NEW NEWTON-TYPE METHOD WITH -ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION

(1)

NEW NEWTON-TYPE METHOD WITH 

^k^²

 -ORDER

CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION

R. Thukral

Padé Research Centre, 39 Deanswood Hill, Leeds West Yorkshire, LS17 5JS, ENGLAND

ABSTRACT

The objective of this paper is to define a new Newton-type method for finding simple root of a polynomial. It is proved that the new one-point method has the convergence order of



^k^²



requiring n function evaluations per iteration, where k is the number of terms in the generating series. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve maximum convergence order 2^n¹, but the new method produces convergence order of



^k^²



, which is better than the expected maximum convergence order of eight. Therefore, we show that the conjecture fails for a particular set of polynomial equations. We will demonstrate that the new method is very simple to construct.

Keywords: Newton-type method; Polynomial equation; Kung-Traub’s conjecture; Efficiency index; Optimal order of convergence.

Subject Classifications: AMS (MOS): 65H05.

INTRODUCTION

In this paper, we present a new one-point



^k^²



-order iterative method to find a simple root of a polynomial equation. Let p x( _n)a₀a x₁ _na x₂ _n² a x₃ _n³ a x_n _n^q be a polynomial of degree q and a are real constants. Finding zeros of a polynomial has been interest in _i theoretical and many areas of applied mathematics. It is well established that many higher order multi-point variants of the Newton-type method have been developed based on the Kung and Traub conjecture [4]. Here we present a new iterative method which has a better efficiency index than the classical Newton method [3,5,6,7,10]. This paper is actually a continuation form the previous studies [8,9], hence the new one-point method is applied to higher order polynomial equation.

For the purpose of this paper, we construct a new Newton-type iterative method of



^k^²



^-

order for finding simple root of polynomial equations. The new one-point method presented in this paper only uses n evaluations of the function per iteration. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve optimal convergence order 2 .^n¹ In fact, we have obtained a higher order of convergence than the maximum order of convergence suggested by Kung and Traub conjecture [4]. We demonstrate that the Kung and Traub conjecture fails for a particular case.

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Preliminaries

In order to establish the order of convergence of the iterative method [3,5,6,10], some of the definitions are stated:

Definition 1 Let f x be a real function with a simple root

 

 and let

 

xn be a sequence of real numbers that converge towards . The order of convergence p is given by

 

lim ⁿ 1 _p 0

n n

x x

 







  

 (1)

where p ^ and  is the asymptotic error constant (AEC). Let e_n x_n be the error in the nth iteration, then the relation

 

¹

1 ^p ^p ,

n n n

e_ e   e ^ (2)

is the error equation. If the error equation exists, then p is the order of convergence of the iterative method, [3,5,6,10].

Definition 2 Let n be the number of function evaluations of the iterative method. The efficiency of the iterative method is measured by the concept of efficiency index and defined

as ^{E n p}



^,



^ⁿ ^p ⁽³⁾

where p is the order of convergence of the method, [6].

Definition 3 (Kung and Traub conjecture) Let xn_1g x

 

n define as an iterative function without memory with n-evaluations. Then

 

opt ²ⁿ ¹^,

p g  p  ^ (4)

where ^p^opt is the maximum order, [4].

Convergence Analysis

In this section we define a new class of one-point



^k^²



-order method for finding simple root of a polynomial equation. In fact, the new iterative method is an extension of the Thukral’s quadratic and cubic methods, given in [8,9]. Hence, we will demonstrate that the new one-point method can be constructed to find a simple root of any order of polynomial equation. The order of convergence the new iterative method is determined by the k, that is number of terms in the generating series, hence depending on k we can construct any desired order of convergence.

The Method

The new one-point



^k^²



-order Newton-type method is expressed by

 

1 , , , ,1 2 3

n n k p

x_ x rH r t t t t (5)

where



1 2 3

 

1 2 3



1

, , , , 1 , , , , ;

k

i

k p p

i

H r t t t t AEC r t t t t i r



 



⁽⁶⁾

(3)

     

   

 

1 2

3 4 5

6 7 8

, , ,

2! 3!

, , ,

4! 5! 6!

, , ,

7! 8! 9!

n n n

iv v vi

n n n

vii viii ix

n n n

f x f x f x

r t t

f x f x f x

t t t

f x f x f x

t t t

f x f x f x



 

      



      



  



   

(7)

where x is the initial guess and provided that denominators of (7) are not equal to zero. ₀ Now, we shall verify the convergence property of the new one-point



^k^²



-order iterative method (5).

Theorem 1

Let D be a simple zero of a sufficiently smooth function f :D    for an open interval D. If the initial guess x is sufficiently close to ,₀  then the convergence order of the new one-point iterative method defined by (5) is



^k^²



^.

Proof

Let  be a simple root of f x , i.e.

 

^f

 

^ ^⁰^and ^f^

 

^ ^⁰, and the error is expressed as e x  .

The Taylor series expansion and taking into account ^f

 

^ ^⁰^{, we have}

  

² ² ³ ³ ⁴ ⁴ ⁵ ⁵ ⁶ ⁶ ⁷ ⁷ ⁸ ⁸



( _n) _n _n _n _n _n _n _n _n

f x  f  e c e c e c e c e c e c e c e  . (8)

  

² ³ ² ⁴ ³ ⁵ ⁴ ⁶ ⁵ ⁷ ⁶ ⁸ ⁷



( _n) 1 2 _n 3 _n 4 _n 5 _n 6 _n 7 _n 8 _n

f x  f   c e  c e  c e  c e  c e  c e  c e  . (9)

  

² ³ ⁴ ² ⁵ ³ ⁶ ⁴ ⁷ ⁵ ⁸ ⁶



( _n) 2 6 _n 12 _n 20 _n 30 _n 42 _n 56 _n

f x  f  c  c e  c e  c e  c e  c e  c e  . (10)

  

³ ⁴ ⁵ ² ⁶ ³ ⁷ ⁴ ⁸ ⁵



( _n) 6 24 _n 60 _n 120 _n 210 _n 336 _n

f x  f  c  c e  c e  c e  c e  c e  . (11)

 ^iv ⁽ ⁿ⁾

  

²⁴ ⁴ ¹²⁰ ⁵ ⁿ ³⁶⁰ ⁶ ⁿ² ⁸⁴⁰ ⁷ ⁿ³ ¹⁶⁸⁰ ⁸ ⁿ⁴



f x  f  c  c e  c e  c e  c e  .

(12)

 ^v ⁽ ⁿ⁾

  

¹²⁰ ⁵ ⁷²⁰ ⁶ ⁿ ²⁵²⁰ ⁷ ⁿ² ⁶⁷²⁰ ⁸ ⁿ³



f x  f  c  c e  c e  c e  .

(13)

 ^vi ⁽ ⁿ⁾

  

⁷²⁰ ⁶ ²⁵²⁰ ⁷ ⁿ ²⁰¹⁶⁰ ⁸ ⁿ²



f x  f  c  c e  c e  .

(14)

 ^vii ( _n)

 

2520 7 40320 8 _n



f x  f  c  c e  .

(15) where

(4)

     

 

2 , 3 , 4 , 5 , 6 ,

iv v vi

f f f f f

c c c c c

f f f f f

    

 

    

    

(16)

Dividing (8) by (9), we have

   

ⁿ ⁿ ² ⁿ² ²



²² ³



ⁿ³ ^.

n

f x e c e c c e

f x     



(17) and

   

ⁿ ² ² ^{2 2}



²² ³ ³



ⁿ ² ²



⁴ ²² ⁹ ³ ⁶ ⁴



ⁿ² ^.

n

f x

c c c e c c c c e

f x

  

     

 

  

 

(18)

   

ⁿ 6 3 12 2



4 2 3



_n .

n

f x

c c c c e

f x

  

   

 

  

 

(19)

 

24 4 24 5



5 2 2 4



.

iv n

n n

f x

c c c c e

f x

 

   

 

  

 

(20)

 

120 5 240 3



6 2 5



.

iv n

n n

f x

c c c c e

f x

 

   

 

  

 

(21)

 

720 6 720 7



7 2 2 6



.

vi n

n n

f x

c c c c e

f x

 

   

 

  

 

(22)

 

5040 7



40320 8 10080 2 7



.

vii n

n n

f x

c c c c e

f x

 

   

 

  

 

(23)

 

40320 8



362880 9 80640 2 8



.

viii n

n n

f x

c c c c e

f x

 

   

 

  

 

(24)

Substituting (19) in (1), we obtain

   

²



²



³

1 ⁿ 2 2 2 3 ,

n n n n

n

e e f x c e c c e

f x

      



(25)

Therefore, the asymptotic error constant given by (25) is expressed as

 

0 2 _n² . AEC c e 

(26)

(5)

It is well known that (26) is the asymptotic error constant for the classical Newton method.

Therefore, we take error equation (26) as our next term of the generating series given in (5).

Furthermore, we obtain a family of higher iterative method by increasing the terms of summation series of (6). Hence, we show the asymptotic error constant ^{AEC k}

 

^{for the}



^k^²



-order Newton-type method. In order to obtain the following terms of the generating series, we must substitute the essential parts in ^{AEC k}

 

^{, thus}

2 1, 3 2, 4 3, 5 4, 6 5,

c t c t c t c t c t

(27)

n . e r

(28)

where and r t are given by (7). We use this principle to obtain the following one-point _i



^k^²



-order iterative method;

Second-order Polynomial equation

This method has been recently presented in [8] and we review the basic expansion, 1. k1: One-point third-order iterative method is given by

   

1 1

1

1 1 1

k

i

n n n

i

x _ x r AEC i r x r t r



 

   



   

(29)

and the error equation

 

1 2 2^{2 3}_n

AEC  c e  (30)

2. k2: One-point fourth-order iterative method is given by

2 2

1 1 1 21

n n

x_ x r t r t r  (31)

 

2 5 ^{3 4}2 _n . AEC  c e 

(32)

3. k3: One-point fifth-order iterative method is given by

2 2 3 3

1 1 1 21 51

n n

x_ x r t r t r  t r  (33)

 

3 14 2^{4 5}_n . AEC  c e 

(34)

4. k4: One-point sixth-order iterative method is given by

2 2 3 3 4 4

1 1 1 21 51 141

n n

x_ x r t r t r  t r  t r  (35)

 

4 42 2^{5 6}_n . AEC  c e 

(36)

5. k5: One-point seventh-order iterative method is given by

(6)

2 2 3 3 4 4 5 5

1 1 1 21 51 141 421

n n

x_ x r t r t r  t r  t r  t r  (37)

 

5 132 2^{6 7}_n . AEC  c e 

(38)

6. k6: One-point eighth-order iterative method is given by

2 2 3 3 4 4 5 5 6 6

1 1 1 21 51 141 421 1321

n n

x_ x r t r t r  t r  t r  t r  t r  (39)

 

6 429 2^{7 8}_n .

AEC  c e 

(40)

It is well established that the maximum order of convergence of optimal methods with three functions evaluations is four. As illustrated above we have obtained order of convergence greater than four, hence the Kung and Traub conjecture fails for k3. To produce the next one-point higher order of convergence with only three function evaluations, we use the following principle. The process is very simple, using the coefficient of the error equation

 

AEC k as our coefficient of the next term of the generating series, thus we can calculate the next higher order of convergence method. Hence, ^{AEC k}

 

is the error equation of the



^k^²



-order Newton-type method.

Third-order Polynomial equation

This cubic equation method has been recently presented in [9] and review some of the results of the method.

1. k1: One-point third-order iterative method is given by

 

1 1 1

n n

x_ x r t r (41)

 

¹



² ²² ³



ⁿ³

AEC  c c e  (42)



²



²

1 1 1 21 2

n n

x _ x r t r t t r  (43)

 

2 5 2



2² 3



_n⁴

AEC  c c c e  (44)



²



²



²



³

1 1 1 21 2 51 1 2

n n

x  x r t r t t r  t t t r  (45)

 

³



¹⁴ ²⁴ ³ ³² ²¹ ²² ³



ⁿ⁵

AEC  c  c  c c e  (46)

(7)



²



²



²

 

³ ⁴ ² ²



⁴

1 1 1 21 2 51 1 2 141 32 211 2

n n

x _ x r t r t t r  t t t r  t  t  t t r  (47)

 

⁴ ¹⁴ ²



³ ²⁴ ² ³² ⁶ ²² ³



ⁿ⁶

AEC  c c  c  c c e  (48)



²



²



²

 

³ ⁴ ² ²



⁴



⁴ ² ²



⁵

1 1 1 21 2 51 1 2 141 32 211 2 141 31 22 61 2

n n

x_ x r t r t t r  t t t r  t  t  t t r  t t  t  t t r 

(49) and the error equation

 

⁵ ^{6 22}



²⁶ ² ³³ ³⁰ ²² ³² ⁵⁵ ²⁴ ³



⁷ⁿ

AEC  c  c  c c  c c e  (50)



²



²



²

 

³ ⁴ ² ²



⁴

1 1 1 21 2 51 1 2 141 32 211 2

n n

x _ x r t r t t r  t t t r  t  t  t t r



⁴ ² ²



⁵



⁶ ³ ^{2 2} ⁴



⁶

1 1 2 1 2 1 2 1 2 1 2

14t 3t 2t 6t t r 6 22t 2t 30t t 55t t r 

       

(51)

 

⁶ ³ ²



¹⁴³ ²⁶ ⁵⁵ ³² ³³⁰ ²² ³² ⁴²⁹ ²⁴ ³



ⁿ⁸

AEC  c c  c  c c  c c e  (52)

Due to Kung and Traub conjecture the maximum order of convergence of optimal methods with four functions evaluations is eight. As illustrated above when k7 we have obtained order of convergence greater than eight, hence the Kung and Traub conjecture fails.

Fourth-order Polynomial equation

 

1 1 1

n n

x_ x r t r (53)

 

¹



² ²² ³



ⁿ³

AEC  c c e  (54)



²



²

1 1 1 21 2

n n

x _ x r t r t t r  (55)

 

2



5 2³ 5 2 3 4



_n⁴

AEC  c  c c c e  (56)



²

 

² ³



³

1 1 1 21 2 51 51 2 3

n n

x  x r t r t t r  t  t t t r  (57)

 

³



¹⁴ ²⁴ ³ ³² ²¹ ²² ³ ⁶ ^{2 4}



⁵ⁿ

AEC  c  c  c c  c c e  (58)

(8)



²

 

² ³

 

³ ⁴ ² ²



⁴

1 1 1 21 2 51 51 2 3 141 32 211 2 61 3

n n

x _ x r t r t t r  t  t t t r  t  t  t t  t t r  (59)

 

⁴



⁴² ²⁵ ²⁸ ^{2 3}² ⁸⁴ ^{2 3}³ ⁷ ^{3 4} ²⁸ ²² ⁴



ⁿ⁶

AEC  c  c c  c c  c c  c c e  (60)



²

 

² ³



³

1 1 1 21 2 51 51 2 3

n n

x _ x r t r t t r  t  t t t r



14t1⁴ 3t2² 21t t1 2² 6t t r1 3

 

⁴ 42t1⁵ 28t t1 2² 84t t1 2³ 7t t2 3 28t t r1 3²



⁵

         

(61)

 

5



132 2⁶ 12 3³ 180 2² 3² 330 2⁴ 3 4 4² 120 2 4³ 72 2 3 4



_n⁷

AEC  c  c  c c  c c  c  c c  c c c e  (62)



²

 

² ³

 

³ ⁴ ² ²



⁴

1 1 1 21 2 51 51 2 3 141 32 211 2 61 3

n n

x  x r t r t t r  t  t t t r  t  t  t t  t t r



⁴²^t¹⁵ ²⁸^{t t}^{1 2}² ⁸⁴^{t t}^{1 2}³ ⁷^{t t}^{2 3} ²⁸^{t t}^{1 3}²



^r⁵



¹³²^t¹⁶ ¹²^t²³ ¹⁸⁰^{t t}^{1 2}^{2 2} ³³⁰^{t t}^{1 2}⁴ ⁴^t³² ¹²⁰^{t t}^{1 3}³ ⁷²^{t t t}^{1 2 3}



^r⁶^

            

(63)

 

6



429 2⁷ 165 2 3³ 990 2 3³ ² 1287 2 3⁵ 495 2⁴ 4 45 2 4² 45 3² 4 495 2² 3 4



_n⁸

AEC  c  c c  c c  c c  c c  c c  c c  c c c e  (64)

The maximum order of convergence of optimal methods with five functions evaluations is sixteen. To establish the order of convergence greater than sixteen we have to display next fifteen tedious terms, hence we have omitted them and it is clear that the Kung and Traub conjecture fails when k15.

Fifth-order Polynomial equation

 

1 1 1

n n

x_ x r t r (65)

 

¹



² ²² ³



ⁿ³

AEC  c c e  (66)



²



²

1 1 1 21 2

n n

x _ x r t r t t r  (67)

 

2



5 2³ 5 2 3 4



_n⁴

AEC  c  c c c e  (68)

(9)



²

 

² ³



³

1 1 1 21 2 51 51 2 3

n n

x _ x r t r t t r  t  t t t r  (69)

 

³



¹⁴ ²⁴ ³ ³² ²¹ ²² ³ ⁶ ^{2 4} ⁵



ⁿ⁵

AEC  c  c  c c  c c c e  (70)



²

 

² ³

 

³ ⁴ ² ²



⁴

1 1 1 21 2 51 51 2 3 141 32 211 2 61 3 4

n n

x _ x r t r t t r  t  t t t r  t  t  t t  t t t r  (71)

 

4



42 2⁵ 28 2 3² 84 ³2 3 7 3 4 28 2² 4 7 2 5



⁶_n

AEC  c  c c  c c  c c  c c  c c e  (72)



²

 

² ³



³

1 1 1 21 2 51 51 2 3

n n

x_ x r t r t t r  t  t t t r



14t1⁴ 3t2² 21t t1 2² 6t t1 3 t r4

 

⁴ 42t1⁵ 28t t1 2² 84t t1 2³ 7t t2 3 28t t1 3² 7t t r1 4



⁵

           

(73)

 

⁵



¹³² ²⁶ ¹² ³³ ¹⁸⁰ ²² ³² ³³⁰ ²⁴ ³ ⁴ ⁴² ¹²⁰ ^{2 4}³ ⁷² ^{2 3 4} ⁸ ^{3 5} ³⁶ ²² ⁵



ⁿ⁷

AEC  c  c  c c  c c  c  c c  c c c  c c  c c e 

(74)



²

 

² ³

 

³ ⁴ ² ²



⁴

1 1 1 21 2 51 51 2 3 141 32 211 2 61 3 4

n n

x_  x r  t r t t r  t  t t t r  t  t  t t  t t t r



42t1⁵ 28t t1 2² 84t t1 2³ 7t t2 3 28t t1 3² 7t t r1 4



⁵

     



132t1⁶ 12t2³ 180t t1 2^{2 2} 330t t1 2⁴ 4t3² 120t t1 3³ 72t t t1 2 3 8t t2 4 36t t1 4²



r⁶

         

(75)

 

⁶



⁴²⁹ ²⁷ ¹⁶⁵ ^{2 3}³ ⁹⁹⁰ ^{2 3}^{3 2} ¹²⁸⁷ ⁵^{2 3} ⁴⁹⁵ ^{2 4}⁴ ⁴⁵ ^{2 4}²

AEC  c  c c  c c  c c  c c  c c



2 2 3 8

3 4 2 3 4 4 5 2 5 2 3 5

45c c 495c c c 9c c 165c c 90c c c e_n

     

(76)

The maximum order of convergence of optimal methods with six functions evaluations is thirty-two. To establish the order of convergence greater than thirty-two we have to display next thirty-one tedious terms, hence we have omitted them and it is clear that the Kung and Traub conjecture fails when k31.

Sixth-order Polynomial equation

 

1 1 1

n n

x_ x r t r (77)

 

¹



² ²² ³



ⁿ³

AEC  c c e  (78)