On Family of Simultaneous Method for Finding Distinct as Well as Multiple Roots of Non-linear Equation

Abstract

We construct a family of 2-step simultaneous methods for determining all the distinct roots of single variable non-linear equations. We further extend this family of simultaneous methods to the case of multiple roots. It is proved that both the family of methods are of convergence order eight and has better computation efficiency as compared to some other simultaneous methods in the literature. At the end, numerical test examples are given to demonstrate the efficiency and performance of the newly constructed simultaneous methods.

Full text

Punjab University Journal of Mathematics
(ISSN 1016-2526)
Vol. 52(6)(2020) pp. 31-44
On Family of Simultaneous Method for Finding Distinct as Well as Multiple Roots of
Non-linear Equation
aNazir Ahmad Mir, bMudassir Shams,cNaila Rafiq, dSaima Akram,eRafiq Ahmed
a,bDepartment of Mathematics and Statistics, Riphah International University I-14,
Islamabad 44000, Pakistan
cDepartment of Mathematics, NUML, Islamabad, Pakistan.
dCentre for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya
University, Multan 60000, Pakistan.
eDepartmnet of Mathematics, University of Sindh, Jamshoro, Pakistan.
Email: anazir.ahmad@riphah.edu.pk, bmudassir.shams@riphah.edu.pk,
cnrafique@numl.edu.pk,dsaimaakram@bzu.edu.pk, era memon@yahoo.com
Received: 24 February, 2020 / Accepted: 07 May, 2020 / Published online: 01 June,
2020
Abstract.: We construct a family of 2-step simultaneous methods for de-
termining all the distinct roots of single variable non-linear equations. We
further extend this family of simultaneous methods to the case of multiple
roots. It is proved that both the family of methods are of convergence or-
der eight and has better computation efficiency as compared to some other
simultaneous methods in the literature. At the end, numerical test exam-
ples are given to demonstrate the efficiency and performance of the newly
constructed simultaneous methods.
AMS (MOS) Subject Classification Codes:
Key Words: Distinct roots, Multiple roots, Non-Linear equation, Iterative methods, Si-
multaneous Methods, Computational Efficiency.
1. INTRODUCTION
One of the primeral problems in mathematics is the determination of roots of non-linear
equation. There are number of applications of non-linear equation in science and engi-
neering. Newton’s method is a numerical method which finds a single root at a time. The
simultaneous iterative methods such as, Weierstrass method is used to find all the distinct
roots. The iterative methods for finding single and multiple roots of non linear polynomial
equation have been studied by Wang [35], Li [15], Osda [19], Chun and Neta [5], Homeier
[8], Bi [2], Proinov and Ivanov [30], Ivanov [11] and many others. On the other hand, there
are lot of numerical iterative methods devoted to approximate all roots of polynomial equa-
tion simultaneously (see, e.g. Weierstrass’ [36], Kanno [12], Proinov [23], Sendov [34],
Petkovi´c [20], Mir [16], Nourein [18], Aberth [1], Cholakov [4], Iliev [10], Kyncheva [13]
31

32 N. A. Mir, M.Shams, N. Rafiq, S. Akram and R. Ahmed
and the references therein). The simultaneous iterative methods are popular as compared
to single root finding methods due to their wider range of convergence, are more stable
and can be implemented for parallel computing as well. Further details on simultaneous
methods, their convergence analysis, efficiency and parallel implementations can be seen
in [6, 24, 35, 25, 18, 26, 22, 7, 27, 28, 3, 29, 31, 32, 33, 9, 14] and references cited there in.
The main objective of this paper is to develop simultaneous methods which have a higher
convergence order and can find distinct as well as multiple roots of non-linear polynomial
equation, say:
f(x)=0.(1. 1)
2. CONSTRUCTIONS OF SIMULTANEOUS METHODS
We construct here a family of higher order simultaneous methods which are more effi-
cient than the methods existing in literature.
2.1. Construction of Family of Simultaneous Methods for Distinct Roots. Consider
2-step method proposed by Li et al. [36]:



y(k)=x(k)−f(x(k))
f0(x(k)),
z(k)=y(k)−f(y(k))
f0(y(k))−αf(y(k)),(k= 0,1,2, ...)(2. 2)
where αis any arbitrary real parameter. The 2-step method ( 2. 2 ), is of fourth order
convergence, if α= 0,then it is well-known 2-step Newton’s method for calculating single
root at a time. We would like to convert it into simultaneous method for approximating all
the distinct roots of ( 1. 1 ).
Method ( 2. 2 ) can be written as:
y(k)=x(k)−f(x(k))
f0(x(k)),
z(k)=y(k)−Ã1
f0(y(k))
f(y(k))−α!.(2. 3)
Let
w(xi) = f(xi)
n
Π
j6=i
j=1
(xi−xj)
, i = 1,2,3, ..., n, (Weierstrass’ Correction [8]).(2. 4)
Taking natural logarithm of ( 2. 4 ) and then differentiating, we obtaine (see [14]):
w0(xi)
w(xi)=f0(xi)
f(xi)−
n
X
j6=i
j=1
1
(xi−xj).(2. 5)
Let x1, . . . , xnbe the distinct approximations to the roots ξ1,...,ξnof non-linear equation (
1. 1 ).Replacing xjby x∗
jin ( 2. 5 ), we have:
w0(xi)
w(xi)=f0(xi)
f(xi)−
n
X
j6=i
j=1
1
(xi−x∗
j),(2. 6)

On Family of Simultaneous Method for Findng Distinct as Well as Multipe Roots of Non-Linear Equation 33
or equivalently
w(xi)
w0(xi)=1
f0(xi)
f(xi)−
n
P
j6=i
j=1
1
(xi−x∗
j)
=1
1
N(xi)−
n
P
j6=i
j=1
1
(xi−x∗
j)
,(2. 7)
where x∗
j=xj−f(xj)
f0(xj)and N(xi) = f(xi)
f0(xi).
Replacing f(xi)
f0(xi)by w(xi)
w0(xi)in ( 2. 3 ), we have:
yi=xi−1
1
N(xi)−
n
P
j6=i
j=1
1
(xi−x∗
j)
,(i, j = 1,2, ..., n)
zi=yi−1
1
N(yi)−
n
P
j6=i
j=1
1
(yi−yj)−α
,(2. 8)
where αis a real parameter.
Thus, we have constructed a new family of 2-step simultaneous methods ( 2. 8 ),abbrevi-
ated as MM N 8D, for extracting all the distinct roots of non-linear equation.
2.1.1. Construction of Family of Simultaneous Methods for Multiple Roots. Now, family
of method ( 2. 8 ) for extracting all the distinct roots of non-linear equations modified for
finding multiple roots of ( 2. 4 ) as given by
yi=xi−σi
σi
N(xi)−
s
P
j6=i
j=1
σj
(xi−x∗
j)
,(i, j = 1,2, ..., s ≤n)
zi=yi−σi
σi
N(yi)−
s
P
j6=i
j=1
σj
(yi−yj)−α
,(2. 9)
where
x∗
j=xj−σj
f(xj)
f0(xj),
and σiis the multiplicity of actual multiple roots ζi.It should be noted that we denote y(xi)
by yiand z(xi)by zi.
We denote the method ( 2. 9 ) by MMN8M.
2.2. Convergence Analysis. In this section, the convergence analysis of a family of two-
step simultaneous methods ( 2. 9 ) given in form of the following theorem. Obviously,
convergence for the method ( 2. 8 ) will follow from the convergence of the method ( 2. 9
) from theorem (1) when the multiplicities of the roots are simple.
Theorem 2.3. :Let ξ1, ..., ξsbe all distinct roots of non-linear polynomial equation ( 1. 1
) with multiplicites σ1, ..., σs,respectively. If x(0)
1, ..., x(0)
sbe the initial approximations
of the roots respectively and sufficiently close to actual roots, the order of convergence of
method ( 2. 9 ) equals eight.

34 N. A. Mir, M.Shams, N. Rafiq, S. Akram and R. Ahmed
Proof. Let
²i=xi−ξi,
²0
i=yi−ξi,
and ²00
i=zi−ξi
be the errors in xi, yiand ziapproximations respectively. Considering the first step of ( 2.
9 ), which is
yi=xi−σi
σi
N(xi)−
s
P
j6=i
j=1
σj
(xi−x∗
j)
,
where
N(xi) = f(xi)
f0(xi).
Then, obviously for distinct roots:
1
N(xi)=f0(xi)
f(xi)=
n
X
j=1
1
(xi−ξj)=1
(xi−ξi)+
n
X
j6=i
j=1
1
(xi−ξj).
Thus, for multiple roots we have from ( 2. 9 ):
yi=xi−σi
σi
(xi−ξi)+
s
P
j6=i
j=1
σj
(xi−ξj)−
s
P
j6=i
j=1
σj
(xi−x∗
j)
,
yi−ξi=xi−ξi−σi
σi
(xi−ζi)+
s
P
j6=i
j=1
σj(xi−x∗
j−xi+ζj)
(xi−ζj)(xi−x∗
j)
,
²0
i=²i−σi
σi
²i+
s
P
j6=i
j=1
−σj(x∗
j−ξj)
(xi−ξj)(xi−x∗
j)
,
=²i−σi²i
σi+²i
s
P
j6=i
j=1
−σj(x∗
j−ξj)
(xi−ξj)(xi−x∗
j)
,
=²i−σi.²i
σi+²i
s
P
j6=i
j=1
Ei²2
j
,

On Family of Simultaneous Method for Findng Distinct as Well as Multipe Roots of Non-Linear Equation 35
where x∗
j−ξj=²2
j([17]) and Ei=−σj
(xi−ξj)(xi−x∗
j).
Thus,
²0
i=
²2
i
s
P
j6=i
j=1
Ei²2
j
σi+²i
s
P
j6=i
j=1
Ei²2
j
.(2. 10)
If it is assumed that absolute values of all errors ²j(j= 1,2,3, ...)are of the same order
as, say |²j|=O|²|, then from ( 2. 10 ), we have:
²0
i=O(²)4.(2. 11)
From second equation of ( 2. 9 ),
zi=yi−σi
σi
N(yi)−
s
P
j6=i
j=1
σj
(yi−yj)−α
,
zi−ξi=yi−ξi−σi
σi
yi−ξi+
s
P
j6=i
j=1
σj
(yi−ξj)−
s
P
j6=i
j=1
σj
(yi−yj)−α
.
This implies,
²00
i=²0
i−σi
σi
²0
i+
s
P
j6=i
j=1
σj
(yi−ξj)−
s
P
j6=i
j=1
σj
(yi−yj)−α
=²0
i−σi.²0
i
σi+²0
i


s
P
j6=i
j=1
σj.(yi−yj−yi+ξj)
(yi−ξj)(yi−yj)

−²0
iα
,
=²0
i−σi.²0
i
σi+²0
i


s
P
j6=i
j=1
−σj.(yj−ξj)
(yi−ξj)(yi−yj)

−²0
iα
,
=²0
i−σi²0
i
σi+²0
i
s
P
j6=i
j=1
²0
jFi−²0
iα
,where Fi=−σj
(yi−ξj)(yi−yj).

36 N. A. Mir, M.Shams, N. Rafiq, S. Akram and R. Ahmed
This implies,
²00
i=²0
i−σi.²0
i
σi+²0
i


s
P
j6=i
j=1
²0
jFi−α


,
= (²0
i)2










s
P
j6=i
j=1
²0
jFi−α
σi+²0
i


s
P
j6=i
j=1
²0
jFi−α












.
Since from ( 2. 11 ) ²0
i=O(²)4, thus,
²00
i=O((²)4)2,
²00
i=O(²)8,
which shows convergence order of method ( 2. 9 ) is eight. Hence, it proves the theorem.
3. COMPUTATIONAL ASPECT
Here we compare the computational efficiency and convergence behavior of the M. S.
Petkovi´c, L. D. Petkovi´c, J. D. Zunic ,methods [10, 3] and the new method( 2. 9 ). As
presented in [3], the efficiency of an iterative method can be estimated using the efficiency
index given by:
EL(m) = logr
D,(3. 12)
where Dis the computational cost and ris the order of convergence of the iterative
method. Using arithmetic operation per iteration with certain weight depending on the
execuation time of operation to evaluate the computational cost D. The weights used for
division, multiplication and addition plus subtraction are wd,wm,wAS respectively. For a
given polynomial of degree mand nroots, the number of division, multiplication addition
and subtraction per iteration for all roots are denoted by Dm, Mmand ASm. The cost of
computation can be calculated as:
D=D(m) = wasASm+wmMm+wdDm,(3. 13)
thus ( 3. 12 ) becomes:
EL(m) = log r
wasASm+wmMm+wdDm
(3. 14)
Considering the number of operations of a complex polynomial with real and complex
roots reduce to operation of real arithmetic, given in Table 3.1 as polynomial degree m

On Family of Simultaneous Method for Findng Distinct as Well as Multipe Roots of Non-Linear Equation 37
EL(9)/EL(PJ10D)
EL(9)/EL(PJ6M)
10
15
20
25
30
35
40
24
26
28
30
32
FIGURE 1
taking the dominant term of order (m2).Apply ( 3. 14 ) and data given in Table 3.1, we
calculate the percentage ratio ρ((2.9),(X)) [3] given by:
ρ((2.9),(X)) = µEL(2.9)
EL(X)−1¶×100 (in percent), (3. 15)
where X is Petkovi´c methods. Figure 1 graphically illustrates these percentage ratios.
It is evident from Figure 1 that the newly constructed simultaneous method 2.9is more
efficient as compared to the M. S. Petkovi´c, L. D. Petkovi´c, J. D. Zunic methods [10, 3].
Table 3.1: The number of basic operations
Methods ASmMmDm
Petkovic Method (PJ8M) 15m2+O(m)13m2+O(m)2m2+O(m)
Petkovic Method(PJ10D) 22m2+O(m)18m2+O(m)2m2+O(m)
New Method( 2. 9 ) 8m2+O(m)10m2+O(m)2m2+O(m)
We also calculate the CPU execuation time, as all the calculations are done using maple
18 on (Processor Intel(R) Core(TM) i3-3110m CPU@2.4GHz with 64-bit Operating Sys-
tem. We observe that CPU time of the method MMN8M is less than M. S. Petkovic meth-
ods [10, 3], showing the dominance efficiency of our method ( 2. 9 ) as compared to them.
4. NUMERICAL RESULTS
Here, some numerical test examples are considered in order to demonstrate the perfor-
mance of our family of two-step eighth order simultaneous methods, namely MMN8D(
2. 8 ) and MMN8M( 2. 9 ). We compare our family of methods with M. S. Petkovi´c,
L. D. Petkovi´c, J. D. Zunic [10] method of order six for multiple roots (abbreviated as
PJ6M method) and M. S. Petkovi´c, L. D. Petkovi´c, J. D. Zunic [3] method of order ten

38 N. A. Mir, M.Shams, N. Rafiq, S. Akram and R. Ahmed
for multiple roots (abbreviated as PJ10D method). All the computations are performed us-
ing Maple 18 with 64 digits floating point arithmetic. We take ∈= 10−30 as a tolerance
and approximating the roots the following stopping criteria are used:
(i)ei=¯¯¯f³x(k+1)
i´¯¯¯<∈,
(ii)ei=°
°
°x(k+1)
i−ξi°
°
°2=Ãn
X
i=1 ¯¯¯x(k+1)
i−ξi¯¯¯!1
2
,(k= 0,1, ...),
where eirepresents the absolute error of function values in (i)and norm-2 in (ii)[10].
In all the examples for MMN8M, we have taken α= 0.001. Numerical tests examples
from [15, 17, 4] are provided in Tables 4.1(a),4.1(b),4.2(a),4.2(b),4.3(a),4.3(b),4.4(a)
and 4.4(b)In Table 4.1(a),4.2(a),4.3(a),and Table 4.4(a)the stopping criteria (i)is used
while in Table 4.1(b),4.2(b),4.3(b)and Table 4.4(b),stopping criteria (ii)is used. In all
Tables CO represents the convergence order, n represents the number of iterations, γ=1
represents for multiplicity is equal to one , γ6=1 represents for multiplicity not equal to one and
CPU represents computational time in seconds. For multiplicity unity,we get numerical
results for distinct roots. We observe that numerical results of the methods MMN8M and
MMN8D are better than PJ6M and PJ10D methods on second iteration.The Figure 2 ,3, 4,
5 shows the residual fall of different methods for the examples 4.1,4.2,4.3,4.4, shows that
MMN8M is more efficient as compared to the other methods.
Algorithm of simultaneous iterative methods (MMN8M)
Step 1: For given x(0)
1, x(0)
2, x(0)
3, ..., x(0)
s, calculate x(1)
1, x(1)
2, x(1)
3, ..., x(1)
ssuch that
yi=xi−σi
σi
N(xi)−
s
P
j6=i
j=1
σj
(xi−x∗
j)
,(i, j = 1,2, ..., s ≤n),
zi=yi−σi
σi
N(yi)−
s
P
j6=i
j=1
σj
(yi−yj)−α
,
where x∗
j=xj−σj
f(xj)
f0(xj),and σiis the multiplicity of actual multiple roots ζi.
Step 2: For a given ∈>0,¯¯¯f³x(k+1)
i´¯¯¯<∈or µn
P
i=1 ¯¯¯x(k+1)
i−ξi¯¯¯¶1
2
<∈, then stop.
Step 3: Set k=k+ 1 and go to step 1.
Example 4.1[15]:Consider
f(x)=(x+ 1)2(x+ 3)3(x2−2x+ 2)2(x−1)3(x2−4x+ 5)2(x2+4x+ 5)2,
with multiple exact roots (γ6=1):
ξ1=−1, ξ2=−3, ξ3,4= 1 ±i, ξ5= 1, ξ6,7=−2±I , ξ8,9= 2 ±I.
The initial approximations have been taken as:
(0)
x1=−1.3 + .2i, (0)
x2=−2.8−.2i, (0)
x3= 1.2 + 1.3i, (0)
x4= 0.8−1.2i, (0)
x5= 0.8−0.3i,
(0)
x6,7=−1.8±1.2i, (0)
x8,9= 1.8±.8i.
For distinct roots (γ= 1):
f(x)=(x+ 1) (x+ 3)(x2−2x+ 2)(x−1)(x2−4x+ 5)(x2+4x+ 5).

On Family of Simultaneous Method for Findng Distinct as Well as Multipe Roots of Non-Linear Equation 39
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root1HáLMMN8M
root2HáL
root3HáL
root4HáL
root5HáL
root6HáL
root7HáL
root8HáL
root9HáL
root1H L PJ10D
root2H L
root3H L
root4H L
root5H L
root6H L
root7H L
root8H L
root9H L
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root2HOL
root3HOL
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root6HOL
root7HOL
root8HOL
root9HOL
0
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10-101
10-83
10-65
10-47
10-29
10-11
Iterations
Log of Residual
FIGURE 2. Residual fall for Example 4.1.
Table.4.1(a)
Method CO CPU γne1 e2 e3 e4 e5 e6 e7 e8 e9
PJ6M 6 1.547 γ6=1 2 0.2e-46 0.4e-81 0.3 e-35 0.2e-35 0.2 e-55 0.8 e-48 0.5 e-49 0.1 e-36 0.4e-35
PJ6M 6 1.156 γ= 1 23.6e-1 4.5e-1 4.3 e-0 8.8 e-0 2.9 e-0 5.0 e-1 7.2 e-1 1.5 e-0 4.3e-0
PJ10D 10 0.797 γ=1 2 1.0e-9 6.9e-13 4.0 e-7 4.2 e-8 2.6 e-7 8.0 e-11 1.2 e-10 1.3 e-8 6.0e-7
MMN8M 8 0.203 γ6=1 2 0.2e-44 0.3e-101 0.1 e-30 0.4 e-32 0.1 e-59 0.3 e-50 0.6 e-54 0.7 e-42 0.5e-29
MMN8D 8 0.234 γ=1 2 3.8e-28 1.0 e-33 2.2 e-21 5.5 e-24 3.3 e-20 3.5e-35 1.8e-31 7.3 e-24 1.4e-21
Table.4.1(b)
Method CO CPU γne1 e2 e3 e4 e5 e6 e7 e8 e9
PJ6M 6 1.641 γ6= 1 20.6e-26 0.1 e-29 0.2 e-20 0.1e-20 0.6e-20 0.1 e-27 0.4 e-28 0.6 e-22 0.4e-21
PJ6M 6 1.172 γ= 1 23.6e-1 6.4 e-6 9.5 e-3 1.9e-2 1.8e-2 2.0 e-5 2.9 e-5 8.6 e-3 2.5e-2
PJ10D 10 0.766 γ= 1 20.6e-26 0.1 e-29 0.2 e-20 0.1e-20 0.6e-20 0.1 e-27 0.4 e-28 0.6 e-22 0.4e-21
MMN8M 8 0.234 γ6= 1 20.6e-25 0.2 e-36 0.4 e-18 0.7e-19 0.2e-21 0.1 e-28 0.1 e-30 0.1 e-24 0.4e-18
MMN8D 8 0.156 γ= 1 29.7e-31 1.5 e-37 5.12 e-24 1.2e-26 2.0e-22 1.4 e-38 7.6 e-35 4.0 e-27 8.1e-21
Example4.2[15]:Consider
f(x) = (x+1) 2(x+2) 3(x2-2 x+2) 2(x2+1) 2(x-2) 3(x+2i)2,
with multiple exact roots (γ6=1):
ξ1=−1, ξ2=−2, ξ3,4=−2±i, ξ5,6=±i, ξ7= 2, ξ8=−2 + i.
The initial approximations valve have been taken as:
(0)
x1=−1.3 + .2i, (0)
x2=−2.2−.3i, (0)
x3= 1.3+1.2i, (0)
x4= 0.7−1.2i, (0)
x5=−0.2 + .8i,
(0)
x6= 0.2−1.3i, (0)
x7= 2.2−.3i, (0)
x8=−2.2+0.7i.

40 N. A. Mir, M.Shams, N. Rafiq, S. Akram and R. Ahmed
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root4HáL
root5HáL
root6HáL
root7HáL
root8HáL
root1H LPJ6M
root2H L
root3H L
root4H L
root5H L
root6H L
root7H L
root8H L
root1H0LPJ10D
root2H0L
root3H0L
root4H0L
root5H0L
root6H0L
root7H0L
root8H0L
0
1
2
3
4
10-121
10-100
10-79
10-58
10-37
10-16
Iterations
Log of Residual
FIGURE 3. Residual fall for Example 4.2
For distinct roots (γ= 1):
f(x) = (x+1) (x+2) (x2-2 x+2) (x2+1) (x-2) (x+2i).
Table.4.2(a)
Method CO CPU γne1 e2 e3 e4 e5 e6 e7 e8
PJ6M 6 0.922 γ6=1 2 2.6e-7 1.8e-7 5.5e-0 1.1e-7 8.4e-9 8.0e-8 1.2e-7 1.0e-7
PJ6M 6 1.078 γ=1 2 2.0e-9 1.2e-7 6.3e-12 5.7e-10 3.4e-12 5.4e-10 5.4e-19 5.5e-11
PJ10D 10 0.594 γ=1 2 1.2e-1 3.0e-2 2.7e-1 7.9e-2 1.8e-1 3.1e-2 1.2e-1 8.2e-3
MMN8M 8 0.250 γ6=1 2 3.5e-47 2.7e-88 0.0 1.4e-45 3.7e-62 7.7e-47 2.2e-97 1.8e-49
MMN8D 8 0.156 γ=1 2 2.6e-35 3.8e-36 0.0 0.0 5.1e-38 4.6e-32 1.9e-37 0.0
Table.4.2(b)
Method CO CPU γne1 e2 e3 e4 e5 e6 e7 e8
PJ6M 6 1.047 γ6=1 2 6.2e-7 4.3e-8 8.9e-9 7.0e-8 1.3e-8 1.1e-7 8.2e-9 1.0e-8
PJ6M 6 0.906 γ=1 2 6.2e-9 9.4e-10 4.1e-12 7.4e-10 1.3e-10 8.9e-10 2.5e-10 2.9e-10
PJ10D 10 0.578 γ=1 2 3.9e-3 1.5e-4 2.1e-1 4.9e-4 2.8e-3 3.5e-4 2.4e-3 2.3e-4
MMN8M 8 0.203 γ6=1 2 0.8e-25 0.2e-30 0.1e-32 0.1e-24 0.2e-32 0.4e-25 0.5e-33 0.6e-27
MMN8D 8 0.156 γ=1 2 6.6e-37 2.0e-38 1.1e-39 3.6e-34 1.6e-39 5.2e-34 4.2e-42 5.4e-37
Example4.3[4]:Consider
f(x) = (ex(x−1) (x−2) (x−3) −1) 4,
with multiple exact roots (γ6=1):
ξ1= 0, ξ2= 1, ξ3= 2, ξ4= 3.
The initial approximations have been taken as:
(0)
x1= 0.1,(0)
x2= 0.9,(0)
x3= 1.8,(0)
x4= 2.9,

On Family of Simultaneous Method for Findng Distinct as Well as Multipe Roots of Non-Linear Equation 41
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
á
á
á
á
á
á
á
á
á
á
á
á
root1HáLMMN8M
root2HáL
root3HáL
root4HáL
root1H0LPJ6M
root2H0L
root3H0L
root4H0L
root1H L PJ10D
root2H L
root3H L
root4H L
0
1
2
3
4
10-35
10-29
10-23
10-17
10-11
10-5
Iterations
Log of Residual
FIGURE 4. Residual fall for Example 4.3
For distinct roots (γ= 1):
f(x) = (ex(x−1) (x−2) (x−3) −1).
Table.4.3(a)
Method CO CPU γne1 e2 e3 e4
PJ6M 6 0.234 γ6=1 2 7.7e-9 2.6e-4 1.1e-3 9.3e-3
PJ6M 6 0.156 γ=1 2 9.3e-3 2.5e-4 2.0e-3 9.3e-3
PJ10D 10 0.125 γ=1 2 9.3e-3 2.7e-4 1.2e-3 9.3e-3
MMN8M 8 0.062 γ6=1 2 0.0 0.0 0.0 0.0
MMN8D 8 0.062 γ=1 2 1.1e-9 0.0 0.0 1.1e-9
Table.4.3(b)
Method CO CPU γne1 e2 e3 e4
PJ6M 6 0.176 γ6=1 2 1.5e-3 1.3e-4 6.5e-4 1.5e-3
PJ6M 6 0.110 γ=1 2 1.5e-3 1.3e-4 6.3e-4 1.5e-3
PJ10D 10 0.094 γ=1 2 1.5e-3 1.3e-4 6.5e-4 1.5e-3
MMN8M 8 0.063 γ6=1 2 6.6e-11 6.0e-16 4.7e-14 3.8e-10
MMN8D 8 0.062 γ=1 2 1.7e-10 5.0e-16 2.4e-13 1.5e-10
Example 4.4[17]:Consider
f(x) = (x3+5 x2-4 x-20+ cos (x3+5 x2-4 x-20)-1)5,
with multiple exact roots (γ6=1):
ξ1=- 5, ξ2=- 2, ξ3= 2,
The initial approximations have been taken as:
(0)
x1=−5.1,(0)
x2=−1.8,(0)
x3= 1.9.

42 N. A. Mir, M.Shams, N. Rafiq, S. Akram and R. Ahmed
For distinct roots (γ= 1):
f(x) = (x3+5 x2-4 x-20+ cos (x3+5 x2-4 x-20)-1).
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
á
á
á
á
á
á
á
á
á
200
200
200
O
O
O
O
O
O
root1HáLMMN8M
root2HáL
root3HáL
root1H0LPJ6M
root2H0L
root3H0L
root1H L PJ10D
root2H L
root3H L
0
1
2
3
4
5
6
10-51
10-42
10-33
10-24
10-15
10-6
Iterations
Log of Residual
FIGURE 5. Residual fall for Example 4.4.
Table.4.4(a)
Method CO CPU γne1 e2 e3
PJ6M 6 0.125 γ6=1 2 2.7e-12 1.3e-11 1.2e-3
PJ6M 6 0.140 γ=1 2 4.9e-3 6.1e-3 2.6e-1
PJ10D 10 0.250 γ=1 2 4.9e-3 6.0e-3 2.5e-1
MMN8M 8 0.141 γ6=1 2 7.2e-11 1.7e-10 2.3e-4
MMN8D 8 0.094 γ=1 2 5.1e-51 1.0e-50 4.8e-19
Table.4.4(b)
Method CO CPU γne1 e2 e3
PJ6M 6 0.031 γ6= 1 22.3e-4 5.5e-4 8.4e-3
PJ6M 6 0.047 γ= 1 22.3e-4 5.1e-4 8.4e-3
PJ10D 10 0.062 γ= 1 22.3e-4 5.5e-4 8.4e-3
MMN8M 8 0.047 γ6= 1 24.7e-12 8.3e-12 8.3e-6
MMN8D 8 0.016 γ= 1 24.7e-12 8.3e-12 7.7e-6
5. CONCLUSION
In this article, a new two-step eighth order family of iterative methods for the simulta-
neous approximations of all roots of a polynomial equation was introduced and discussed.
Our method MMN8M, determines all the multiple roots and as a special case for mul-
tiplicity unity all the distinct roots of polynomial equation. The results of numerical test

On Family of Simultaneous Method for Findng Distinct as Well as Multipe Roots of Non-Linear Equation 43
examples, CPU time, residual error, computational efficiency corroborate theoretical analy-
sis, illustrate the effectiveness and rapid convergence of our proposed family of iterative
method as compared to the methods PJ6M and PJ10D [21, 22].
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