IMPROVING THREE-POINT ITERATIVE METHODS FOR SOLVING ...

IMPROVING THREE-POINT ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS

arXiv:1302.3095v3 [math.GM] 9 Apr 2013

FAYYAZ AHMAD AND D. GARCA-SENZ

Abstract. In this article, we report on sixth-order and seventh-order iterative methods for solving nonlinear equations. In particular sixth-order derivative-based and derivative-free iterative families are constructed in such a way that they comprise a wide class of sixth-order methods which were developed in the past years. Weighting functions are introduced to enhance the algorithmic efficiency whereas an appropriate parametric combination gives weight-age flexibility in between those weighting functions. The usage of weighting factors and weighting functions define a wide class of iterative schemes for solving nonlinear equations. The freedom to construct different parametric combinations as well as different forms of weighting functions makes the iterative schemes more accurate and flexible, It means that one can easily modify the scheme by changing weight functions and parametric combination.

1. Introduction Nonlinear algebraic equations are at the heart of many problems of nonlinear science. However, it is not always possible to find exact analytical solutions of these nonlinear equations which very often have to be solved numerically. Iterative methods are numerical algorithms which, starting from an initial guess in the neighborhood of a root of the nonlinear equation, manage to refine it and achieve convergence to the true root after several iterations . We can divide them in two classes, namely memory-based and without-memory iterative methods. From the point of view of efficiency and stability, multipoint iterative methods are superior than single-point methods. Kung and Traub conjectured [13] that multipoint iterative methods without-memory have at most 2n order of convergence for n + 1 function evaluations per iteration. The further classification falls into derivative-free and non-derivative-free iterative methods. The multipoint iterative methods for nonlinear equations, which use derivatives, are efficient. The most famous derivative-based iterative method is Newton’s method (NM) which is written as (1)

xn+1 = xn −

f (xn ) , f ′ (xn )

n = 0, 1, 2, 3, · · · .

It is quadratically convergent in some neighborhood of the root α of f (x) = 0. The computational effort invested in the derivative of a function is not always comparable to the evaluation of the function itself [2, 3, 4, 5, 6, 7, 8]. Steffensen [1] gave a quadratically convergent derivative-free iterative method for nonlinear equations. The Steffensen’s method (SM) is given by   = xn − κf (xn ),  wn df (xn )   xn+1

(2)

f (xn )−f (wn ) κf (xn ) n) . xn − dff(x (xn )

=

=

f ′ (x

Clearly, df (xn ) is a finite difference approximation of n ). The SM approximation merely replaces the derivative evaluation by a function evaluation. Actually both iterative method NM and SM achieve the optimal order of convergence 2, which is in agreement with the conjecture of Kung and Traub. The computational efficiency of an iterative method (IM) is frequently calculated by the Ostrowski-Traub’s [14, 15] efficiency index E(IM ) = p1/d ,

(3)

where p is order of convergence and d is function evaluations per iteration. The optimal computational efficiency is defined by expression (n)

Eopt = 2n/(n+1) .

(4)

We now briefly review a number of three step sixth-order convergent methods. Many three steps sixth-order convergent methods use three function and one derivative evaluations. Sharma and Guha [9] proposed the following sixth-order convergent family by introducing one parameter (SG)  f (x )  = xn − f ′ (xn ) ,   yn n f (yn ) (xn ) , zn = yn − f (x f)−2f (5) (yn ) f ′ (xn ) n   f (x )+af (y f (zn ) n n) x =z − , n+1

n

f (xn )+(a−2)f (yn ) f ′ (xn )

Key words and phrases. Non-linear equations, Iterative methods, Derivative free, Steffensen’s method, Convergence order. 1

2


where a ∈ ℜ. Neta [10] uni-parametric sixth-order family consists of three step (NT1)  n)  , y = xn − ff′(x  (xn )  n f (xn )+af (yn ) f (y ) zn = yn − f (x )+(a−2)f (y ) f ′ (xn ) , (6) n n n   f (xn )−f (yn ) f (zn ) x =z − , n

n+1

where a ∈ ℜ. In 2011, Neta [11] provided a    yn     z   n (7)    xn+1    

f (xn )−3f (yn ) f ′ (xn )

more efficient sixth-order scheme (NT2) = xn − = yn −

f (xn ) , f ′ (xn ) f (yn ) 1 2 , ′ f (xn ) f (yn ) 1− f (x ) n

= zn −

f (zn ) f ′ (xn )

1

f (y ) f (z ) 1− f (xn ) − f (xn ) n n

2 .

Chun and Ham [12] also presented a family of sixth-order iterative methods (CH)  n)  y = xn − ff′(x ,  (xn )  n f (xn ) f (yn ) (8) zn = yn − f (x )−2f , (yn ) f ′ (xn ) n   f (z ) x n , = z − H(µ ) n n f ′ (x ) n+1 n

where real valued function H(s) satisfies H(0) the sixth-order (GR)   y   n (9) zn   x n+1

= 1,

H ′ (0)

= xn − = yn − = zn −

= 2, and µn = f (yn )/f (xn ). Grau et al. [16] developed

f (xn ) , f ′ (xn ) f (xn ) f (xn )−2f (yn ) f (xn ) f (xn )−2f (yn )

f (yn ) , f ′ (xn ) f (zn ) . f ′ (xn )

Alicia cordero et al. [17] constructed the following sixth-order scheme (AL)  n)  , y = xn − ff′(x  (xn )  n f (xn )+f (yn ) f (x ) f (xn ) (10) zn = xn − θ − (1 − θ) f ′ (xn ) f (x )−f , f ′ (xn ) (yn ) n n   f (z ) x n , =z − n

n+1

f [zn ,yn ]+f [zn ,xn ,xn ](zn −yn )

where θ ∈ ℜ.Finally, Sanjay K. Khattri et al. [21] presented a paper, Unification of sixth-order iterative methods, which covers all the features of sixth-order iterative methods (SK1)  n) yn = xn − ff′(x ,  (xn )   j  m  P  f (yn ) f (yn ) zn = yn − f ′ (x ) 1 + aj , where a1 = 2, f (xn ) n (11) j=1  k   l P  µ1 f (yn )+µ2 f (zn )  xn+1 = zn − f′(zn ) 1 + bk , where b1 = 2/µ1 , f (x ) f (x ) n

+

n

k=1 +

where, {aj , bk , µ1 , µ2 } ∈ ℜ, m ∈ Z , and l ∈ Z are independent parameters. Sanjay K. Khattri et al. [18] also developed four parameter sixth-order derivative free family (SK2)

(12)    yn    zn      xn+1

= xn − κ f (x

f (xn )f (yn ) n )−f (xn −κf (xn ))

1+

f (yn ) f (xn )

f (xn )f (zn ) n )−f (xn −κf (xn ))

1+

f (yn ) f (xn )

= yn − κ f (x = zn − κ f (x

f (xn )2 , n )−f (xn −κf (xn ))

2 f (y ) + α f (xn ) + n 2 f (y ) + α f (xn ) + n

f (yn ) f (xn −κf (xn ))

+β

f (yn ) f (xn −κf (xn ))

+β

2 , 2 f (z ) + η f (yn ) . ))

f (yn ) f (xn −κf (xn )) f (yn ) f (xn −κf (xn

n

In [19], R. Thukral constructed Steffensen-Six.one method (TS1) and Steffensen-six.two method (TS2), which are respectively   wn = xn + f  (xn ),    f (xn )2   yn , = x −  n f (wn )−f (xn )   −λ (13) xn −yn −1 f (yn )  zn = y f (yn ), n− 1−λ  f (wn ) f (xn )−f (yn )    −λ    (yn ) xn −yn  xn+1 = zn − 1 − λ−1 ff(w f (zn ), ) f (x )−f (y ) n

n

n


3

and  wn        yn  

(14)

= xn + f (xn ),

f (xn )2 f (wn )−f (xn )

= xn −

 zn        xn+1

= yn − = zn −

f (y )

1 − λ−1 f (wn ) n zn −yn f (zn )−f (yn )

, −λ

xn −yn f (xn )−f (yn )

f (yn ),

f (zn ),

where λ ∈ ℜ/{0}. F. Soleymani et al. [20] discussed the following sixth-order derivative-free method (FS1)   = xn + f (xn ), wn   f (xn )   y = xn − f [x ,w ,  n  n n] f (x ) f (y ) f (y ) n (15) zn = xn − f [x ,w 1 + f (xn ) 1 + 2 f (xn ) ,  n n] n n      1+f [xn ,wn ] f (zn ) f (zn )  xn+1 = zn − f [y 1 − , ,z ] f [x ,w ] f (w ) n

(yn ) , f [xn , wn ] where f [yn , zn ] = f (zzn )−f n −yn derivative-free scheme (FS2)  wn      yn (16)    zn   xn+1

=

n

f (wn )−f (xn ) . wn −xn

n

n

n

F. Soleymani [22] also presented an other sixth-order

= xn − βf (xn ), = xn − = yn − = zn −

f (xn ) , f [xn ,wn ] f (yn ) , f [wn ,yn ]

f (zn ) , f [wn ,zn ]+f [zn ,yn ]−f [wn ,yn ]

where β ∈ ℜ/{0}.In 2013, F. Soleymani [26] developed some efficient seventh-order derivative-free methods. First is (FS3-1)   wn = xn + f (xn ),   f (xn )   yn = xn − f [xn ,wn ] ,  f (yn ) , zn = yn − f [x ,y ]+f [y (17) n n " n ,wn ]−f [xn ,wn ]  ( !2 # )     f (zn ) 2+f [xn ,wn ] f (zn ) f (zn ) f (zn ) f (yn ) f (yn )   x = z − + γ + + + δ 1 + , n  n f [xn ,zn ] f (wn ) f (yn ) f (xn ) f (xn ) f (wn ) (1+f [xn ,wn ])2 where γ and δ are real numbers. Second is (FS3-2)   wn = xn − f (xn ),    f (xn )   y = xn − f [x ,w ,   n n n] f (yn ) , zn = yn − f [x ,y ]+f [y (18) n n " n ,wn ]−f [xn ,wn ]  ) (     f (zn ) 2−f [xn ,wn ] f (zn ) f (yn )   x = z − + + 1 + n n 2  f [xn ,zn ] f (wn ) f (yn ) (1−f [xn ,wn ])

f (yn ) f (xn )

!2

f (z )

f (z )

#

+ ρ f (xn ) + τ f (wn ) , n

n

where ρ and τ are real numbers. Third is (FS4-1) (19)   wn      y   n zn        xn

= xn + f (xn ), = xn −

f (xn ) , f [xn ,wn ]

= yn −

f (yn ) , f [xn ,yn ]+f [yn,wn ]−f [xn ,wn ]

= zn −

f (zn ) f [wn ,zn ]

"

1+

f (zn ) f (yn )

+

f (yn ) f (xn )

n o + 2 + f [xn , wn ](3 + f [xn , wn ])

f (yn ) f (wn )

!2

where ω and φ are real numbers. Fourth is (FS4-2)   wn = xn − f (xn ),   f (xn )   yn = xn − f [xn ,wn ] ,  f (yn ) , zn = yn − f [x ,y ]+f [y (20) n n " n ,wn ]−f [xn ,wn ]    n o   f (zn ) f (yn ) f (zn )   xn = zn − f [w ,z ] 1 + f (y ) + f (x ) + 2 + f [xn , wn ](−3 + f [xn , wn ]) n

n

n

n

f (z )

f (z )

#

+ ω f (xn ) + φ f (wn ) , n

f (yn ) f (wn )

n

!2 #

.

4


2. A More general formulation of sixth-order iterative methods In this section, we go beyond the iterative method (SK1) and present a more general formulation which covers most of the sixth-order iterative methods based on one derivative evaluation f ′ (x). The proposed iterative scheme is (FD1)

(21)

    yn zn   x

n+1

where t1 =

f (yn ) , t2 f (xn )

=

f (zn ) f (xn )

and t3 =

f (zn ) . f (yn )

f (xn ) , f ′ (xn ) f (y ) yn − G(t1 ) f ′ (xn ) , n n) , zn − H(t1 , t2 , t3 ) ff′(z (xn )

= xn − = =

we state the following theorem.

Theorem 1. Let f : D ⊆ ℜ → ℜ be a sufficiently differentiable function, and α ∈ D is a simple root of f (x) = 0, for an open interval D. If x0 is chosen sufficiently close to α, then iterative scheme (21) converges to α. If G and H satisfy (22)

G(0) = 1, dG/dt1 (0) = 2, H(0, 0, 0) = 1, ∂H/∂t1 (0, 0, 0) = 2,

and both G, H have bounded higher order derivatives then the iterative scheme (21) has convergence order at-least six and along with conditions (22) if ∂ 2 H/∂t21 (0, 0, 0) = d2 G/dt21 (0) + 2, ∂H/∂t3 (0, 0, 0) = 1

(23)

then the iterative scheme (21) has convergence order at least seven.

Proof. let α be the simple root of f (x) = 0 and en = xn − α. Further we denote c1 = f ′ (α) and ck = where k = 2, 3, · · · . The Tyalor’s expression of f and f ′ around α in terms of en :

(k) ek (α) n f , k! f ′ (α)

f (xn ) = c1 [en + c2 e2n + c3 e3n + c4 e4n + c5 e5n + c6 e6n + O(e7n )]

(24)

f ′ (xn ) = c1 [1 + 2c2 en + 3c3 e2n + 4c4 e3n + 5c5 e4n + 6c6 e5n + O(e6n )]

(25)

Substituting (24) and (25) into yn , we obtain yn = x n −

f (xn ) f (xn ) f (xn ) , yn − α = x n − α − ′ = en − ′ f ′ (xn ) f (xn ) f (xn )

= c2 e2n + (2c3 − 2c22 )e3n + (3c4 − 7c2 c3 + 4c32 )e4n + (4c5 − 10c2 c4 − 6c23 + 20c3 c22 − 8c42 )e5n (26)

+ (−17c3 c4 + 33c2 c23 − 52c3 c32 + 28c4 c22 − 13c2 c5 + 5c6 + 16c52 )e6n + O(e7n )

Expansion of f (yn ) around α gives f (yn ) = c1 [c2 e2n − 2(−c3 + c22 )e3n + (3c4 − 7c2 c3 + 5c32 )e4n − 2(−2c5 + 5c2 c4 + 3c23 − 12c3 c22 + 6c42 )e5n (27)

c1 (37c2 c23 − 73c3 c32 + 28c52 + 34c4 c22 − 17c3 c4 − 13c2 c5 + 5c6 )e6n + O(e7n )]

We define G(t1 ) = 1 + 2t1 + M0 t21 + M1 t31 + M2 t41 + higher powers in t1 , where M0 , M1 and M2 are finite real numbers. By using (24)-(27), the expressions for zn and f (zn ) are zn = yn − G(t1 )

f (yn ) f ′ (xn )

zn − α = −c2 (−5c22 + M0 c22 + c3 )e4n + (−2c2 c4 − 2c23 + 32c3 c22 − 36c42 − M1 c42 − 6M0 c22 c3 + 10M0 c42 )e5n (−12M0 c2 c23 − 9M0 c22 c4 + 74M0 c32 c3 − 62M0 c52 − 7c3 c4 + 66c2 c23 − 262c3 c32 + 48c4 c22 − 3c2 c5 (28)

+ 170c52 − M2 c52 + 13M1 c52 − 8M1 c32 c3 )e6n + O(e7n ) f (zn ) = −c1 c2 (−5c22 + M0 c22 + c3 )e4n + c1 (−2c2 c4 − 2c23 + 32c3 c22 − 36c42 − M1 c42 − 6M0 c22 c3 + 10M0 c42 )e5n + c1 (−12M0 c2 c23 − 9M0 c22 c4 + 74M0 c32 c3 − 62M0 c52 − 7c3 c4 + 66c2 c23 − 262c3 c32

(29)

+ 48c4 c22 − 3c2 c5 + 170c52 − M2 c52 + 13M1 c52 − 8M1 c32 c3 )e6n + O(e7n )


5

Now, we define H(t1 , t2 , t3 ) = 1 + 2t1 + R0 t2 + R1 t3 + R2 t21 + R3 t22 + R4 t23 + higher powers in t1 , t2 , t3 . Again, substituting (25), (28) and (29) in (21), we obtain xn+1 − α = zn − α − H(t1 , t2 , t3 )

f (yn ) f ′ (xn )

= −c2 (−5c22 + M0 c22 + c3 )(R1 M0 c22 − 5R1 c22 + 6c22 − R2 c22 + R1 c3 − c3 )e6n + (10R0 c62 M0 − R0 c62 M02 + 10R1 M1 c62 − 172R1 c62 M0 + 20R1 M02 c62 + 2M0 c32 c4 + 6M0 c22 c23 − 68M0 c42 c3 + 88M0 c62 + 124R1 M0 c42 c3 − 12R1 M0 c22 c23 − 10R1 M02 c42 c3 − 2R1 M1 c62 M0 − 2R1 M1 c42 c3 − 2R0 c42 M0 c3 − 4R1 c4 M0 c32 − 22c4 c32 − 66c23 c22 + 366c3 c42 + 2c33 − 356c62 + 4c4 c2 c3 − 6M1 c62 − 25R6 c62 + 10R6 c62 M0 + 10R6 c42 c3 − R6 c62 M02 − R6 c22 c23 + M1 c42 c3 + 76R2 c62 − 2R6 c42 M0 c3 − 4c2 R1 c4 c3 − 342R1 c3 c42 + 20R1 c4 c32 + 10R0 c42 c3 + 64c22 R1 c23 − R0 c22 c23 − 25R0 c62 + 360R1 c62 − 2R1 c33 − 60R2 c42 c3 + 6R2 c22 c23 + 2R2 c32 c4 − 18R2 c62 M0 + R2 c62 M1 + 10R2 c42 M0 c3 )e7n + O(e8n )

(30)

By equating one of the factor of coefficient of e6n to zero, we have R1 M0 c22 − 5R1 c22 + 6c22 − R2 c22 + R1 c3 − c3 = 0.

(31)

Now by equating the coefficient of c3 and c22 in (31), we find ( R1 − 1 = 0, (coefficient of c3 ) (32) R1 M0 − 5R1 + 6 − R2 = 0, (coefficient of c22 ) where M0 = 1/2 d2 G/dt21 (0), R1 = ∂H/∂t3 (0, 0, 0) and R2 = 1/2 ∂ 2 H/∂t21 (0, 0, 0). Which completes the proof.

Remark 2.1. It is obvious that sixth-order family (21) can construct all the presented iterative methods in (5)-(9) and (11). (10) can be written as (FD2)

(33)

   yn zn   x

= xn − = xn −

n+1

where t1 =

f (yn ) f (xn )

and A(t1 ) =

1−θt1 1−t1

= zn −

f (xn ) , f ′ (xn ) f (yn ) A(t1 ) f ′ (x ) , n f (zn ) , f [zn ,yn ]+f [zn ,xn ,xn ](zn −yn )

, with θ ∈ ℜ and f [zn , xn , xn ] =

f [zn ,xn ]−f ′ (xn ) . zn −xn

The more general form

of (33) is stated in the following theorem. Theorem 2. Let f : D ⊆ ℜ → ℜ be a sufficiently differentiable function, and α ∈ D is a simple root of f (x) = 0, for an open interval D. If x0 is chosen sufficiently close to α, then iterative scheme (33) converges to α. If A satisfies (34)

A(0) = 1

then the iterative scheme (33) has convergence order at least six and if along with (34), it satisfies (35)

dA/dt1 (0) = 2

then (33) has convergence order at least seven. Proof. We define A(t1 ) = 1 + M0 t1 + M1 t21 + M2 t31 + higher powers in s. By using the (24)-(27), we obtain zn − α = yn − α − (1 + M0 t1 + M1 t21 + M2 t31 )

f (yn ) f ′ (xn )

= (−c22 (−2 + M0 ))e3n + c2 (−9c22 + 7M0 c22 − M1 c22 + 7c3 − 4M0 c3 )e4n + (10c2 c4 + 6c23 − 44c3 c22 + 30c42 − 6M0 c2 c4 + 38M0 c22 c3 − 33M0 c42 − M2 c42 − 6M1 c22 c3 + 10M1 c42 − 4M0 c23 )e5n + (17c3 c4 − 70c2 c23 + 188c3 c32 − 62c4 c22 + 13c2 c5 − 88c52 + 74M1 c32 c3 − 9M1 c22 c4 − 12M1 c2 c23 − 62M1 c52 + 13M2 c52 − 8M2 c32 c3 + 129M0 c52 − 12M0 c3 c4 + 68M0 c2 c23 − 225M0 c3 c32 + 55M0 c4 c22 − 8M0 c2 c5 )e6n (36)

+ O(e7n ), f (zn ) = c1 [(−c22 (−2 + M0 ))e3n + c2 (−9c22 + 7M0 c22 − M1 c22 + 7c3 − 4M0 c3 )e4n + (10c2 c4 + 6c23 − 44c3 c22 + 30c42 − 6M0 c2 c4 + 38M0 c22 c3 − 33M0 c42 − M2 c42 − 6M1 c22 c3 + 10M1 c42 − 4M0 c23 )e5n + (17c3 c4 − 70c2 c23 + 188c3 c32 − 62c4 c22 + 13c2 c5 − 88c52 + 74M1 c32 c3 − 9M1 c22 c4 − 12M1 c2 c23 − 62M1 c52 + 13M2 c52 − 8M2 c32 c3 + 129M0 c52 − 12M0 c3 c4 + 68M0 c2 c23 − 225M0 c3 c32 + 55M0 c4 c22 − 8M0 c2 c5 )e6n

(37)

+ O(e7n )].

6


By substituting values from (24)-(27), (36) and (37) in xn+1 from (33), we get xn+1 − α = zn − α −

f (zn ) f [zn , yn ] + f [zn , xn , xn ](zn − yn )

en+1 = c32 (−2 + M0 )(M0 c22 + 2c3 − 2c22 )e6n − c22 (36c42 − 46M0 c42 + 4M1 c42 − 2M1 c42 M0 + 14M02 c42 − 10M02 c3 c22 − 2M1 c22 c3 − 64c3 c22 + 57M0 c22 c3 + 6c2 c4 − 3M0 c2 c4 + 22c23 − 12M0 c23 )e7n + O(e8n ).

(38)

Clearly, if dA(t1 )/dt1 (0) = M0 = 2 then convergence order is seven otherwise is six.

Now we present derivative-free optimal fourth-order iterative schemes (FD3):   w = xn − κf (xn ),   n f (xn ) yn = xn − f [x , (39) h n ,wn ] i   g0 xn+1 = yn − G (t ) + f [yg1,x ] G1 (t2 ) + f [x g2,w ] G2 (t1 , t2 ) , f [y ,w ] 0 1 n

where t1 =

f (yn ) , t2 f (xn )

=

f (yn ) . f (wn )

n

n

n

n

n

The convergence of (39) is given in the following theorem.

Theorem 3. Let f : D ⊆ ℜ → ℜ be a sufficiently differentiable function, and α ∈ D is a simple root of f (x) = 0, for an open interval D. If x0 is chosen sufficiently close to α, then iterative scheme (39) converges to α. If G0 (t1 ), G1 (t2 ), G2 (t1 , t2 ), g0 , g1 and g2 satisfy    G0 (0)) = 1, G1 (0) = 1, G2 (0, 0) = 1, (40) dG0 /dt1 (0) = 1, dG1 /dt2 (0) = 1, ∂G2 /∂t1 (0, 0) = 1, ∂G/∂t2 (0, 0) = 1,   g0 + g1 + g2 = 1 and all higher order derivatives of Gi , i = 0, 1, 2 are bounded then the iterative scheme (39) has convergence order at least four. Proof. Error term at nth-step is denoted by en = xn −α. we call c1 = f ′ (α) and ck = Taylor’s expressions give: (41)

(k) (α) ek n f , k! f ′ (α)

where k = 2, 3, · · · .

f (xn ) = c1 [en + c2 e2n + c3 e3n + c4 e4n + c5 e5n + c6 e6n + O(e7n )]. f (wn ) = −c1 (−1 + κc1 )en + c1 c2 (−3κc1 + 1 + κ2 c21 )e2n − c1 (4κc1 c3 − c3 − 3c3 κ2 c21 + c3 κ3 c31 + 2c22 κc1 − 2κ2 c21 c22 )e3n + c1 (−5κc1 c4 − 5c2 κc1 c3 + 8κ2 c21 c2 c3 + κ2 c21 c32 + c4 + 6c4 κ2 c21 − 4c4 κ3 c31 + c4 κ4 c41

(42)

− 3c2 κ3 c31 c3 )e4n + · · · + O(e7n ).

By using (41) and (42), we get f (xn ) − f (wn ) = −c1 c2 (−2 + κc1 )en + c1 (3c3 − 3κc1 c3 + c3 κ2 c21 − c22 κc1 )e2n − c1 (−4c4 xn − w n + 4c2 κc1 c3 + 6κc1 c4 − 4c4 κ2 c21 + c4 κ3 c31 − 2κ2 c21 c2 c3 )e3n + c1 (2κ2 c21 c23 + 8κ2 c21 c4 c2

f [xn , wn ] =

− 10κc1 c5 + 5c5 − 7c4 κc1 c2 − 3κc1 c23 + 10c5 κ2 c21 − 5c5 κ3 c31 + c5 κ4 c41 − 3c4 κ3 c31 c2 + κ2 c21 c22 c3 )e4n (43)

+ · · · + O(e7n )

By substituting (41) and (43) in y = x − f (xn )/f [xn , wn ], we obtain yn − α = −c2 (−1 + κc1 )e2n + (2c3 − 3κc1 c3 + c3 κ2 c21 + 2c22 κc1 − 2c22 − κ2 c21 c22 )e3n + (3c4 + 10c2 κc1 c3 − 6κc1 c4 (44)

+ 4c4 κ2 c21 − c4 κ3 c31 − 7κ2 c21 c2 c3 − 7c2 c3 − 5κc1 c32 + 2c2 κ3 c31 c3 + 3κ2 c21 c32 + 4c32 − κ3 c31 c32 )e4n + · · · + O(e7n )

Taylor’s Expansion of f around yn by using (44) is f (yn ) = c1 (−c2 (−1 + κc1 )e2n + (2c3 − 3κc1 c3 + c3 κ2 c21 + 2c22 κc1 − 2c22 − κ2 c21 c22 )e3n + (3c4 + 10c2 κc1 c3 − 6κc1 c4 (45)

+ 4c4 κ2 c21 − c4 κ3 c31 − 7κ2 c21 c2 c3 − 7c2 c3 − 5κc1 c32 + 2c2 κ3 c31 c3 + 3κ2 c21 c32 + 4c32 − κ3 c31 c32 )e4n + · · · + O(e7n ))

Now, we define

(46)

  G0 (t1 ) G1 (t2 )   G2 (t1 , t2 )

= 1 + t1 + L1 t21 + higher powers in t1 , = 1 + t2 + M1 t22 + higher powers in t2 , = 1 + t1 + t2 + N1 t21 + N2 t22 + higher powers in t1 , t2 ,


where L1 , M1 , N1 and N2 are real numbers. By substituting (41)-(45) in xn+1 = yn − i g1 G (t ) + f [x g2,w ] G2 (t1 , t2 ) , we obtain f [y ,x ] 1 2 n

n

n

h

g0 G (t ) f [yn ,wn ] 0 1

7

+

n

en+1 = (−1 + g0 + g1 + g2 )c2 (−1 + κc1 )e2n +

(−1 + g0 + g1 + g2 )(−2c3 + 3κc1 c3 − c3 κ2 c21 − 2c22 κc1 + 2c22 + κ2 c21 c22 )e3n + (−3g0 c4 − g0 c32 + 6g0 c2 c3 − 7κ2 c21 c2 c3 − 6κc1 c4 + 3c4 + 3g2 N1 c32 κc1 − 3g2 N1 c32 κ2 c21 + g2 N1 c32 κ3 c31 + g2 N2 c32 κc1 + g1 M1 c32 κc1 + g0 κ3 c31 L1 c32 + 3g0 L1 c32 κc1 − 3g0 L1 c32 κ2 c21 + 3κ2 c21 c32 + 4c4 κ2 c21 − c4 κ3 c31 − 5κc1 c32 − κ3 c31 c32 + 10c2 κc1 c3 + 2c2 κ3 c31 c3 − 7c2 c3 + 4c32 − 5g2 κc1 c32 + g1 κ3 c31 c32 + 6g0 κc1 c4 − 4g0 c4 κ2 c21 + g0 c4 κ3 c31 − 2g0 κc1 c32 + 2g0 κ2 c21 c32 − g2 N2 c32 − g1 M1 c32 − g2 N1 c32 − g0 L1 c32 − 8g0 c2 κc1 c3 + 6g0 κ2 c21 c2 c3 − 2g0 c2 κ3 c31 c3 + 6g1 κc1 c4 + 6g1 κ2 c21 c2 c3 − 8g1 c2 κc1 c3 − 2g1 c2 κ3 c31 c3 + 6g2 κ2 c21 c2 c3 − 8g2 c2 κc1 c3 − 2g2 c2 κ3 c31 c3 + 6g1 c2 c3 + 6g2 c2 c3 − 3g1 c4 − g1 c32 − 3g2 c4 + g2 c32 − g1 κ2 c21 c32 − 4g1 c4 κ2 c21 + g1 c4 κ3 c31 + 6g2 κc1 c4 + 3g2 κ2 c21 c32 − 4g2 c4 κ2 c21 + g2 c4 κ3 c31 )e4n + · · · + O(e7n )

(47)

Clearly, if g0 + g1 + g2 = 1 then the coefficients of e2n and e3n are zero and hence the convergence order is four which is optimal in the sense of Kung and Traub. Remark 2.2. In (39), if we equate g0 = 0, g1 = 0, g2 = 1 and define G2 (t1 , t2 ) = 1 + t1 + t2 + αt21 + βt22 , then the resulted iterative scheme is the first two steps of iterative method (12). Similarly if we equate g0 = 0, g2 = 0, g1 = 1, κ = −1 and define G1 (t2 ) = (1 − λ−1 t2 )−λ , then it reproduces the first three step of iterative schemes (13) and (14). In order to   wn     yn    zn (48)    xn+1     

extend the iterative scheme (39), we add one more step and get (FD4) = xn − κf (xn ), = xn −

f (xn )

hf [xn ,wn ]

, i

1 −g2 ) G0 (t1 ) + f [yg1,x ] G1 (t2 ) + f [x g2,w ] G2 (t1 , t2 ) , = yn − (1−g n n n n h f [yn ,wn ] (1−h1 −h2 −h3 −h4 −h5 ) h1 S (t , t , t ) + S (t , t , t , t ) + f [zh2,x ] S2 (t2 , t3 , t4 , t5 ) = zn − 0 3 4 5 f [yn ,zn ] f [zn ,wn ] 1 1 3 4 5 n n i h3 h4 + f [x ,y ] S3 (t2 , t3 , t4 , t5 ) + f [y ,w ] S4 (t1 , t3 , t4 , t5 ) + f [x h5,w ] S5 (t1 , t2 , t3 , t4 , t5 ) f (zn ), n

n

n

n

n

n

(yn ) (zn ) (zn ) (zn ) (yn ) , t2 = ff(w , t3 = ff (x , t4 = ff(w , t5 = ff (y and h1 , h2 , h3 , h4 , h5 , κ 6= 0 are real numbers. where t1 = ff (x n) n) n) n) n) The convergence of (48) is discussed in the following theorem.

Theorem 4. By including the statement of (Theorem 3), iterative scheme (48) has convergence order at least six if    S0 (0, 0, 0) = 1, S1 (0, 0, 0, 0) = 1, S2 (0, 0, 0, 0) = 1, S3 (0, 0, 0, 0) = 1, S4 (0, 0, 0, 0) = 1,    S (0, 0, 0, 0, 0) = 1, ∂S /∂t (0, 0, 0, 0) = 1, ∂S /∂t (0, 0, 0, 0) = 1, 5 1 1 2 2 (49)  ∂S3 /∂t2 (0, 0, 0, 0) = 1, ∂S4 /∂t1 (0, 0, 0, 0) = 1,     ∂S /∂t (0, 0, 0, 0, 0) = 1, ∂S /∂t (0, 0, 0, 0, 0) = 1 5

1

5

2

and all higher order derivatives of all functions are bounded.

Proof. Taylor’s expansion of various orders for Si , i = 0, 1, · · · , 5 is given below. (50)   S0 (t3 , t4 , t5 )    S1 (t1 , t3 , t4 , t5 )    S (t , t , t , t ) 2

1

3

4

5

 S3 (t2 , t3 , t4 , t5 )    S (t , t , t , t )  4 1 3 4 5    S5 (t1 , t2 , t3 , t4 , t5 )

= 1 + a1 t3 + a2 t4 + a3 t5 + higher powers in t3 , t4 , t5 , = 1 + t1 + b1 t21 + b2 t3 + b3 t4 + b4 t5 + higher powers in t1 , t3 , t4 , t5 , = 1 + t2 + b5 t22 + b6 t3 + b7 t4 + b8 t5 + higher powers in t2 , t3 , t4 , t5 , = 1 + t2 + b9 t22 + b10 t3 + b11 t4 + b12 t5 + higher powers in t2 , t3 , t4 , t5 , = 1 + t1 + b13 t21 + b14 t3 + b15 t4 + b16 t5 + higher powers in t1 , t3 , t4 , t5 , = 1 + t1 + t2 + b17 t21 + b18 t22 + b19 t3 + b20 t4 + b21 t5 + higher powers in t1 , t2 , t3 , t4 , t5 ,

where a1 , a2 , a3 , b1 , b2 , b3 , b4 , b5 , b6 , b7 , b8 , b9 , b10 , b11 , b12 , b13 , b14 , hb15 , b16 , b17 , b18 , b19 , b20 , and b21 (1−h1 −h2 −h3 −h4 −h5 ) S0 (t3 , t4 , t5 ) + are real numbers. By using calculations from (41)-(47) for xn+1 = zn − f [y ,z ] n

n

h1 S (t , t , t , t ) f [zn ,wn ] 1 1 3 4 5 i h2 4 S4 (t1 , t3 , t4 , t5 ) + f [x h5,w ] S5 (t1 , t2 , t3 , t4 , t5 ) f (zn ), + f [z ,x ] S2 (t2 , t3 , t4 , t5 ) + f [xh3,y ] S3 (t2 , t3 , t4 , t5 ) + f [y h,w n n n n n n] n n

8


we obtain the following error equation. en+1 = −(2h4 b13 c22 κc1 − h4 b13 c22 κ2 c21 + a3 g1 M1 c22 − h5 b18 c22 − h5 b17 c22 − h3 b9 c22 + h5 b21 g1 κ2 c21 c22 + h5 b21 g2 c22 κc1 − 2h5 b21 L1 c22 κc1 + h5 b21 L1 c22 κ2 c21 + 5c22 h5 + h4 κ2 c21 c22 + h4 κc1 c3 − 4h4 c22 κc1 + h1 κ2 c21 c22 + h1 κc1 c3 − 3h1 c22 κc1 − c22 h2 κc1 − 2c22 h3 κc1 − 5c22 h5 κc1 + h2 κc1 c3 + h3 κc1 c3 + h5 κ2 c21 c22 + h5 κc1 c3 − a3 κ2 c21 c22 − a3 κc1 c3 + 4a3 c22 κc1 + 2a3 h1 g2 c22 + · · · + h3 b12 L1 c22 κ2 c21 − 2h2 b8 g1 c22 κc1 + h2 b8 g1 κ2 c21 c22 + h2 b8 g2 c22 κc1 − 2h2 b8 L1 c22 κc1 + h2 b8 L1 c22 κ2 c21 )c2 (−1 + κc1 )(−g1 κ2 c21 c22 − L1 c22 κ2 c21 − g2 N1 c22 κ2 c21 + κ2 c21 c22 + g1 L1 c22 κ2 c21 + g2 L1 c22 κ2 c21 − 4c22 κc1 + 2g1 c22 κc1 + 2g2 N1 c22 κc1 − g2 c22 κc1 − 2g1 L1 c22 κc1 + κc1 c3 + 2L1 c22 κc1 − 2g2 L1 c22 κc1 − c3 − g2 N2 c22 − g2 N1 c22 + 2g2 c22 + 3c22 − g1 M1 c22 − L1 c22 + g1 L1 c22 (51)

+ g2 L1 c22 )e6n + O(e7n )

Remark 2.3. In (48), by equating κ = −1, g1 = 1, g2 = 0, h3 = 1, h1 = h2 = h4 = h5 = 0 and defining G2 (t2 ) = (1 − λ−1 t2 )−λ , S3 (t2 , t3 , t4 , t5 ) = (1 − λ−1 t2 )−λ gives the iterative scheme (13), by setting κ = −1, g1 = 1, g2 = 0, h1 = h2 = h3 = h4 = h5 = 0 and defining G2 (t2 ) = (1 − λ−1 t2 )−λ ,S0 (t3 , t4 , t5 ) = 1, we obtain the iterative schemes (14). (12) can be constructed by choosing parameters g1 = 0, g2 = 1, h5 = 1, h1 = h2 = h3 = h4 = 0 and weight functions G2 (t1 , t2 ) = 1 + t1 + t2 + αt21 + βt22 , S5 (t1 , t2 , t3 , t4 , t5 ) = 1 + t1 + t2 + αt21 + βt22 + ηt5 . F. Soleymani constructs derivative-free sixth-order iterative scheme (16). In the following theorem, we present a modified iterative family with convergence order seven. Theorem 5. Let f : D ⊆ ℜ → ℜ be a sufficiently differentiable function, and α ∈ D is a simple root of f (x) = 0, for an open interval D. If x0 is chosen sufficiently close to α, then iterative scheme (FD5)  = xn − κf (xn ),  wn  f (xn )   , y = xn − f [x ,w n   # " n n] (52) 1−g1 −g2 g1 g2  = yn − f [y ,w ] G0 (t1 ) + f [x ,y ] G1 (t2 ) + f [x ,w ] G2 (t1 , t2 ) f (yn ),   zn n n n n n n     f (zn ) , xn+1 = zn − H(t3 , t4 , t5 ) f [z ,y ]−(h−1)f [z ,w ]+hf [z ,x ]−hf [y ,x ]+(h−1)f [y ,w ] n

n

n

n

n

n

n

n

n

n

where

  G0 (t1 )    G (t ) 1 2  G2 (t1 , t2 )   H(t , t , t ) 3 4 5

(53)

= 1 + t1 + a1 t21 + higher powers in t1 , = 1 + t2 + a2 t22 + higher powers in t2 , = 1 + t1 + t2 + a3 t21 + a4 t22 + a5 t1 t2 + higher powers in t1 , t2 , = 1 + a6 t3 + a7 t4 + a8 t5 + higher powers in t3 , t4 , t5 ,

ti , i = 1, 2, .., 5 are consistent with previous definition in (48), g1 , g2 , h, aj , j = 1, · · · , 8 are real numbers. has convergence order at-least six if ( G0 (0) = 1, G1 (0) = 1, G2 (0, 0) = 1, H(0, 0, 0) = 1, (54) dG0 /dt1 (0) = 1, dG1 /dt2 (0) = 1, ∂G2 /∂t1 (0, 0) = 1, ∂G2 /∂t2 (0, 0) = 1, and has convergence order seven if along with (54), the following condition is valid (55)

∂H/∂t5 (0, 0, 0) = 0.

Proof. The terminology of (Theorem 3) and expressions in (41), (42) assist to perform all necessary calculations by using Maple 13 [23] software-package in order to get the following error equation: en+1 = a8 c2 (−1 + κc1 )(κ2 c21 c22 − c3 + κc1 c3 + 3c22 − 4c22 κc1 + g1 a1 c22 − a1 c22 + 2a1 c22 κc1 − a1 c22 κ2 c21 − g2 c22 κc1 + 2g1 c22 κc1 − g1 κ2 c21 c22 + g2 a1 c22 − g1 a2 c22 − g2 a3 c22 − g2 a5 c22 − g2 a4 c22 + 2g2 c22 − 2g1 a1 c22 κc1 + g1 a1 c22 κ2 c21 − 2g2 a1 c22 κc1 + g2 a1 c22 κ2 c21 + 2g2 a3 c22 κc1 − g2 a3 c22 κ2 c21 + g2 a5 c22 κc1 )2 e6 n + −(κ2 c21 c22 − c3 + κc1 c3 + 3c22 − 4c22 κc1 + g1 a1 c22 − a1 c22 + 2a1 c22 κc1 − a1 c22 κ2 c21 − g2 c22 κc1 + 2g1 c22 κc1 − g1 κ2 c21 c22 + g2 a1 c22 − g1 a2 c22 − g2 a3 c22 − g2 a5 c22 − g2 a4 c22 + 2g2 c22 − 2g1 a1 c22 κc1 + g1 a1 c22 κ2 c21 + ···+ − a6 a1 c42 − 2a8 c23 − 30a8 c42 + 3a6 c42 − 3a6 κ2 c21 c22 c3 + a6 c3 κ3 c31 c22 + 3a6 κc1 c22 c3 − 5a6 g1 κ2 c21 c42 + 2a6 g1 κc1 c42 + 4a6 g1 κ3 c31 c42 + 4a6 g2 κ2 c21 c42 − 5a6 g2 κc1 c42 − a6 g2 κ3 c31 c42 − a6 g1 κ4 c41 c42 − a6 a1 c42 κ4 c41 (56)

8 + 4a6 a1 c42 κ3 c31 )e7 n + O(en )


9

Clearly, a8 = ∂H/∂t5 (0, 0, 0) is a factor in asymptotic error constant of e6n , if a8 = 0 then (53) has convergence order seven otherwise convergence order is six. Remark 2.4. Iterative scheme (53) reproduces F. Soleymani derivative-free sixth-order iterative scheme (16) if (57)

h = 0, g1 = g2 = 0, G0 (t1 ) = 1, H(t3 , t4 , t5 ) = 1.

The derivative-free version of (33) is (FD6):  wn = xn − κf (xn ),     f (xn )  , yn = xn − f [x ,w   " n n]  (58) 1−g1 −g2 zn = yn − G0 (t1 ) +  f [y,w]       x = zn − H(t , t , t ) n+1

3

4

5

g1 G (t ) f [xn ,yn ] 1 2

+

#

g2 G (t , t ) f [xn ,wn ] 2 1 2

f (zn ) zn −yn f [zn ,yn ]+(f [zn ,xn ]−f [wn ,xn ]) z −x n

,

,

n

where

   G0 (t1 )  G (t ) 1 2  G (t1 , t2 ) 2    H(t , t , t ) 3 4 5

(59)

= 1 + t1 + a1 t21 + higher powers in t1 , = 1 + t2 + a2 t22 + higher powers in t2 , = 1 + t1 + t2 + a3 t21 + a4 t22 + a5 t1 t2 + higher powers in t1 , t2 , = 1 + a6 t3 + a7 t4 + a8 t5 + higher powers in t3 , t4 , t5 ,

ti , i = 1, 2, · · · , 5 are consistent with previous definition in (48), g1 , g2 , h, aj , j = 1, · · · , 8 are real numbers. Theorem 6. Let f : D ⊆ ℜ → ℜ be a sufficiently differentiable function, and α ∈ D is a simple root of f (x) = 0, for an open interval D. If x0 is chosen sufficiently close to α, then iterative scheme (59) has convergence order at-least six if ( G0 (0) = 1, G1 (0) = 1, G2 (0, 0) = 1, H(0, 0, 0) = 1, (60) dG0 /dt1 (0) = 1, dG1 /dt2 (0) = 1, ∂G2 /∂t1 (0, 0) = 1, ∂G2 /∂t2 (0, 0) = 1. Proof. We obtain the following error equation en+1 = (−1 + κc1 )(κ2 c21 c22 − c3 + κc1 c3 + 3c22 − 4c22 κc1 + g1 a1 c22 + g2 a1 c22 − g1 a2 c22 − g2 c22 a5 − g2 a4 c22 − g2 a3 c22 − a1 c22 − 2g1 a1 c22 κc1 + g1 a1 c22 κ2 c21 − 2g2 a1 c22 κc1 + g2 a1 c22 κ2 c21 + 2g2 a3 c22 κc1 + 2a1 c22 κc1 − a1 c22 κ2 c21 + 2g1 c22 κc1 − g1 κ2 c21 c22 − g2 c22 κc1 + 2g2 c22 − g2 a3 c22 κ2 c21 + g2 c22 a5 κc1 )(κ2 c21 c22 − c22 κc1 + 3c22 a8 − a8 c3 + 2c22 a8 g2 − a1 c22 a8 − c22 a8 a2 g1 − c22 a8 g2 a3 − c22 a8 g2 a5 − c22 a8 g2 a4 + c1 a8 c3 κ + a1 c22 a8 g1 + a1 c22 a8 g2 − c21 c22 a8 κ2 g1 + 2c1 c22 a8 κg1 − c1 c22 κa8 g2 − c21 c22 κ2 a8 g2 a3 + c1 c22 κa8 g2 a5 + 2c1 c22 κa8 g2 a3 − a1 c21 c22 a8 κ2 + 2a1 c1 c22 a8 κ + a1 c21 c22 a8 κ2 g1 + a1 c21 c22 κ2 a8 g2 − 2a1 c1 c22 a8 κg1 − 2a1 c1 c22 κa8 g2 + c21 c22 a8 κ2 (61)

− 4c1 c22 a8 κ)c2 e6n + O(e7n ) 3. Numerical computations

Definition 1. Let xn−1 , xn and xn+1 be successive iterations closer to the root α of f (x) = 0, the computational order of convergence (COC) [24], can be approximated by COC ≈

(62)

ln|(xn+1 − α)(xn − α)−1 | . ln|(xn − α)(xn−1 − α)−1 |

A set of twelve functions is listed in Table 1 which is taken from [25] to validate the iterative methods and performance. In all methods, twelve total number of function evaluations (TNFE) are used. 3.1. Sixth-order convergence performance evaluation of FD1. For the purpose of comparison between FD1 and SG, NT1, NT2, CH, GR, AL, we derive an iterative method from (21) which is given as (FD1-M1 )  n)  , = xn − ff′(x  yn (xn )  f (yn ) (63) zn = yn − G(t1 ) f ′ (x ) , n   x = z − H (t )H (t )H (t ) f (zn ) , n+1

where t1 = 1 . 1−1.1t3

f (yn ) ,t f (xn ) 2

=

f (zn ) ,t f (xn ) 3

=

n

f (zn ) , G(t1 ) f (yn )

1

=

1

2

2

3

1 , H1 (t1 ) 1−2t1

3 f ′ (x ) n

=

1 , H2 (t2 ) 1−2t1 −t2 1

= 1 + 2t2 and H3 (t3 ) =

Table 2 shows the performance of different listed iterative methods in terms of absolute error (|xn − α|). Clearly FD1-M1 is competitive with other developed methods and freedom to choose different weight functions, makes it more accurate and stable in the third step of iterative scheme (21).

10


Table 1. List of test functions Functions f1 (x) = exp(x) sin(x) + ln(1 + x2 ) f2 (x) = x15 + x4 + 4x2 − 15 f3 (x) = (x − 2)(x10 + x + 1) exp(−x − 1) f4 (x) = exp(−x2 + x + 2) − cos(x + 1) + x3 + 1 f5 (x) = (x + 1) exp(sin(x)) − x2 exp(cos(x)) − 1 f6 (x) = sin(x)2 − x2 + 1 f7 (x) = 10 exp(−x2 ) − 1 f8 (x) = (x2 − 1)−1 − 1 f9 (x) = ln(x2 + x + 2) − x + 1 f10 (x) = cos(x)2 − x/5 f11 (x) = x10 − 2x3 − x + 1 f12 (x) = exp(sin(x)) − x + 1

Roots α=0 α = 1.148538... α=2 α = −1 α=0 α = 1.40449165... α = 1.517427... α = 1.414214... α = 4.15259074... α = 1.08598268... α = 0.591448093... α = 2.63066415...

Table 2. Numerical comparison between absolute errors(|xn − α|, T N F E = 12)

fn (x),x0

FD1-M1

f1 , 0.25 f2 , 1.1 f3 , 2.1 f4 , -0.5 f5 , 0.25 f6 , 1.2 f7 , 1.0 f8 , 1.6 f9 , 4.4 f10 , 1.5 f11 , 0.25 f12 , 2.0

2.71 e-142 3.90 e-190 2.31e-181 5.28 e-218 1.96e-247 3.79e-215 5.26e-123 2.27e-181 2.92e-402 2.33e-56 8.99e-115 3.53e-173

SG a = −1 2.82e-121 3.72 e-153 3.75e-135 8.50 e-181 4.05e-184 1.41e-181 2.88e-95 1.18e-157 3.13e-366 2.22e-53 1.30e-97 1.80e-132

NT1 a = −1 1.22e-127 1.23 e-122 6.27e-135 5.30 e-143 2.51e-165 7.22e-173 2.10e-113 1.65e-97 4.80e-387 3.42e-64 1.54e-91 3.42e-138

NT2

CH

GR

4.25e-84 1.47 e-95 2.89e-93 2.70 e-173 3.77e-213 3.90e-123 3.87e-69 6.69e-59 2.35e-344 3.58e-31 1.31e-53 1.84e-117

5.12e-82 2.45 e-104 1.07e-94 1.27 e-174 7.33e-162 1.69e-123 6.28e-67 1.23e-103 6.52e-341 10.1 3.86e-45 2.78e-115

9.23 e-93 4.43 e-141 6.52e-109 5.89 e-207 2.97e-243 6.80e-154 1.81e-76 2.26e-147 1.89e-353 5.45e-56 4.77e-89 3.52e-126

AL θ = −1.01 6.22e-141 1.60 e-123 1.80e-147 8.99 e-191 2.20e-215 2.42e-180 2.55e-118 4.14e-60 2.41e-432 8.42e-58 1.72e-92 8.05e-162

Table 3. Computational order of convergence (COC) fn (x) f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12

FD1-M1 5.9999 6.0000 6.0000 6.0000 6.0000 6.0000 5.9999 6.0000 6.0000 5.9806 5.9999 6.0000

SG 5.9994 6.0000 6.0000 6.0000 6.0000 6.0000 5.9992 6.0000 6.0000 6.0038 5.9981 6.0000

NT1 6.0003 5.9999 5.9999 6.0000 5.9999 5.9999 6.0001 6.0001 6.0000 6.0098 6.0001 6.0000

NT2 5.9989 5.9997 5.9995 6.0000 6.0000 5.9999 5.9974 5.9985 6.0000 5.8343 5.9909 6.0002

CH 5.9987 5.9998 5.9996 6.0000 5.9999 5.9999 5.9972 5.9999 6.0000 3.4125 5.9856 6.0002

GR 5.9995 6.0000 5.9999 6.0000 6.0000 6.0000 5.9991 6.0000 6.0000 5.9880 6.0000 6.0001

AL 5.9965 5.9933 6.0012 5.9998 5.9999 5.9998 5.9860 6.0971 6.0000 6.1207 6.0120 5.9981

3.2. Seventh-order convergence performance evaluation of FD1. To compare the results, we construct seventh-order iterative scheme (FD1-M2) from (21):  n)  , = xn − ff′(x  yn (xn )  f (yn ) (64) zn = yn − G(t1 ) f ′ (x ) , n   f (zn ) x n+1 = zn − H1 (t1 )H2 (t2 )H3 (t3 ) f ′ (x ) , n

where t′i s are defined previously, G(t1 ) =

(33), if we define A(t1 ) = if A(t1 ) =

1 (1−t1 )2

1+t1 1−t1

1 , H1 (t1 ) 1−2t1

=

1 , H2 (t2 ) 1−2t1 −t2 1

= 1 + 2.1t2 and H3 (t3 ) =

1 . 1−t3

In

then (33) is a seventh-order iterative scheme (AL1) which is developed in [17] and

then again (33) is a seventh-order iterative scheme (FD2-M1).


11

Table 4. Numerical comparison between absolute errors(|xn − α|, T N F E = 12) and COC fn (x),x0

FD1-M2

FD2-M1

AL1

f1 , 0.25 f2 , 1.1 f3 , 2.1 f4 , -0.5 f5 , 0.25 f6 , 1.2 f7 , 1.0 f8 , 1.6 f9 , 4.4 f10 , 1.5 f11 , 0.25 f12 , 2.0

3.60e-182 2.99e-263 8.60e-223 2.78e-297 1.31e-307 5.02e-304 1.51e-156 4.95e-240 5.10e-662 1.08e-77 8.67e-162 1.30e-242

8.55e-177 6.71e-172 3.95e-177 4.95e-247 1.45e-306 2.70e-246 1.83e-145 3.58e-102 3.74e-567 1.33e-82 4.51e-141 4.29e-196

6.80e-167 1.63e-145 3.11e-178 1.45e-233 2.81e-273 1.27e-222 3.98e-136 2.35e-67 1.07e-561 1.73e-64 1.31e-103 9.45e-193

FD1-M2 (COC) 6.9998 7.0000 7.0000 7.0000 7.0000 7.0000 6.9997 6.9995 7.0000 6.8846 7.0000 7.0000

FD2-M1 (COC) 6.9999 7.0000 7.0000 7.0000 7.0000 7.0000 6.9998 7.0001 7.0000 7.0083 7.0003 7.0000

AL1 (COC) 6.9997 6.9999 7.0000 7.0000 7.0000 7.0000 6.9996 6.9978 7.0000 6.9701 7.0019 6.9999

Table 4 shows absolute error due to method FD2-M1 has small magnitude in comparison with other describe methods for the set of given twelve test functions and initial guesses. Only in the case of f9 and f10 , AL has better absolute error magnitude. 3.3. Sixth-order convergence performance evaluation of FD4, FD5, FD6. For comparison, we develop f (y ) some particular cases of derivative-free methods. In all calculations the value of parameter κ = 1/100 and t1 = f (xn ) , n

t2 =

f (yn ) , t3 f (wn )

=

f (zn ) , t4 f (xn )

=

f (zn ) , t5 f (wn )

=

f (zn ) . f (yn )

• FD4  wn      yn

(65)

• FD5

= xn − = yn − = zn −

  wn      yn

(66)

• FD6

(67)

 zn     xn+1

= xn − κf (xn ),

zn     xn+1

+

1−2t2 1 1−3t2 f [zn ,xn ]

i f (zn ).

= xn − κf (xn ), f (xn ) , f [xn ,wn ] f (yn ) 1 yn − 1−t , ,yn ] 2 f [xn−1 f (zn ) t5 . zn − 1 − 10 f [zn ,yn ]+f [zn,wn ]−f [yn ,wn ]

= xn −

zn       xn+1  wn      yn

f (xn ) , f [xn ,wn ] 1+t1 −t2 f (yn ) , 1−2t h 2 f [xn ,wn ] 1 1 1 2 1−t5 −2t3 −2t4 f [zn ,yn ]

= =

= xn − κf (xn ), f (xn ) , f [xn ,wn ] 1−t1 +t2 f (yn ) yn − 1−2t f [x ,w ] , n n 1 −1 f (zn ) t5 zn − 1 − 10 zn −yn f [zn ,yn ]+(f [zn ,xn ]−f [xn ,wn ]) z −x

= xn − = =

n

.

n

• SK2M1, SK2M2 In [18], Authors developed two methods by selecting different values of parameters. If α = β = then SK2 is SK2M1 and if α = β = η = 1 then SK2 is SK2M2.

5 , 2

η=1

Table 5 shows overall performance of derivative-free sixth-order iterative methods FD4, FD5, FD6. Absolute error |xn − α|, for developed sixth-order derivative-free methods, is comparatively better than referenced sixth-order derivative-free iterative methods. Computational order of convergence is given in Table 6. dgt stands for divergent and X is for no information. 3.4. Seventh-order convergence performance evaluation of FD7. FD7 is seventh-order convergent iterative method which is deduced from FD5 and is defined as  wn = xn − κf (xn ),     f (xn )  yn = xn − f [x , n ,wn ] (68) 1−2t1 f (yn )  = yn − 1−3t f [y ,w ] ,   zn n n 1   f (zn ) 1 . xn+1 = zn − 1−t f [z ,y ]+f [z ,x ]−f [x ,y ] 3

n

n

n

n

n

n

12


Table 5.

fn (x) f1 , 0.25 f2 , 1.1 f3 , 2.1 f4 , 0.5 f5 , 0.25 f6 , 1.2 f7 , 1.0 f8 , 1.6 f9 , 4.4 f10 , 1.5 f11 , 0.25 f12 , 2.0

FD4 4.e-172 1.e-190 2.e-107 3.e-220 3.e-247 1.e-200 3.e-140 3.e-168 5.e-466 1.e-70 7.e-118 1.e-189

Numerical comparison between absolute errors(|xn − α|, T N F E = 12)

FD5 2.e-158 2.e-297 2.e-203 2.e-198 7.e-248 2.e-186 8.e-130 1.e-90 3.e-392 6.e-70 4.e-118 3.e-149

FD6 8.e-172 6.e-221 4.e-168 8.e-213 6.e-215 5.e-247 8.e-146 8.e-165 1.e-404 8.e-65 4.e-169 1.e-144

TS1 9.e-54 0.01 3.e-18 3.e-127 2.e-129 1.e-132 dgt 5.e-55 2.e-395 1.e-138 2.e-71 8.e-137

TS2 2.e-66 3.e-14 4.e-24 2.e-202 8.e-172 5.e-153 dgt 2.e-71 2.e-424 3.e-168 1.e-105 2.e-194

SK2M1 3.e-103 7.e-162 2.e-197 1.e-217 4.e-157 8.e-112 3.e-78 2. 1.e-383 0.06 1. 2.e-173

SK2M2 4.e-83 7.e-241 6.e-145 3.e-175 1.e-197 2.e-161 1.e-65 5.e-25 1.e-366 1. 8.e-29 2.e-144

FS1 1.e-59 2.e-7 3.e-7 9.e-196 2.e-171 8.e-194 dgt 2.e-68 2.e-429 9.e-156 1.e-118 5.e-215

FS2 2.e-104 9.e-333 5.e-204 4.e-171 3.e-215 2.e-161 6.e-87 3.e-145 5.e-378 5.e-68 3.e-85 4.e-191

SK2M2 6.00 6.00 6.00 6.00 6.00 6.00 6.00 5.78 6.00 0.00299 5.91 6.00

FS1 5.98 3.33 3.81 6.00 6.00 6.00 X 6.00 6.00 6.00 6.00 6.00

FS2 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 5.99 6.00 6.00

Table 6. Computational order of convergence (COC)

fn (x) f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12

FD4 6.01 6.00 6.00 6.00 6.00 6.00 6.00 6.01 6.00 6.30 6.05 5.98

FD5 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.01 6.00 6.00

FD6 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0

TS1 5.99 -1.15 5.32 6.00 6.00 6.00 X 5.99 6.00 6.00 6.00 6.00

TS2 6.00 6.24 5.63 6.00 6.00 6.00 X 6.00 6.00 6.00 6.00 6.00

SK2M1 5.98 6.00 6.00 6.00 6.00 6.00 5.98 5.23 6.00 1.51 2.57 6.00

Table 7. Numerical comparison between absolute errors(|xn − α|, T N F E = 12) and COC fn (x) f1 , 0.25 f2 , 1.1 f3 , 2.1 f4 , 0.5 f5 , 0.25 f6 , 1.2 f7 , 1.0 f8 , 1.6 f9 , 4.4 f10 , 1.5 f11 , 0.25 f12 , 2.0

FD7(κ)(COC) 2.45e-378(1.0)(10) 1.22e-388(0.01)(7) 3.29e-266(0.01)(7) 8.61e-262(0.01)(7) 1.54e-303(0.01)(7) 5.15e-353(0.01)(7) 3.65e-213(0.01)(7) 1.10e-174(0.01)(7) 9.09e-689(0.01)(7) 3.81e-262(-1.0)(7) 7.21e-212(0.01)(7) 3.52e-271(0.01)(7)

FS3-1(COC) 3.23e-99(7) 3.33e-31(6.42) 1.37e-40(6.80) 1.89e-221(7) 2.47e-220(7) 1.71e-251(7) 0.320(4.96) 1.09e-234(7) 4.55e-617(7) 6.26e-174(7) 1.33e-125(7) 7.93e-257(7)

FS3-2(COC) 4.83e-318(9) 8.14e-24(7.02) 1.32e-26(7.42) 1.17e-144(7) 6.43e-397(7) 6.17e-74(7.01) 3.03(X) 7.46e-22(7.09) 1.71e-517(7) 3.87e-5(2.04) 5.57e-77(6.99) 6.79e6(-7.79)

FS4-1(COC) 2.96e-85(7) 2.37e-7(3.30) 3.96e-8(4.18) 2.06e-210(7) 1.33e-206(7) 1.34e-257(7) dgt(X) 2.68e-233(7) 5.06e-640(7) 1.96e-214(7) 6.28e-147(7) 3.40e-222(7)

FS4-2(COC) 6.60e-311(10) 2.04(-0.0211) divergent(X) 8.75e-75(6.97) 1.25e-384(7) 4.81e-44(7.05) 3.03(X) 290(1.43) 4.45e-506(7) 7.16(X) 1.58e-52(6.97) 2.64(-0.261)

All parameters set to zero value namely γ = 0, δ = 0 in FS3-1, ρ = 0, τ = 0 in FS3-2 and ω = 0, φ = 0 in FS4. Table 7 represents the comparison between absolute errors as well as computational order of convergence. Clearly FD7 is superior than other methods in comparison. 4. Conclusions We have presented derivative-based and derivative-free iterative methods by introducing free weighting parameters g0 , g1 , g2 and weight functions. Weighting parameters, in the second step of derivative-free iterative methods, give flexibility for selecting different forms of approximation for first order derivative. A combination of derivative approximations can be constructed by fixing weighting parameters. The proper selection weight functions for both


13

derivative-free and derivative-based iterative methods help to improve the accuracy. The presented iterative methods are defined in very general form which actually reproduce many existing iterative methods. By introducing different form of weighting functions and selection of parameters produce new families of iterative methods. We explore some of them to show the effectiveness of newly constructed iterative methods. Acknowledgement. This research was supported by Spanish MICINN grants AYA2010-15685 and MEC grants AYA2010-15685, AYA2008-04211-C02-C01. References [1] J. F. steffensen, ”Remarks on iteration”, Skand Aktuar Tidsr, vol. 16, pp. 64-72, 1933. [2] M.A. Hernndez, Second-derivative-free variant of the Chebyshev method for nonlinear equations, J. Optim. Theory Appl., 104 (3) (2000), pp. 501?515. [3] M.A. Hernndez, Chebyshev?s approximation algorithms, applications, Comput. Math. Appl., 4 (2001), pp. 433?445. [4] J. Kou, Y. Li, X. Wang, A uniparametric Chebyshev-type method free from second derivatives, Appl. Math. Comput., 179 (2006), pp. 296?300. [5] Q.B. Wu, Y.Q. Zhao, The convergence theorem for a family deformed Chebyshev method in Banach space, Appl. Math. Comput., 182 (2006), pp. 1369?1376. [6] M. Frontini, E. Sormani, Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput., 149 (2004), pp. 771?782. [7] J. Kou, Y. Li, On Chebyshev-type methods free from second derivative, Comm. Numer. Method Eng., 24 (2008), pp. 1219?1225. [8] J. Kou, Y. Li, Modified Chebyshev?s method free from second derivative for non-linear equations, Appl. Math. Comput., 187 (2) (2007), pp. 1027?1032. [9] J.R. Sharma, R.K. Guha, A family of modified Ostrowski methods with accelerated sixth-order convergence, Appl. Math. Comput., 190 (2007), pp. 111?115. [10] B. Neta, A sixth-order family of methods for nonlinear equations, Int. J. Comput. Math., 7 (1979), pp. 157?161. [11] C. Chun, B. Neta, A new sixth-order scheme for nonlinear equations, Appl. Math. Lett. (2011), doi:10.1016/j.aml.2011.08.012. [12] C. Chun, Y. Ham, Some sixth-order variants of Ostrowski root-finding methods, Appl. Math. Comput., 193 (2003), pp. 389?394. [13] H. T. Kung, J. F. Traub: Optimal order of one-point and multipoint iteration. Journal of the ACM, 21 (1974), 643?651. [14] A. M. Ostrowski: Solution of Equations and Systems of Equations, Academic Press, New York-London, 1966. [15] J. F. Traub: Iterative Methods for the Solution of Equations, Prentice Hall, New York, 1964. [16] M. Grau, J.L. Diaz-Barrero, An improvment of Ostrowski root-finiding method, J. Math. Anal. Appl., 173 (2006) 450-456. [17] Alicia Cordero and Jos L. Hueso and Eulalia Martnez and Juan R. Torregrosa, A family of iterative methods with sixth and seventh order convergence for nonlinear equations, J. Mathematical and Computer Modelling, 52 (2010) 1490-1496. [18] Sanjay K. Khattri, Joannis K. Argyros, Sixth order derivative free family of iterative methods, J. Appl. Math. Comput., 217 (2011), 5500-5507 [19] R. Thukral, New family of higher order Steffensen-type methods for solving nonlinear equations, J. of Mod. Meth. in Numer. Math., Vol. 3, No. 1, 2012, 1-10 [20] F. Soleymani, V. Hosseinabadi, New third- and tixth-order derivative-free techniques for nonlinear equations, J. of Math. Research, Vol. 3 N0. 2; May 2011. [21] Sanjay Khattri, Ioannis Argyros, Unification of sixth-order iterative methods, J. of Mathematical Sciences 2013, 7:5, doi:10.1186/2251-7456-7-5. [22] F. Soleymani, Efficient Sixth-Order Nonlinear Equation Solvers Free from Derivative, World Applied Sciences Journal 13 (12): 2503-2508, 2011. [23] http://www.maplesoft.com/products/maple/ [24] Laila M. Assas et al., Eighth-order Derivative-Free Family of Iterative Methods for Nonlinear Equations, J. of Mod. Meth. in Numer. Math. (2013), in press. [25] R. Thukral. A family of three-point derivative-free methods of eight-order for solving nonlinear equations. J. of Mod. Meth. in Numer. Math., 3(2):11-21, 2012. [26] Fazlollah Soleymani, Some efficient seventh-order derivative-free families in root-finding, Opuscula Math. 33, no. 1 (2013), 163?173, http://dx.doi.org/10.7494/OpMath.2013.33.1.163 Dept. de Fsica i Enginyeria Nuclear, Universitat Politcnica de Catalunya, Comte d’Urgell 187, 08036 Barcelona, Spain E-mail address: [email protected], [email protected]