Modification of three-steps iteration method with third

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Modification of three-steps iteration method with third-order hermite interpolation approach To cite this article: I Suryani et al 2018 J. Phys.: Conf. Ser. 1116 022045

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SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022045

IOP Publishing doi:10.1088/1742-6596/1116/2/022045

Modification of three-steps iteration method with third-order hermite interpolation approach I Suryani*, Wartono, Rahmadeni, and Khairudin Department of Mathematics, Faculty of Science and Technology, UIN Sultan Syarif Kasim, Pekanbaru, 28293, Indonesia *E-mail: [email protected] Abstract. Newton-Steffensen-Potra-Ptak methods is one of two steep iteration methods by Jisheng [3] that can be used to solve nonlinear equation involving the parameter  . In this paper, the author develop Newton-Steffensen-Potra-Ptak methods using hermite’s interpolation. Based on research result, obtained the new iteration equation involving the parameter  having eighth-order of convergence for   1 and sixth-order of convergence for   1 and involves four evaluation functions. Numerical simulation performed several function to show the performance of modification Newton-Steffensen-Potra-Ptak methods.

1. Introduction The iteration method is one way to calculate the root of a nonlinear equation. One of the iteration methods always used in the calculation is the Newton method. The general form of the Newton method is

xn 1  xn 

f ( xn ) f ' ( xn )

(1) Newton method is the simple iteration method and have quadratic convergence order. Numerical experts are trying to get new iteration methods. In the hope of getting a new method that has higher convergence order than the previous method. As the time progress, numerical experts try to get new iterative methods. With get new method have better order convergen from before method. Some of methods have been developed and have greater convergence order than the newton method. The examples are Newton Steffensen Method and Potra-Ptak Method have third-order convergence. Forms of Newton Steffensen Method and Potra-Ptak Method are :

xn1  xn 

f ( xn ) 2 f ' ( xn )( f ( xn )  f ( y n )) (2)

and

xn1  xn 

f ( xn )  f ( y n ) f ' ( xn )

(3)

with

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SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022045

y n  xn 

IOP Publishing doi:10.1088/1742-6596/1116/2/022045

f ( xn ) f ' ( xn )

(4) Many researchers have developed the Newton Steffensen method and the Potra-Ptak method. One of them is that of Jisheng et al. [3] where they combine the two methods in Equation (2) and Equation (3) with the form

xn1

f ( xn )  f ( y n ) f ( xn ) 2  xn    (1   ) f ' ( xn ) f ' ( xn )( f ( xn )  f ( y n ))

(5)

with,

y n  xn 

f ( xn ) f ' ( xn )

(6) By taking value then Eq.(4) has four convergence order. Furthermore, this paper will has develope two-step iteration method from [3] with adds Newton Method to third-steps. Then, the thirdsteps will approximated by using thirdorder Hermite polynomial interpolation. Some of the following basic definitions will be used for further discussion. Definition 1. Let are a simple root of a nonlinear equations ( ) and let * + be a sequence of real numbers that converges towards . We say that order of convergence of the sequence is ,if there exist and such that | | | | (7) | | | | If or , the sequence is said to have quadratic convergence or cubic convergence, respectively. Definition 2. Let is the error in the n-th iteration, we call the relation ( ) (8) Definition 3. Suppose that are three successive iterations closer to the root . Then, the computational order of convergence COC ( ) is approximated by using as |

||

| | |

| |

(9)

|

|

2. Results and discussions From the Equation (1) - (5) obtained the new iteration method such as,

y n  xn  z n  xn  

xn1  z n 

f ( xn ) f ' ( xn ) f ( xn )  f ( y n ) f ( xn ) 2  (1   ) f ' ( xn ) f ' ( xn )( f ( xn )  f ( y n ))

f (zn ) f ' (zn )

(10) Then, f ' ( z n ) will be approximated by using third order Hermite interpolation as practiced by Zhao [11]. By defining the three-order Hermite interpolation

H 3 ( x) 

( x  y n )( x  z n )  ( x  x n )(2 x n  y n  z n )  1   f ( xn ) ( x n  y n )( x n  z n )  ( x n  y n )( x n  z n ) 2 

2

SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022045

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( x  z n )( x  x n ) 2 ( x  xn ) 2 ( x  y n ) f ( y )  f (zn ) n ( y n  xn ) 2 ( y n  z n ) ( z n  xn ) 2 ( z n  y n ) ( x  xn )( x  y n )( x  z n )  f ' ( xn ) ( x n  y n )( x n  z n )



If Equation (11) is derived and by substituting the value

(11)

with z n obtained

(3x n  2 y n  z n )( y n  z n ) ( xn  z n ) 2 H 3 ' (zn )   f ( xn )  f ( yn ) ( xn  y n ) 2 ( xn  z n ) ( y n  xn ) 2 ( y n  z n ) x  2 y n  3z n y  zn  n f (zn )  n f ' ( xn ) ( z n  xn )( z n  y n ) y n  xn (12) Equation

(12)

can

be

simplified

to

be

f ( z n )  f ( xn ) f ( z n )  f ( y n ) f ( y n )  f ( xn )  2 z n  xn zn  yn y n  xn y  z n f ( y n )  f ( xn ) y n  z n  n  f ' ( xn ) y n  xn y n  xn y n  xn

H 3 ' (zn )  2

 2 f [ xn , z n ]  f [ y n , z n ]  2 f [ xn , y n ]  ( y n  z n ) f [ y n , xn , xn ]

(13)

with

f ( z n )  f ( xn ) z n  xn f ( y n )  f ( xn ) f [ xn , y n ]  f [ y n , xn ]  y n  xn

f [ xn , z n ] 

f ( zn )  f ( yn ) zn  yn f [ y n , xn ]  f ' ( xn ) f [ y n , xn , xn ]  y n  xn Then, obtained an approach f ' ( z n ) f [ yn , zn ] 

f ' ( z n )  H 3 ' ( z n )  2 f [ xn , z n ]  f [ y n , z n ]  2 f [ xn , y n ]  ( y n  z n ) f [ y n , xn , xn ]

(14)

Then, obtained

y n  xn 

f ( xn ) f ' ( xn )

f ( xn )  f ( y n ) f ( xn ) 2  (1   ) f ' ( xn ) f ' ( xn )( f ( xn )  f ( y n )) f (zn )  zn  2 f [ xn , z n ]  f [ y n , z n ]  2 f [ xn , y n ]  ( y n  z n ) f [ y n , xn , xn ] z n  xn  

xn1

(15) Equation (15) is a modification of the Newton-Steffensen-Potra-Ptak method using three-order Hermite interpolation and has four function evaluations namely f ( x n ) , f ( y n ) , f ( z n ) dan f ' ( x n ) .

3

SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022045

IOP Publishing doi:10.1088/1742-6596/1116/2/022045

The following is given a convergence analysis related to the results obtained. Theorem 1. Let be a real-valued function which has a derivative in the open interval and has completion roots. If the initial point sufficiently close to , then the method in Equation (15) converges to at sixth order. If is any real number and order of eight. If with the error function is

en1  (  1) 2 c25 en  ((  1)c4 c23  (5  13  8 2 )c24 c3  (4  16  12 2 )4c26 )en7 6

 (24 2 c32 c23  46c32 c23  84 2 c27  47c27  58c24 c4  22c27  113 2 c25 c3  66c24 c4  68c23 c32  11 2 c24 c4  162c25 c3  60c25 c3  39c4 c22 c3  40c4 c22 c3 )en8

 O(en9 ) Proof. Suppose

(16) is root of f (x) , then f ( )  0 . Then, assume ( )(

and

)

( )

,

,

,

then using Taylor series expansion obtained

( )

f ( xn ) 2 2 3 3 4  en  c2 en  2(c2  c3 )en  (7c2 c3  4c2  3c4 )en f ' ( xn )

 ...  (en ) 9

(17)

From The equation (6) dan (17) obtained

y n    c2 en  2(c2  c3 )en  (7c2 c3  4c2  3c4 )en  ...  (en ) 2

2

3

3

4

9

(18)

or , with

c2en 2  2(c22  c3 )en3  (7c2c3  4c23  3c4 )en 4  ...  (en9 ) Furthermore, using Taylor series and ( )

obtained

f ( y n )  f ' ( )(c2 en  2(c3  c2 )en  (3c4  3c2  7c2 c3 )en 2

2

3

3

4

  (en )) 9

(19)

Then,

f ( xn )  f ( y n )  en  2c22 en3  (3c23  5c2 c3  7c2 c3  12c2 c22 )en4 f ' ( xn )  ...  O(en9 )

(20)

and,

f ( xn )  1  c2 en  (2c3  2c22 )en2  (3c23  3c4  6c2 c3 )en3 f ( xn )  f ( y n )  (13c24  11c22 c3  8c2 c4  4c32 )en4  ...  O(en9 ) .

(21)

Then, multiply Equation (21) and (17) obtain

f ( xn ) f ( xn )  en  c22 en3  (3c23  3c2 c3 )en4  ...  O(en9 ) f ( xn )  f ( y n ) f ' ( xn ) Next, found value z n ,

4

(22)

SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022045

z n  xn  

IOP Publishing doi:10.1088/1742-6596/1116/2/022045

f ( xn )  f ( y n ) f ( xn ) 2  (1   ) f ' ( xn ) f ' ( xn )( f ( xn )  f ( y n ))

   (c22  c22 )en3  (3c2 c3  3c23  4c2 c3  6c23 )en4  ...  O(en9 ) Using Taylor series, will be found (

(23)

) and obtained

f ( z n )  f ' ( )((c  c )e  (3c2 c3  3c23  4c2 c3  6c23 )en4 2 2

2 2

3 n

  (en )) Furthermore, to get 2 f [ xn , z n ] will be found 9

(24)

z n  xn  en  (c22  c22 )en3  (3c2 c3  3c23  4c2 c3  6c23 )en4  ...  O(en9 ) (24) and

f ( z n )  f ( xn )  f ' ( )(en  c2 en2  (c22  c22  c3 )en3  (3c2 c3  3c23  4c2 c3  6c23  c4 )en4  ...  O(en9 )

(25)

From Equation (24) and (25) obtained

2 f [ xn , z n ]  2

f ( z n )  f ( xn ) z n  xn

 f ' ( )(2  2c2 en  2c3 en  (2c2 (c22  c22 )  2c4 )en 2

3

 (2c24  2 2 c24  4c24  2(c22  c22  c3 )(c22  c22 )  2c2 (3c23  6c23  3c2 c3  4c2 c3 ))en  ...  O(en )) 4

9

(26)

From Equation (19) and (24) obtained

f [ yn , zn ] 

f ( zn )  f ( yn ) zn  yn

 f ' ( )(1  c22 en2  (2c2 c3  c23  c23 )en  (3c2 c4  4c22 c3  6c24 3

 c24  4c22 c3 )en  ...  (en )) 4

Then, obtained

2 f [ xn , y n ]  2

9

(27)

f ( y n )  f ( xn ) y n  xn  f ' ( )(2  2c2 en  (2c22  2c3 )en2  (2c4  4c23  6c2 c3 )en3

 (8c2 c4  16c3c22  4c32  6c24 )en4  ...  (en9 )) Furthermore, to get value 2 f [ x n , y n ] will be found

(28)

y n  xn  en  c2 en2  (2c22  2c3 )en3  (3c4  7c2 c3  4c23 )en4

 ...  O(en9 )

(29)

and

f ( y n )  f ( xn )  f ' ( )(en  (c3  2c22 )en3  (5c23  2c4  73c2 c3 )en4

 ...  O(en9 ))

(30)

Then, from Equation (29) and(30) obtained

2 f [ xn , y n ]  2

f ( y n )  f ( xn ) y n  xn

5

SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022045

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 f ' ( )(2  2c2 en  (2c22  2c3 )en2  (2c4  4c23  6c2 c3 )en3

 (8c2 c4  16c3c22  4c32  6c24 )en4  ...  (en9 ))

(31)

Next, obtained

yn  z n  sn  d n  c2 en2  (2c3  c22  3c22 )en3  (3c4  10c2 c3  6c23  4c2 c3 )en

4

 ...  (en ) 9

(32)

From Equation (29), (30) and (32) obtained

y n  z n f ( y n )  f ( xn )  f ' ( )(c2 en  (c22  c22  2c3 )en2  (4c2 c3  4c23 y n  xn y n  xn  2c23  3c2 c3  3c4 )en3  (8c22 c3  13c24  21c22 c3  6c2 c4  2c32  12c24  4c32  4c2 c4 )en4

 ...  O(en9 ))

(33)

Furthermore, from Equation (29) and (30) obtained

yn  z n f ' ( xn )  f ' ( )(c2 en  (c22  2c3 )en2  (3c4  4c2 c3  c2 c3  c23 y n  xn  6c2 c4  15c22 c3  3c23 )en3  (7c22 c3  2c2 c4  19c24  7c24  4c32  2c32 )en4  ...  O(en9 ))

(34)

From Equation (33) and (34) obtained,

( y n  z n ) f [ y n , xn , xn ] 

f [ y n , xn ]  f ' ( xn ) y n  xn

 f ' ( )(c22 en2  (3c23  4c2 c3  c23 )en3  (4c32  6c2 c4  6c22 c3  6c24  15c22 c3  7c24 )en4

 ...  O(en9 )

(35)

From Equation (26), (27), (28) and (35) obtained,

2 f [ xn , z n ]  f [ y n , z n ]  2 f [ xn , y n ]  ( y n  z n ) f [ y n , xn , xn ]  f ' ( )(1  (2c23  2c23 )en  (c2 c4  8c22 c3  4c24  5c22 c3  12c24 )en4 ) 3

 ...  O(en9 )

(36)

Next, from Equation (24) and (36) obtained,

f (zn ) 2 f [ xn , z n ]  f [ y n , z n ]  2 f [ xn , y n ]  ( y n  z n ) f [ y n , xn , xn ]  (c22  c22 )en  (3c23  3c2 c3  4c2 c3  6c23 )en    O(en ) 3

4

9

Then, The error equation from Equation (15) is

xn1  z n 

f (zn ) 2 f [ xn , z n ]  f [ y n , z n ]  2 f [ xn , y n ]  ( y n  z n ) f [ y n , xn , xn ]

en1  (c22  c22 )en3  (3c2 c3  3c23  4c2 c3  6c23 )en4  O(en5 )  ((c22  c22 )en  (3c23  3c2 c3  4c2 c3  6c23 )en    O(en )) 3

4

9

 ( 2 c25  2c25  c25 )en6  (c23c4  5c24 c3  16c26  8 2 c24 c3  4c26  c23c4

6

(37)

SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022045

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 c23c4  12 2 c26  13c24 c3 )en7  (47c27  46c32 c23  22c27  162c25 c3  39c22 c3 c4  40c22 c4 c3  66c24 c4  84 2 c27  58c24 c4  113 2 c25 c3

 60c25 c3  24 2 c32 c23  68c32 c23 )en  (en ) . 8

9

(38) In Equation (38), it can be seen that the three-step iteration method of modification of the NewtonSteffensen-Potra-Ptak method using three-order Hermite interpolation has a sixth convergence order and is still dependent on the parameter . If parameters are taken obtained that the best iteration with the highest convergence order is when with an eight-convergence order with four function evaluationsie f ( x n ) , f ( y n ) , f ( z n ) and f ' ( x n ) then its efficiency index value is 1

8 4  1,682 . Furthermore, to test the performance of new iteration methods, a numerical simulation of some real functions using maple 13 and 800 floating arithmethics. Computational calculations are performed by taking initial values x0 as close as possible to the roots of nonlinear equations ( ) and using eight real functions. As for the functions used are :

f 2 ( x)  sin ( x)  x  1 f 3 ( x)  cos( x)  x

  1,365230013414   1,4044916482 153411   0,7390851332 1516067

f 4 ( x )  ( x  2) e x  1 f 5 ( x)  sin( x)e x  ln(x 2  1)

  0,4428544010 023  0

f 6 ( x)  x  x

 1

f1 ( x)  x 3  4 x 2  10 2

f 7 ( x)  e  x

2

 x2

2

  1   2,1544346900 31883

 cos( x  1)  x 3  1

f 8 ( x)  x 3  10

Furthermore, there will be a comparison of the number of iterations with several other iteration methods, namely MN, MPP, MNS, MNSPP, Ostrowski (MMO) methodology and iteration methods as in Equation (15) with =-1 , =0, and =1. Here is a table of comparison iterations based on the function that has been given.

f (x)

x0

f1 ( x) f 2 ( x) f 3 ( x)

-1

Table 1. Comparison of number of iterations Number of Iterations (MNSPPH) (MNSPPH) (MNSPPH) MN MPP MNS MNSPP MMO (  0 ) (   1) (   1 ) 10 6 6 5 4 4 4 3

2

10

7

7

7

4

4

4

3

1

9

5

6

4

3

3

3

3

f 4 ( x) f 5 ( x)

0,8

11

6

7

6

4

4

4

4

0,8

12

8

7

6

4

4

4

4

f 6 ( x)

1,5 0,65 2,4

9

6

6

4

3

3

3

3

10

6

6

5

3

3

3

3

9

7

7

4

3

3

3

3

f 7 ( x) f 8 ( x)

7

SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022045

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Based on table 1, it can be concluded that Newton Steffensen Potra-Ptak Hermite Method is better than the other seven methods because it has fewer iterations.Then, it will be shown the convergence order of the Iteration Equation in Table 1 by using the Computational Order of Convergence (COC) calculation. Table2. Comparison of COC values COC Values

f (x)

x0

f1 ( x)

-1

f 2 ( x)

2

f 3 ( x)

1

f 4 ( x)

0,8

f 5 ( x)

0,8

f 6 ( x)

1,5

f 7 ( x)

-0,65

f 8 ( x)

2,4

f (x)

x0

f1 ( x)

-1

f 2 ( x)

2

f 3 ( x)

1

f 4 ( x)

0,8

f 5 ( x)

0,8

f 6 ( x)

1,5

f 7 ( x)

-0,65

f 8 ( x)

2,4

MN 1,9999999999 999 1,9999999999 999 2,0000000000 000 1,9999999999 999 1,9999999999 999 1,9999999999 999 2,0000000000 000 1,9999999999 999 MMO 5,9999951330 005 5,9999999999 999 5,9999999755 081 5,9999999999 994 5,9999999884 044 5,9999995123 504 5,9999999450 232 5,9999999490 445

MPP MNS 3,000000000000 3,0000000000 0 000 2,999999999999 2,9999999999 9 999 2,999999999999 2,9999999999 9 999 2,999999999999 2,9999999999 9 999 2,999999999999 2,9999999999 9 999 2,999999999999 3,0000000000 9 000 3,000000000000 3,0000000000 0 000 2,999999999999 2,9999999999 9 999 COC Values (MNSPPH) (MNSPPH) (  0 ) (   1) 5,999992751355 5,9999999999 29 999 5,999999999999 5,9999999999 99 999 5,999999975508 5,9999999891 13 042 5,999999999427 5,9999996182 66 044 5,999999998840 5,9999999515 44 162 5,999999838628 5,9999993480 62 249 5,999999811475 6,0000004596 66 816 5,999999916985 5,9999995213 07 631

MNSPP 3,999999999999 99 3,999999999999 99 3,999999999999 99 3,999999999999 99 3,999999999999 99 3,999999999952 81 3,999999999999 99 3,999999999921 72 (MNSPPH) (   1 ) 7,999992100156 8 7,999950700079 8 7,999999999410 7 7,999999999999 9 7,999999999999 2 7,999999952722 5 7,999999985694 7 7,999999993843 5

From Table 2, based on COC value, it can be concluded that steffensen method has eight convergence order. 3. Conclusion

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SEMIRATA- International Conference on Science and Technology 2018 IOP Conf. Series: Journal of Physics: Conf. Series 1116 (2018) 022045

IOP Publishing doi:10.1088/1742-6596/1116/2/022045

Based on the results of numerical simulations in Table 1 and Table 2 MNSPPH ( ) generally have fewer iterations than others and have higher COC values. And the MNSPPH Efficiency index value ( ) is greater than other comparative methods which means that the method is more effective in solving nonlinear equations. References [1] Fang, Liang and He G 2009 Journal of Computational and Applied Mathematics 228 296 [2] Jaiswal J P and Panday S 2013 Universal Journal of Computational Mathematics 1 83 [3] Kou J 2006 Applied Mathematics and Computation 10 10 [4] Khattri and Kumar S 2012 Global-Science Press 4 592 [5] Munir R 2008 Metode Numerik Edisi Revisi (Bandung: Penerbit Informatika) [6] Salas and Saturnino 2003 Calculus: One Variabel Ninth Edition (United States: John Wiley and Sons) [7] Sharma J R. 2011 Plagia Research Library 1 240 [8] Singh S and Gupta D K 2014 The Scientific World Corporation http://dx.doi.org/10.1155/2014/890138 [9] Wang, Xia and Liu L 2010 Applied Mathematics Letters 23 549 [10] Weerakon, S and Fernando T G I 2000 Applied Mathematics Letters 13 87 [11] Zhao and Lingling 2012 WSEAS Transactions on Mathematics 11 2448

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