-6 6.4e-14 2.6e-4 3.05 3.03 2.23 0.01

1 2.3 2.3 1.5 0.4 - - - -

NMS 5 0.012 2 0.3 0.2 0.02 0.05 -1.30 -1.90 -9.61 3.23

3 2.1e-1 2.6e-4 1.1e-6 3.1e-6 7.61 5.129 3.52 4.22

4 7.1e-17 4.5e-17 2.8e-20 9.0e-20 4.39 4.51 3.21 3.45

Example 2 [19]: Consider

f5(x) = (x+ 1)(x− 2)(x− 1− i)(x− 1 + i)

with exact roots:

�1 = −1; �2 = 2; �3 = 1 + i; �4 = 1− i:

The initial approximations have been taken as:

(0)x1 = −1:1 + 0:2i;

(0)x2 = 2:1− 0:2i;

(0)x 3 = 0:8 + 1:2i;

(0)x 4 = 0:9− 1:2i:

470 Saima Akram, Mudassir Shams, Naila Rafiq and Nazir Ahmad Mir

Table.4

Method CO CPU n e(k)1 e

(k)2 e

(k)3 e

(k)4 �

(k)1 �

(k)2 �

(k)3 �

(k)4

1 0.2 0.1 0.1 0.09 - - - -

ZPH 5 0.049 2 2.7e-5 3.9e-5 3.9e-5 7.1e-5 6.53 4.40 4.40 3.96

3 8.5e-16 3.9e-15 2.6e-15 1.0e-14 3.29 3.28 3.30 3.37

1 0.1 0.04 0.06 0.04 - - - -

NMS 5 0.047 2 3.2e-7 2.0e-7 1.4e-7 1.5e-7 6.4 4.79 5.60 4.88

3 4.4e-29 1.3e-29 3.0e-29 4.2e-29 4.3 4.31 4.16 4.15

Example 3 [20]: Consider

f6(x) = (ex(x−1)(x−2)(x−3) − 1);

with exact roots:�1 = 0; �2 = 1; �3 = 2; �4 = 3:

The initial approximations have been taken as:

(0)x1 = 0:1;

(0)x2 = 0:9;

(0)x 3 = 1:8;

(0)x 4 = 2:9;

Table.5

Method CO CPU n e(k)1 e

(k)2 e

(k)3 e

(k)4 �

(k)1 �

(k)2 �

(k)3 �

(k)4

1 0.1 0.01 0.08 0.09 - - - -

ZPH 5 0.094 2 7.0e-3 2.0e-3 4.0e-2 5.0e-3 2.15 1.18 2.2 3.15

3 2.5e-4 2.0e-8 8.4e-6 1.2e-5 1.67 2.11 2.13 1.49

4 3.2e-10 2.0e-16 3.5e-11 8.2e-11 2.63 2.03 2.04 2.04

1 0.13 0.6e-2 0.2e-2 0.13 - - - -

NMS 5 0.047 2 0.1e-2 0.1e-3 0.5e-3 0.1e-2 2.25 2.4 1.94 2.25

3 5.6e-5 1.4e-7 1.6e-7 6.1e-5 2.12 6.0 2.05 2.10

4 1.6e-19 0.0 0.0 0.0 4.47 4.66 4.41 7.11

Fig. 3(a)

On the stability of Weierstrass type method with King’s correction 471

Fig. 3(b)

Fig. 3(b)

Fig. 3(a-c), shows residual fall of iterative methods (NMS) and ZPH fornon-linear function f4(s), f5(s)and f6(s) respectively.

5. CONCLUSION

We have developed here a family of simultaneous method of order five fordetermination of all the distinct roots of non-linear equation, namely NMS.From Tables 3 − 6 and Figures 1-3, we observe that our method is superiorin term of efficiency, CPU time and residual error as compared to the existingmethod ZPH.

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Received: March 30, 2020; Published: July 4, 2020