A Sixth-Order Two-Step Method for Finding a Multiple Root of Nonlinear Equations

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https://doi.org/10.12988/ams.2019.9795

A Sixth-Order Two-Step Method for Finding a Multiple Root of Nonlinear Equations

Siti Rahma¹, M. Imran and Syamsudhuha

Computational Mathematics Group, Department of Mathematics University of Riau, Pekanbaru 28293, Indonesia

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

Copyright c2019 Hikari Ltd.

Abstract

This article discusses a two-step method for finding multiple roots of nonlinear equations. We apply Newton’s method for the first step and Osada’s method for the second. This method has a six-order convergence and requires five function evaluations per iteration. From numerical simulation, we conclude that the proposed method is competitive to the compared methods and it can be used as an alternative method for two-step methods.

Mathematics Subject Classification: AMS9795

Keywords: Two-step method, Newton’s method, Osada’s method, multiple roots

1 Introduction

Numerical analysis is the area of mathematics that creates, analyzes and im- plements algorithms for solving numerically the problems of mathematics. One of the most familiar mathematical problems is how to find roots of a nonlinear equation

f(x) = 0, (1)

wheref :I ⊆R→Ris a differentiable function in an open interval I.

1Corresponding author

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A basic and important method to obtain multiple roots of nonlinear equations is Newton’s method [14],

x_n+1 =x_n−mf(x_n)

f⁰(xn), f⁰(x_n)6= 0. (2) which converges quadratically and requires the multiplicitym to be known.

Many modified methods for multiple roots have been developed, such as Behl et al. [1] [2], Cui et al. [3] , Dong [4] [5], Geum et al. [6], [7], Homeier [8], Kim et al. [9], Li et al. [10], Li et al. [11], Qudsi et al. [13], Sharma and Sharma [15], Sharma and Bahl [16], Victory and Neta [17], Zafar et al. [19]

and Zhou et al. [20] [21].

This paper discusses a two steps method by combining Newton’s method for the first step with Osada’s method for the second. The method and its convergence analysis are discussed in section two and the computation is discussed in section three.

2 The Proposed Two-Step Method

Osada’s method [12] for multiple roots is given by x_n+1 =x_n− 1

2m(m+ 1)f(x_n) f⁰(xn)+ 1

2(m−1)²f⁰(x_n)

f⁰⁰(xn). (3) By combining (2) and (3), we have the following two-step scheme method:

y_n=x_n−mf(x_n) f⁰(x_n), xn+1 =yn−1

2m(m+ 1)f(y_n) f⁰(y_n) +1

2(m−1)²f⁰(y_n) f⁰⁰(y_n).

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The convergence order of iterative method (4) is given by Theorem 2.1.

Theorem 2.1. [Order of Convergence] Let α ∈ I be a multiple root with multiplicity m of a sufficiently differentiable function f : I ⊆ R → R in an open interval I. If x₀ is sufficiently close to α, then the method defined by (4) has a sixth-order convergence.

Proof. Let α be a multiple root of f(x) = 0 with multiplicity m, so that f(α) = 0, f⁰(α) = 0, f⁰⁰(α) = 0, . . . , f^(m−1)(α) = 0 and f^(m)(α) 6= 0. By

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expandingf(x) about x=α by Taylor’s series we have f(x) = f^(m)(α)

m! (x−α)^m+ f^(m+1)(α)

(m+ 1)! (x−α)^m+1+f^(m+2)(α)

(m+ 2)! (x−α)^m+2 + f^(m+3)(α)

(m+ 3)! (x−α)^m+3+f^(m+4)(α)

(m+ 4)! (x−α)^m+4 + f^(m+5)(α)

(m+ 5)! (x−α)^m+5+f^(m+6)(α)

(m+ 6)! (x−α)^m+6 +O

(x−α)^m+7

. (5)

Then by evaluating (5) atx=x_n and letting e_n=x_n−α, (5) becomes f(x_n) = f^(m)(α)

m! e^m_n 1 +c₁e_n+c₂e²_n+c₃e³_n+c₄e⁴_n+c₅e⁵_n+c₆e⁶_n+O(e⁷_n) , (6) where

c_j = m!

(m+j)!

f^(m+j)(α)

f^(m)(α) . (7)

Similarly, we obtain f⁰(x_n) = f^(m)(α)

(m−1)!e^(m−1)_n

1 + m+ 1

m c₁e_n+m+ 2

m c₂e²_n+m+ 3 m c₃e³_n + m+ 4

m c₄e⁴_n+m+ 5

m c₅e⁵_n+m+ 6 m c₆e⁶_n

+O(e⁷_n), (8) where c_j is defined by (7). Dividing (6) by (8) and multiply the resulting equation by m, we have

mf(x_n)

f⁰(x_n) =e_n− c₁

me²_n+ (m+ 1)c²₁+ 2mc₂

m² e³_n+· · ·+O(e⁷_n). (9) Now by substituting (9) into (4) we get

y_n =α+A₂e²_n+A₃e³_n+A₄e⁴_n+A₅e⁵_n+A₆e⁶_n+O(e⁷_n), (10)

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where

A₂ = c1

m,

A₃ = (m+ 1)c²₁+ 2mc₂ m² ,

A₄ = (−(m+ 1)²)c³₁ + (4m+ 3m²)c₁c₂−3m²c₃

m³ ,

A₅ = (6m²+ 4m³)c1c3+ (−4m³−10m²−6m)c²₁c2

m⁴

+(4m²+ 2m³)c²₂+ (m+ 1)³c⁴₁−4m³c₄

m⁴ ,

A₆ =

−m²(5m+ 9)(m+ 1)

c²₁ +m³(5m+ 12)c₂ c₃ m⁵

+(−m²)(m+ 2)(5m+ 6)c₁c²₂+ m(5m+ 8)(m+ 1)² c³₁c₂ m⁵

+ −(m+ 1)⁴

c⁵₁+ (5m⁴+ 8m³)c₁c₄−5m⁴c₅

m⁵ .

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To derive f(y_n), we evaluate (5) at x = y_n and by letting ˆe_n = y_n −α we obtain

f(y_n) = f^(m)(α) m! eˆ^m_n

1 +B₂e²_n+B₃e³_n+B₄e⁴_n+B₅e⁵_n+B₆e⁶_n

+O(e⁷_n), (12) where

B₂ = c²₁

m, B₃ = (−1−m)c³₁+ 2mc₁c₂

m² ,

B₄ = −3m(m+ 1)c²₁c₂+ (m+ 1)²c⁴₁+ 3m²c₁c₃

m³ ,

B₅ = (−6m²−4m³)c²₁c₃−2m³c₁c²₂+ 4m(m+ 1)²c³₁c₂−(m+ 1)³c⁵₁+ 4m³c₁c₂

m⁴ ,

B₆ = (−5m⁴−8m³)c²₁c4+ (10m²+ 14m³+ 5m⁴)c³₁−(−6m³−5m⁴)c1c2

c3

m⁵

+4m³c³₂+m³(5m+ 6)c²₁c²₂− 5m(m+ 1)³

c⁴₁c₂+ (m+ 1)⁴c⁶₁ + 5m⁴c₁c₅

m⁵ .

Thus by evaluating f⁰(x) at x=y_n and simplifying we get f⁰(y_n) = f^(m)(α)

m! eˆ^(m−1)_n

m+D₂e²_n+D₃e³_n+D₄e⁴_n+D₅e⁵_n+D₆e⁶_n

+O(e⁷_n) (13)

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where

D₂ = (m+ 1)c²₁

m , D₃ = −(m+ 1)²c³₁+ 2m(m+ 1)c₁c₂

m² ,

D₄ = 3m²(m+ 1)c₁c₃−(2m+ 3m³+ 6m²)c²₁c₂+ (m+ 1)³c⁴₁

m³ ,

D₅ = (4m³+ 4m⁴)c₁c₄−(4m⁴+ 6m²+ 10m³)c²₁c₃ −(2m⁴ −4m²+ 2m³)c₁c²₂ m⁴

+(12m³+ 10m²+ 4m⁴+ 2m)c³₁c2−(m+ 1)⁴c⁵₁

m⁴ ,

D₆ = (5m⁵+ 5m⁴)c₁c₅−(13m⁴+ 5m⁵ + 8m³)c²₁c₄ + (m+ 1)⁵c⁶₁ m⁵

+· · ·+(−2m−20m⁴−5m⁵−27m³−14m²)c⁴₁c₂

m⁵ .

Now by expanding f⁰⁰(x) about x = α then we evaluate at x = y_n and after simplifying we obtain

f⁰⁰(y_n) = f^(m)(α)

m! eˆ^(m−2)_n

(m²−m) +F₂e²_n+F₃e³_n+F₄e⁴_n+F₅e⁵_n+F₆e⁶_n

+O(e⁷_n), (14) where

F₂ = (m+ 1)c²₁, F₃ = −(m+ 1)²c³₁+ 2m(m+ 1)c₁c₂

m ,

F₄ = (m+ 1)(−3m² −3m+ 2)c²₁c₂+ (m+ 1)³c⁴₁+ 3m²(m+ 1)c₁c₃

m² ,

F5 = −(m+ 1)(6m² + 4m³)c²₁c3 −(m+ 1)(2m³−8m)c1c²₂ m³

+ −(m+ 1)(−4m³−8m²+ 4)c³₁c₂−(m+ 1)⁴c⁵₁+ 4m³(m+ 1)c₁c₄

m³ ,

F₆ = (5m⁵+ 5m⁴)c₁c₅+ (−13m⁴ −5m⁵−8m³)c²₁c₄ m⁴

+· · ·+(6 + 13m−5m⁵−2m²−24m³−20m⁴)c⁴₁c₂+ (m+ 1)⁵c⁶₁

m⁴ .

Dividing (12) by (13) and by using geometric series we get the following:

f(y_n)

f⁰(y_n) =G₂e²_n+G₃e³_n+G₄e⁴_n+G₅e⁵_n+G₆e⁶_n+O(e⁷_n), (15)

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where G₂ = c₁

m², G₃ = (−m−1)c²₁+ 2mc₂

m³ ,

G₄ = (−4−3m)c₁c₂+ (m+ 2)c³₁+ 3mc₃

m³ ,

G₅ = (−4m³−6m²)c₁c₃+ (−4m²−2m³)c²₂+ (2m+ 10m²+ 4m³)c²₁c₂ m⁵

+(−m−m³−3m²+ 1)c⁴₁+ 4c4m³

m⁵ ,

G₆ = (−5m⁴−8m³)c₁c₄+ (3m²+ 14m³+ 5m⁴)c²₁+ (−5m⁴−12m³)c₂ c₃ m⁶

+· · ·+(m⁴−1−m+ 4m³+ 3m²)c⁵₁+ 5c₅m⁴

m⁶ .

Multiplying (15) by m(m+ 1)/2 we have m(m+ 1)

2

f(yn)

f⁰(y_n) =H₂e²_n+H₃e³_n+H₄e⁴_n+H₅e⁵_n+H₆e⁶_n+O(e⁷_n), (16) where

H₂ = (m+ 1)c₁

2m , H₃ = (m+ 1) (−m−1)c²₁+ 2mc₂

2m² ,

H₄ = (m+ 1) (−4−3m)c₁c₂+ (m+ 2)c³₁+ 3mc₃

2m² ,

H₅ = (m+ 1) (−4m³−6m²)c₁c₃+ (−4m²−2m³)c²₂+ 4m³c₄ 2m⁴

+(m+ 1) (2m+ 10m² + 4m³)c²₁c₂ + (−m−m³−3m²+ 1)c⁴₁

2m⁴ ,

H6 = (m+ 1) (−5m⁴−8m³)c₁c₄+ ((3m²+ 14m³+ 5m⁴)c²₁ 2m⁵

+· · ·+ (m+ 1) (m⁴−1−m+ 4m³+ 3m²)c⁵₁+ 5c₅m⁴

2m⁵ .

If we divide (13) by (14) then by using geometric series we can get the following:

f⁰(y_n)

f⁰⁰(y_n) =J₂e²_n+J₃e³_n+J₄e⁴_n+J₅e⁵_n+J₆e⁶_n+O(e⁷_n), (17)

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where

J₂ = c₁

m(m−1), J₃ = (−m−1)c²₁+ 2mc₂ m²(m−1) ,

J₄ = (3m³−3m²)c₃+ (−3m³−m²+ 4m)c₁c₂+ (m³−2m−2 +m²)c³₁

m³(m−1)² ,

J₅ = (4m⁴−4m³)c₄+ (6m²−4m⁴−2m³)c₁c₃+ (4m²−2m³−2m⁴)c²₂ m⁴(m−1)²

+ (−8m²+ 6m³+ 4m⁴−10m)c²₁c₂+ (3−m⁴−2m³+ 2m²+ 6m)c⁴₁

m⁴(m−1)² ,

J₆ = (5m⁴+ 5m⁶−10m⁵)c₅+ (11m⁴−5m⁶−8m³+ 2m⁵)c₁c₄ m⁵(m−1)³

+· · ·+(2m⁵ +m²−4m⁴+m⁶+ 4−9m³+ 9m)c⁵₁ m⁵(m−1)³ . Then multiplying (17) by (m−1)²/2, we have

(m−1)² 2

f⁰(y_n)

f⁰⁰(y_n) =K₂e²_n+K₃e³_n+K₄e⁴_n+K₅e⁵_n+K₆e⁶_n+O(e⁷_n), (18) where

K₂ = (m−1)c₁

2m , K₃ = −(m−1)(m+ 1)c²₁+ 2m(m−1)c₂

2m² ,

K₄ = (3m³−3m²)c₃+ (−3m³−m²+ 4m)c₁c₂ + (m³−2m−2 +m²)c³₁

2m³ ,

K₅ = (4m⁴−4m³)c4+ (6m²−4m⁴ −2m³)c1c3+ (4m²−2m³−2m⁴)c²₂ 2m⁴

+ (−8m² + 6m³+ 4m⁴−10m)c²₁c₂+ (3−m⁴−2m³+ 2m²+ 6m)c⁴₁

2m⁴ ,

K₆ = (5m⁴+ 5m⁶−10m⁵)c₅+ (11m⁴−5m⁶−8m³+ 2m⁵)c₁c₄ 2m⁵(m−1)

+· · ·+(2m⁵+m²−4m⁴+m⁶+ 4−9m³+ 9m)c⁵₁

2m⁵(m−1) .

Then by substituting (16), (18), x_n = e_n−α and y_n = ˆe_n−α into (4) we obtain the following error equation for (4):

en+1 =

(−2m² + 2m)c³₁c₂+ (m²+ 2m+ 1)c⁵₁ e⁶_n

2m⁵(m−1) +O(e⁷_n). (19) Thus from the definition of the order of convergence [18], we know that the

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method has sixth-order convergence and Theorem 2.1 is proven.

3 Numerical Simulation

In this section, some numerical simulation are carried out to compare the number of iterations and COC of Qudsi-Imran-Syamsudhuha (QISM) by [13], Sharma-Bahl Method (SBM) by [16], Newton-Halley Method (NHM) by Cui et al. [3] and the proposed Method (NOM) by (4). To show the comparisons, the following nonlinear equations are used:

i. f₁(x) = x³+ 4x²−103

, α ∈[−3,1];

ii. f2(x) = 8xe^(−x²⁾−2x−38

, α∈[0.3];

iii. f₃(x) = sin²(x)−x²+ 12

, α∈[1,6];

iv. f₄(x) = cos (x)−x3

, α∈[−1,1];

v. f₅(x) = (x−1)³−16

, α∈[−2,2].

All computations have been carried out using the tolerance (tol) of 1.0×10⁻³⁰⁰ where the maximum iteration is 100. The stopping criteria of computation program are|x_n+1−x_n|< toland|f(x_n+1)|< tolfor all comparison methods.

Table 1: The number of iteration comparisons of several methods

f(x) x0

Number of iterations

QISM SBM NHM NOM

f1

−2.7 23 77 20 5

−0.1 19 19 48 10

0.4 12 13 4 4

f₂ 0.3 5 5 7 4

1.2 10 9 5 4

2.1 6 7 13 4

f3 1.4 3 3 3 3

3.8 6 6 4 4

5.3 8 4 4 5

f₄ −0.1 4 4 3 4

0.8 3 3 3 3

1.9 4 4 3 3

f₅ −1.2 5 7 6 4

1.3 5 6 4 5

2.4 4 4 3 3

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From the result present in Table 1 we can see that for the given test functions and initial guesses, the proposed method has the less number of iterations from the compared methods.

Table 2: COC comparisons of several methods

f(x) x0

COC

QISM SBM NHM NOM

f1

−2.7 6.00 6.00 6.00 6.00

−0.1 6.00 6.00 6.00 6.00 0.3 6.00 6.00 6.00 6.00

f₂ 0.3 6.00 6.00 6.00 6.00

1.2 6.00 6.00 6.00 6.00 2.1 6.00 5.36 5.99 6.00

f₃ 1.4 6.00 6.00 6.00 6.00

3.8 6.00 6.00 6.00 6.00 5.3 6.00 6.00 6.00 6.00 f4

−0.1 6.00 6.00 6.00 6.00 0.8 6.00 6.00 6.00 6.00 1.9 6.00 6.00 6.00 6.00

f₅ −1.2 6.00 6.00 6.00 6.00

1.3 6.00 6.00 5.99 6.00 2.4 6.00 6.00 5.99 6.00

Table 2 shows that the computational order of convergence (COC) of the proposed methods is in accordance with the theoretical order of convergence.

Finally we conclude that the proposed method is competitive to the compared methods, so that this method can be alternative method for the sixth- order method.

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