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This paper stresses the theoretical nature of constructing the optimal derivative-free iterations. We give necessary and sufficient conditions for derivative-free three-point iterations with the eighth-order of convergence. We also establish the connection of derivative-free and derivative presence three-point iterations. The use of the sufficient convergence conditions allows us to design wide class of optimal derivative-free iterations. The proposed family of iterations includes not only existing methods but also new methods with a higher order of convergence.

At present, there are a lot of iterative methods for solving nonlinear equations and systems of equations (see [

Methods | τ ˜ n | H ( θ n ) = c + ( d ˜ n c + d ) θ n + ω θ n 2 c + d θ n + b θ n 2 |
---|---|---|

M 1 , M 2 , M 3 . Thukral [ | 1 + γ ϕ n ( 1 + γ ϕ n − θ n ) ( 1 − θ n ) | c = 1 , d = − d ˜ n , b = 1 1 + γ ϕ n , ω = 0 |

Kung-Traub’s method | ||

Thukral [ | ||

P1 Thukral [ | 1 + d ˜ n θ n | c = 1 , d = b = ω = 0 |

Soleymani, Khattri [ | ||

P2 Thukral [ | 1 + θ n 1 − θ n 1 + γ ϕ n | c = 1 , d = − 1 1 + γ ϕ n , b = ω = 0 , |

Sharma [ | ||

M2 [ | ( 1 + θ n 1 + γ ϕ n + θ n 2 ( 1 + γ ϕ n ) 2 ) ⋅ 1 1 − θ n | b = 0 , c = 1 , d = − 1 , ω = 1 ( 1 + γ ϕ n ) 2 |

method in [ | h ( θ n , s n ) = 1 + θ n + s n + a 2 θ n 2 + b θ n s n + c 2 s n 2 | c = 1 , d = b = 0 , ω = ( a + c 2 + a 1 + γ ϕ ) |

Soleymani [ | 1 1 − d ˜ n θ n | c = 1 , d = − d ˜ n , b = ω = 0 |

Zheng et al. [ | ||

Soleymani [ | 1 + θ n 1 + γ ϕ n + θ n 2 ( 1 + γ ϕ n ) 2 | c = 1 , d = b = 0 , ω = 1 ( 1 + γ ϕ n ) 2 |

Cordero et al. [ | ψ 4 ( x n , y n , ω n ) , ( γ = 0 ) | |

Sharifi et al. [ | 1 + β θ n 1 + ( β − 2 ) θ n 1 1 − θ n 1 + γ ϕ n G ( θ n ) | c = 1 , b = 2 − β 1 + γ ϕ n , ω = − β , d = β − d ˜ n − 1 |

Chebyshew-Halley type method [ | 1 1 − 2 α θ n 1 1 − θ n 1 + γ ϕ n H ( θ n ) H ( 0 ) = 1 , H ′ ( 0 ) = 1 − 2 α | c = 1 , d = − ( 2 α + 1 1 + γ ϕ n ) , b = 2 α 1 + γ ϕ n , ω = H ( θ n ) |

Lotfi et al. [ | 1 + θ n + a d ˜ n θ n 2 2 1 − θ n 1 + γ ϕ n | c = 1 , b = 0 , d = − 1 1 + γ ϕ n , ω = a d ˜ n 2 |

Behl. [ | ψ 4 ( x n , y n , ω n ) |

and to obtain the wide class of optimal derivative-free three-point methods. The paper is organized as follows. In Section 2, we give the necessary and sufficient conditions for derivative-free three-point iterations to be optimal order eight. We also establish the connection between derivative presence and derivative-free three-point methods. In Section 3, we apply the sufficient convergence conditions to obtain the optimal derivative-free methods which are dependent on parameters in the third-step of considered iterations. We obtain families of optimal derivative-free three-point methods. They include many existing methods as particular cases as well as new methods with the higher order of convergence. In last section, we present the results of numerical experiments that confirm the theoretical conclusion about the convergence order and make comparison with other known methods of the same order of convergence. Finally, numerical results show that new iterative methods can be significant by its high precision and practical use.

Typically, the optimal three-point iterative methods have a form [

y n = x n − f ( x n ) f ′ ( x n ) , z n = y n − τ ¯ n f ( y n ) f ′ ( x n ) , x n + 1 = z n − α n f ( z n ) f ′ ( x n ) , (1)

in which the parameters τ ¯ n and α n are given by

τ ¯ n = 1 + 2 θ n + β ˜ θ n 2 + γ ˜ θ n 3 + ⋯ , (2)

and

α n = 1 + 2 θ n + ( β ˜ + 1 ) θ n 2 + ( 2 β ˜ + γ ˜ − 4 ) θ n 3 + ( 1 + 4 θ n ) f ( z n ) f ( y n ) + O ( f ( x n ) 4 ) , (3)

where β ˜ , γ ˜ ∈ R , and θ n = f ( y n ) f ( x n ) . In [

Theorem 1. Let the function f ( x ) be sufficiently smooth and have a simple root x * ∈ I . Furthermore, let the initial approximation x 0 be sufficiently close to x * . Then, the convergence order of the iterative method (1) is eight if and only if the parameters τ ¯ n and α n satisfy conditions (2) and (3), respectively.

Remark. The second sub-step in (1) can be rewritten as any two-point optimal fourth-order method

z n = ψ 4 ( x n , y n ) ,

where ψ 4 ( x n , y n ) is a real function using the evaluation of f ( x n ) , f ′ ( x n ) and f ( y n ) . Each method in ψ 4 has a parameter τ ¯ n given by (2) with own β ˜ and γ ˜ .

Now we consider the derivative-free variant of (1)

y n = ψ 2 ( x n , y n ) = x n − f ( x n ) ϕ n ,

z n = ψ 4 ( x n , y n ) = y n − τ ˜ n f ( y n ) ϕ n , (4)

x n + 1 = z n − α ˜ n f ( z n ) ϕ n ,

where

w n = x n + γ f ( x n ) ,

ϕ n = f ( w n ) − f ( x n ) w n − x n = 1 γ ( f ( w n ) f ( x n ) − 1 ) ≈ f ′ ( x n ) , γ ∈ R . (5)

Here ψ 2 ( x n , y n ) is any second-order method. Actually, in Formula (4), the fundamental quantities are

θ n = f ( y n ) f ( x n ) , σ n = f ( y n ) f ( w n ) , and υ n = f ( z n ) f ( y n ) .

Then θ n = O ( f ( x n ) ) , σ n = O ( f ( x n ) ) for x n → x * , where x * is a simple root of f ( x ) . If ψ 4 ( x n , y n ) is any two-point optimal fourth-order method then f ( z n ) = O ( f ( x n ) 4 ) , therefore υ n = O ( f ( x ) 2 ) . The iteration (4) obtained from (1) replacing f ′ ( x n ) by ϕ n . Due to change (5), the parameters in (4) does not remain as before and we denote them by τ ˜ n and α ˜ n . We call the iterations (1) and (4) the derivative presence (DP) and derivative-free (DF) variants respectively. If we use the notations

c ˜ n = 1 1 + γ ϕ n , d ˜ n = 1 + c ˜ n ,

then we have

σ n = c ˜ n θ n , θ n + σ n = d ˜ n θ n . (6)

DP can be derived from DF by substituting σ n = θ n . The following is the main result of our work [

Theorem 2. Let the assumptions of Theorem 1 be fulfilled. Then, the convergence order of the iteration (4) is eight if and only if the parameters τ ˜ n and α ˜ n in (4) are given by formulas

τ ˜ n = 1 + d ˜ n θ n + β ˜ θ n 2 + γ ˜ θ n 3 + ⋯ , (7)

and

α ˜ n = 1 + d ˜ n θ n + ( β ˜ + c ˜ n ) θ n 2 + ( γ ˜ + d ˜ n ( β ˜ − 1 − c ˜ n 2 ) ) θ n 3 + ( 1 + 2 d ˜ n θ n ) υ n + O ( f ( x n ) 4 ) . (8)

The proposed method (4) with parameters given by (7) and (8) is three-point derivative free and optimal in the sense of Kung and Traub. Kung-Traub conjecture [

Define τ ˜ n = h ( θ n , σ n ) , α ˜ n = g ( θ n , σ n , υ n ) as sufficiently smooth functions of θ n , σ n , υ n . It is easy to show that f ( z n ) = O ( f ( x n ) 4 ) if and only if h 00 = h 10 = h 01 = 1 , where h i j = h ( i , j ) ( 0,0 ) , ( i ≥ 0, j ≥ 0 ) . Hence, under the restriction h 11 = h 02 = h 21 = h 12 = h 03 = 1 , (4) is optimal if and only if

g 000 = 1 ,

g 100 = g 010 = g 001 = 1 ,

g 101 = g 011 = 2 ,

g 200 = h 20 , g 110 = 1 , g 020 = 0 ,

g 300 = h 20 + h 30 − 1 , g 210 = h 20 − 1 , g 120 = g 030 = − 1.

Those can be easily checked with using (6). For the optimal formula, the remainder term is O ( f ( x n ) 4 ) in (8) because υ n = O ( f ( x ) 2 ) . In this sense, we can say that (4) is optimal if and only if τ ˜ n , α ˜ n can be written as (7) and (8).

When γ → 0 the Formula (7) leads to (2) and the Formula (8) leads to (3). A query may arise that there exists an optimal (DF) variant (4) for each optimal (DP) variant (1) and vice versa. If yes, how to find its (DF) variant? To respond this we use the connection of formulas (3) and (8). Actually, from (3) and (8) we deduce that

α ˜ n = α n ( x n , y n , ϕ n ) + ( d ˜ n − 2 ) ( 1 + θ n + ( d ˜ n + β ˜ ) θ n 2 + 2 υ n ) θ n , (9)

where α n ( x n , y n , ϕ n ) is obtained replacing f ′ ( x n ) by ϕ n in α n ( x n , y n , f ′ ( x n ) ) in (1). From (9) we find that

α n ( x n , y n , f ′ ( x n ) ) = α ˜ n ( x n , y n , ϕ n ) | ϕ n = f ′ ( x n ) = α ˜ n ( x n , y n , f ′ ( x n ) ) . (10)

These relations (9) and (10) give the rule of mutual transition of (DP) and (DF) variants. There exists the one optimal (DP) variant (4) for each optimal (DF) variant (1). The converse does not true. Namely there are several (DF) variants of (DP).

Now we give the application of Theorem 2 to construct new iterations. The sufficient convergence conditions (7) and (8) allow us to design new derivative-free optimal methods. Depending on the form of α ˜ n we can obtain different iterations. We consider some special cases.

1) Let α ˜ n in (4) be a form

α ˜ n = φ ( θ n ) + ψ ( υ n ) + μ ( f ( z n ) f ( x n ) ) , (11)

where φ , ψ , and μ are smooth enough functions. As regarding the iteration (4) with α ˜ n given by (11) we give the following result.

Theorem 3. The iteration (4) with τ ˜ n given by (7) and with α ˜ n given by (11) have the order of convergence eight, if the following conditions hold:

φ ( 0 ) = 1 , φ ′ ( 0 ) = d ˜ n , φ ″ ( 0 ) = 2 ( β ˜ + c ˜ n ) ,

φ ‴ ( 0 ) = 6 ( γ ˜ + d ˜ n ( β ˜ − 1 − c ˜ n 2 ) ) ,

ψ ( 0 ) = 0 , ψ ′ ( 0 ) = 1 , (12)

μ ( 0 ) = 0 , μ ′ ( 0 ) = 2 d ˜ n .

Proof. Using the Taylor expansion of smooth enough functions φ ( θ n ) , ψ ( υ n ) and μ ( θ n , υ n ) we obtain an expression for (11). The comparison of this expression with sufficient condition (8) gives conditions (12).

When β ˜ = γ ˜ = 0 in (7) the Theorem 3 leads to a theorem in [

τ ˜ n = 1 + d ˜ n θ n . (13)

Therefore, Theorem 3 is more general, than that of [

( f ( z n ) f ( ω n ) ) 2 . By neglecting these terms, α ˜ n can be simplified essentially without loss of the order of convergence. When γ = 0 the condition (12) reduced to

φ ( 0 ) = 1 , φ ′ ( 0 ) = 2 , φ ″ ( 0 ) = 2 ( β ˜ + 1 ) , φ ″ ( 0 ) = 6 ( γ ˜ + 2 β ˜ − 4 ) ,

ψ ( 0 ) = 0 , ψ ′ ( 0 ) = 1 , μ ( 0 ) = 0 , μ ′ ( 0 ) = 4. (14)

It means that the derivative presence variant (1) with parameters given by (7) and (11) has a convergence order eight under conditions (14).

Thukral and Petković considered in [

τ ˜ n = 1 + b θ n 1 + ( b − 2 ) θ n = 1 + 2 θ n + 2 ( 2 − b ) θ n 2 + 2 ( 2 − b ) 2 θ n 3 .

In this case β ˜ = 2 ( 2 − b ) and γ ˜ = 2 ( 2 − b ) 2 and the condition (14) coincides with that of [

τ ˜ n = θ n + 1 1 − θ n = 1 + 2 θ n + θ n 2 + θ n 3 + ⋯ .

In this case β ˜ = γ ˜ = 1 and the condition (14) leads to that of [

φ ( θ n ) = τ ˜ n + c ˜ n θ n 2 + d ˜ n ( β ˜ − 1 − c ˜ n 2 ) θ n 3 . (15)

Due to generating function method [

τ ˜ n = H ( θ n ) = c + ( H ′ ( 0 ) c + d ) θ n + ω θ n 2 c + d θ n + b θ n 2 , b , c , d , ω ∈ R , (16)

satisfying conditions

H ( 0 ) = 1 , H ′ ( 0 ) = d ˜ n , H ″ ( 0 ) = 2 β ˜ , H ‴ ( 0 ) = 6 γ ˜ .

As a result, we have a family of optimal derivative-free three-point methods (4) with (11), (15), and (16). The constants β ˜ and γ ˜ can be expressed through b , c , d and ω as:

β ˜ = ω − b − d H ′ ( 0 ) c , γ ˜ = d ( 2 b − ω ) c 2 + ( H ′ ( 0 ) c + d ) ( d 2 c 3 − b c 2 ) − d 3 c 3 .

That is we have the iterations (4) with τ ˜ n is given by (16) and α ˜ n is given by

α ˜ n = τ ˜ n + c ˜ n θ n 2 + d ˜ n ( β ˜ − 1 − c ˜ n 2 ) θ n 3 + ( 1 + 2 d ˜ n θ n ) υ n . (17)

Note that the choice of parameter τ ˜ n defined by (16) includes almost all the choices listed in

2) Let α ˜ n in (4) be a form

α ˜ n = τ ˜ n + K ( θ n , υ n ) , (18)

where τ ˜ n is given by (7) and K ( θ n , υ n ) is sufficient smooth function of θ n and υ n .

Theorem 4. The iteration (4) with τ ˜ n given by (7) and α ˜ n given by (18) has the order of convergence eight, if the following conditions hold:

K ( 0 , 0 ) = K ′ θ ( 0 , 0 ) = 0 , K ′ υ ( 0 , 0 ) = 1 , K ″ θ υ ( 0 , 0 ) = 2 d ˜ n ,

K ″ θ θ ( 0 , 0 ) = 2 c ˜ n , K ‴ θ θ θ ( 0 , 0 ) = 6 d ˜ n ( β ˜ − 1 − c ˜ n 2 ) . (19)

Proof. From (7) and (8) it is clear that

K ( θ n , υ n ) = c ˜ n θ n 2 + d ˜ n ( β ˜ − 1 − c ˜ n 2 ) θ n 3 + ( 1 + 2 d ˜ n θ n ) υ n + O ( f ( x ) 4 ) (20)

which holds under conditions (19).

The (DP) variant of this iteration is obtained from (4), (7), and (18) when γ → 0 . Note that the similar scheme was considered in [

In some cases, the form

α ˜ n = τ ˜ n ( 1 + K ( θ n , υ n ) τ ˜ n ) = τ ˜ n W ( θ n , υ n ) (21)

obtained from (18) is useful. Using (20) we obtain

W ( θ n , υ n ) = 1 + c ˜ n θ n 2 + ( − c ˜ n d ˜ n + d ˜ n ( β ˜ − 1 − c ˜ n 2 ) ) θ n 3 + ( 1 + d ˜ n θ n ) υ n + O ( f ( x n ) 4 ) . (22)

For the iteration (4) with (7) and (21) we can formulate the following:

Theorem 5. The iteration (4) with (7) and (21) has the order of convergence eight, if the following conditions hold:

W ( 0 , 0 ) = 1 , W ′ θ ( 0 , 0 ) = 0 , W ″ θ θ ( 0 , 0 ) = 2 c ˜ n ,

W ‴ θ θ θ ( 0 , 0 ) = 6 ( − c ˜ n d ˜ n + d ˜ n ( β ˜ − 1 − c ˜ n 2 ) ) , (23)

W ′ υ ( 0 , 0 ) = 1 , W ″ υ θ ( 0 , 0 ) = d ˜ n .

Proof. If we take (22) into account in the Taylor expansion of function W ( θ n , υ n ) we arrive at (23).

When γ = 0 the conditions (23) take a form

W ( 0 , 0 ) = 1 , W ′ θ ( 0 , 0 ) = 0 , W ″ θ θ ( 0 , 0 ) = 2 ,

W ‴ θ θ θ ( 0 , 0 ) = 12 ( β ˜ − 3 ) , W ′ υ ( 0 , 0 ) = 1 , W ″ υ θ ( 0 , 0 ) = 2. (24)

Remark. Obviously, as for τ ˜ n one can take any function H given by (16) in the formulas (17) and (21).

Note that in [

τ ˜ n = h ( θ n , s n ) = 1 + θ n + s n + a 2 θ n 2 + b θ n s n + c 2 s n 2 + ⋯ ,

s n = c ˜ n f ( y n ) f ( x n ) ,

α ˜ n = h ( θ n , s n ) ⋅ μ ( θ n , s n , υ n ) ,

μ ( θ n , s n , υ n ) = 1 + υ n + d 2 υ n 2 + θ n s n + θ n υ n + s n υ n + a − 2 2 θ n 3 + c − 2 2 s n 3 + m 6 υ n 3 + a + 2 b − 4 2 θ n s n 2 + 2 b + c − 4 2 θ n s n 2 = 1 + c ˜ n θ n 2 + ( 1 + d ˜ n θ n ) υ n + ( d ˜ n β ˜ − ( 2 + c ˜ n 3 + 2 c ˜ n ) d ˜ n ) θ n 3 + O ( f ( x n ) 4 ) (25)

that does not coincide with (22). Moreover, the terms d 2 υ n 2 and m 6 υ n 3 seem to be redundant, because it suffices to determine α ˜ n with accuracy O ( f ( x n ) 4 ) .

Note that (DP) methods with (7) and (21) are often used. For example, Kung-Traub’s eighth-order method [

τ ˜ n = 1 ( 1 − θ n ) 2 , W ( θ n , υ n ) = f ( y n ) ( f 2 ( x n ) + f ( y n ) ( f ( y n ) − f ( z n ) ) ) ( f ( x n ) − f ( z n ) ) 2 ( f ( y n ) − f ( z n ) ) = 1 + θ n ( θ n − θ n υ n ) ( 1 − θ n υ n ) 2 ( 1 − υ n ) . (26)

The Bi-Wu-Ren’s optimal eighth-order method [

τ ˜ n = h ( θ n ) , α n = f ′ n ( x n ) ( f ( x n ) + β f ( z n ) ) f ( x n ) + ( β − 2 ) f ( z n ) 1 f [ z n , y n ] + f [ z n , x n , x n ] ( z n − y n ) , (27)

where

f [ z n , y n ] = f ( y n ) − f ( z n ) y n − z n , f [ z n , x n , x n ] = f ′ ( x n − f [ z n , x n ] ) x n − z n .

But (27) is not the example for (21).

The Sharma and Arora’s optimal eighth-order method [

τ ˜ n = 1 1 − 2 θ n , α ˜ n = τ ˜ n 4 θ n 2 ( 1 − θ n ) 2 ( 1 − υ n ) ( − 1 + θ n υ n ) ( ( 1 − 2 θ n ) 2 + ( 3 − 4 θ n ) θ n υ n ) . (28)

Moreover, we suggest that more general theory for τ ˜ n as

τ ˜ n = 1 + θ n + σ n + h 20 θ n 2 + h 11 θ n σ n + h 02 σ n 2 + h 30 θ n 3 + h 21 θ n 2 σ n + h 12 θ n σ n 2 + h 03 σ n 3 + ⋯ .

3) Let α ˜ n in (4) be a form

α ˜ n = ϕ n f [ z n , y n ] + ( z n − y n ) f [ z n , y n , x n ] + ( z n − y n ) ( z n − x n ) f [ z n , y n , x n , ω n ] (29)

that often used in practice, see [

y n = x n − f ( x n ) f ′ ( x n ) ,

z n = ψ 4 ( x n , y n ) ,

x n + 1 = z n − α n f ( z n ) f ′ ( x n ) , (30)

where α n = f ′ ( x n ) f [ z n , y n ] + ( z n − y n ) f [ y n , x n , x n ] .

In [

τ ¯ n = 1 + 1 θ n + ( 1 + 1 1 + f ( x n ) f ′ ( x n ) ) θ n . (31)

Our iteration (1) with (2) and (30) is more general than that of [

4) Let α ˜ n in (4) be a form

α ˜ n = ϕ n ( 1 + A θ n + B θ n 2 + C θ n 3 + ( ω + Δ θ n ) υ n ) a f [ x n , z n ] + b f [ z n , y n ] + c f [ x n , y n ] , a + b + c = 1 , (32)

where a , b , c ∈ R .

We shall find the coefficients A , B , C and ω , Δ such that the iteration (4) with (7) and (32) has the order of convergence eight and state the following:

Theorem 6. The iteration (4) with (7) and (32) has the order of convergence eight, if the following conditions hold:

A = ( 1 − b ) ( d ˜ n − 1 ) , B = ( β ˜ − d ˜ n ) ( 1 − b ) + c ˜ n ( 1 − a ) ,

C = a − 1 + ( b − a ) β + ( 1 − b ) γ + ( a + β − 2 ) c ˜ n + ( c − 2 ) c ˜ n 2 − c ˜ n 3 , (33)

ω = 1 − b , Δ = − a + b − 1 + d ˜ n ( 2 − b ) .

Proof. Using the following relations

f [ x n , y n ] = ϕ n ( 1 − θ n ) , f [ z n , y n ] = ϕ n τ ˜ n ( 1 − υ n ) ,

f [ x n , z n ] = ϕ n 1 + τ ˜ n θ n ( 1 − θ n υ n ) ,

1 τ ˜ n = 1 − d ˜ n θ n + ( d ˜ n 2 − β ˜ n ) θ n 2 + ( 2 β ˜ n d ˜ n − γ ˜ − d ˜ n 3 ) θ n 3 + ⋯ , (34)

1 1 + τ ˜ n θ n = 1 − θ n + ( 1 − d ˜ n ) θ n 2 + ( 2 d ˜ n − β ˜ n − 1 ) θ n 3 + ⋯ ,

1 1 − x = 1 + x + x 2 + x 3 + ⋯ , | x | < 1 ,

we get

ϕ n a f [ x n , z n ] + b f [ z n , y n ] + c f [ x n , y n ] = 1 + ( a + c + b d ˜ n ) θ n + ( a ( d ˜ n − 1 ) + b ( β ˜ − d ˜ n 2 ) + ( a + c + b d ˜ n ) 2 ) θ 2 + ( a ( β ˜ + 1 − 2 d ˜ n ) + b ( γ ˜ + d ˜ n 3 − 2 β ˜ d ˜ n ) + 2 ( a + c + b d ˜ n ) ( a ( d ˜ n − 1 ) + b ( β ˜ − d ˜ n 2 ) ) + ( a + c + b d ˜ n ) 3 ) θ n 3 + ( b + ( a + 2 b ( a + c ) + ( 2 b − 1 ) b d ˜ n ) θ n ) υ n + O ( f ( x n 4 ) ) . (35)

Substituting (35) into (32) and using the sufficient convergence condition (8) we arrive at (33).

Thus, we have a family of optimal three-point (DF) the iteration (4) with (7) and (32) that contains three parameters a, b and c. Now, we consider some particular cases of the iteration (4) with (7) and (32). Let a = b = 1 and c = − 1 . Then from (33) we find that

A = B = 0 , C = ( 1 + γ ϕ n ) β ˜ − d ˜ n − 3 − γ ϕ n ( 1 + γ ϕ n ) 2 , ω = 0 , Δ = c ˜ n .

Hence we obtain

α ˜ n = ( 1 + ( ( 1 + γ ϕ n ) β ˜ − d ˜ n − 3 − γ ϕ n ) c ˜ n 2 θ n 3 + c ˜ n θ n υ n ) ϕ n f [ x n , z n ] + f [ z n , y n ] − f [ x n , y n ] ≈ ( 1 + f ( z n ) f ( ω n ) ) ( 1 + ( ( 1 + γ ϕ n ) β ˜ − d ˜ n − 3 − γ ϕ n ) f 3 ( y n ) f 2 ( ω n ) f ( x n ) ) ϕ n f [ x n , z n ] + f [ z n , y n ] − f [ x n , y n ] , (36)

or

α ˜ n ≈ ϕ n ( 1 − f ( z n ) f ( ω n ) ) ( 1 − C ^ f 3 ( y n ) f 2 ( ω n ) f ( x n ) ) ( f [ x n , z n ] + f [ z n , y n ] − f [ x n , y n ] ) , (37)

where C ^ = ( 1 + γ ϕ n ) β ˜ − d ˜ n − 3 − γ ϕ n . The sign ≈ in (37) indicates that it holds with accuracy O ( f ( x n 4 ) ) . Now, we consider concrete choice of τ ˜ n :

τ ˜ n = 1 1 − d ˜ n θ n + p c ˜ n θ n 2 , p ∈ N . (38)

For the choice (38) we have

β ˜ = d ˜ n 2 − p c ˜ n

and

C ^ = − ( p + 1 ) .

The iteration (4) with (38) and (36) (or (37)) is converted to one given by Soleymani in [

α ˜ n = ( 1 + f ( z n ) f ( ω n ) ) ϕ n f [ x n , z n ] + f [ z n , y n ] − f [ x n , y n ] , (39)

or using 1 + f ( z n ) f ( w n ) = 1 1 − f ( z n ) f ( w n ) + O ( f ( x n ) 6 ) we have

α ˜ n = ϕ n ( 1 − f ( z n ) f ( ω n ) ) ( f [ x n , z n ] + f [ z n , y n ] − f [ x n , y n ] ) . (40)

Let

τ ˜ n = 1 1 − d ˜ n θ n = 1 + d ˜ n θ n + β ˜ θ n 2 + γ ˜ θ n 3 + ⋯ , (41)

with

β ˜ = d ˜ n 2 , γ ˜ = d ˜ n 3 .

Then C = − c ˜ n 2 and we have

α ˜ n = ( 1 − c ˜ n 2 θ n 3 + c ˜ n θ n υ n ) ϕ n f [ x n , z n ] + f [ z n , y n ] − f [ x n , y n ] . (42)

The iteration (4) with (41) and (42) coincides with one given by Soleymani in [

α ˜ n = ( 1 + θ n 4 − ( 1 + γ ϕ n ) f 3 ( y n ) ( 1 + γ ϕ n ) 3 f 3 ( x n ) − υ n 2 + c ˜ n σ n υ n + ( θ n υ n ) 2 ) ϕ n f [ x n , z n ] + f [ z n , y n ] − f [ x n , y n ] ,

here we can neglect the redundant terms θ n 4 − υ n 2 + θ n 2 υ n 2 . Let a = 1 , and b = c = 0 . Then from (33) we find that

A = d ˜ n − 1 , B = β ˜ − d ˜ n , C = β ˜ ( d ˜ n − 2 ) + γ ˜ − c ˜ n d ˜ n 2 ,

ω = 1 , Δ = 2 ( d ˜ n − 1 ) .

The Formula (32) is converted to

α ˜ n = ( 1 + ( d ˜ n − 1 ) θ n + ( β ˜ − d ˜ n ) θ n 2 + ( β ˜ ( d ˜ n − 2 ) + γ ˜ − c ˜ n d ˜ n 2 ) θ n 3 + ( 1 + 2 ( d ˜ n − 1 ) θ n ) υ n ) ϕ n f [ x n , z n ] . (43)

On the other hand, the direct calculation using relations (34) gives

( 1 − f ( z n ) f ( ω n ) ) − 1 ( 1 − η f 3 ( y n ) f ( ω n ) f 2 ( x n ) ) × f [ x n , y n ] f [ z n , y n ] = 1 + ( d ˜ n − 1 ) θ n + ( β ˜ − d ˜ n ) θ n 2 + ( γ ˜ − β ˜ − η c ˜ n ) θ n 3 + ( 1 + 2 c ˜ n θ n ) υ n + O ( f ( x n ) 4 ) , η ∈ R . (44)

We choose parameter η in (44) such that the expression (44) coincides with the numerator of (43) within accuracy O ( f ( x n ) 4 ) . That is to say, that

η = − β ˜ + d ˜ n 2 .

As a result, (43) can be rewritten as

α ˜ n = ( 1 − f ( z n ) f ( ω n ) ) − 1 ( 1 − ( d ˜ n 2 − β ˜ ) f 3 ( y n ) f ( ω n ) f 2 ( x n ) ) f [ x n , y n ] ϕ n f [ x n , z n ] ⋅ f [ z n , y n ] . (45)

Thus, we find a family of optimal (DF) iteration (4) with (7) and (45), that contains some existing iterations as particular cases. Thukral in [

τ ˜ n = 1 1 − d ˜ n θ n + c ˜ n θ n 2 = 1 + d ˜ n θ n + ( d ˜ n 2 − c ˜ n ) θ n 2 + ⋯ .

In this case β ˜ = d ˜ n 2 − c ˜ n and hence η = c ˜ n , the α ˜ n given by (45) leads to that of M 1 and M 3 in [

τ ˜ n = 1 + θ n 1 − c ˜ n θ n . (46)

In this case β ˜ = c ˜ n d ˜ n and η = d ˜ n . Thus, our family of method (4) with (45) converted to P 2 . It means that the ( P 1, P 2 ) methods are also included in our family of (4) with (7) and (45). As stated above for the choice of (41) we have β ˜ = d ˜ n 2 , so (45) is simplified as

α ˜ n = ( 1 − f ( z n ) f ( ω n ) ) − 1 f [ x n , y n ] ϕ n f [ x n , z n ] f [ z n , y n ] . (47)

Thus, we have optimal (DF) methods

y n = x n − f ( x n ) ϕ n ,

z n = y n − τ ˜ n f ( y n ) ϕ n , τ ˜ n = 1 1 − d ˜ n θ n , (48)

x n + 1 = z n − α ˜ n f ( z n ) ϕ n ,

where α ˜ n is defined by (47). This is (DF) variant of Sharma and Sharma’s optimal methods given in [

In this section, we make some numerical experiments to show the convergence behavior of the presented derivative-free method (4) with parameters τ ˜ n and α ˜ n . We also compare them with the ones developed by Soleymani [

f 1 ( x ) = e − x + x 5 − 1 , x * = 4.9651142

f 2 ( x ) = e x 3 − x − cos ( x 2 − 1 ) + x 3 + 1 , x * = − 1

f 3 ( x ) = sin x + e x 2 − 1 , x * = 0

f 4 ( x ) = 1 x − | x | , x * = 1.

All computations are performed using the programming package Maple18 with multiple-precision arithmetic and 2500 significant digits. The test functions have been used with stopping criterion | x n − x * | < 10 − 250 , where x * is a root of f ( x ) and the approximation x n to x * . In all examples, we consider that the parameter γ = − 0.01 and that α = − 2 in Chebyshew-Halley’s method.

Nowadays, high order methods are important due to scientific computations in many areas of science and engineering use. For instance, planetary orbit calculation, radiation calculations and many real life problems demand higher precision for desired results [

f 1 ( x ) = 0 has two zeros. Obviously, one of the roots x = 0 is not taken for discussion. Another root is x * ≈ 4.965114231744276303699 . Now, we give some numerical experiments and compare new methods with some well-known methods for the smooth function f 1 using the initial guess x 0 = 6 . In

COC ≈ l n ( | x n − x * | / | x n − 1 − x * | ) l n ( | x n − 1 − x * | / | x n − 2 − x * | ) ,

we have computed the order of convergence.

From

methods | α ˜ n | τ ˜ n | n | | x * − x n | | COC |
---|---|---|---|---|---|

(4) | (32), ( a = b = 1 , c = − 1 ) | (38), ( p = − 1 ) | 3 | 0.3130e−674 | 8.00 |

(32), ( b = 1 , a = 1 , c = − 1 ) | (13) | 3 | 0.3422e−670 | 8.00 | |

(32), ( b = 1 , a = c = 0 ) | (13) | 3 | 0.1346e−667 | 8.00 | |

(32), ( b = 1 , a = c = 0 ) | (38), ( p = 0 ) | 3 | 0.1078e−670 | 8.00 | |

(32), ( b = 1 , a = − 1 , c = 1 ) | (13) | 3 | 0.3285e−665 | 8.00 | |

(32), ( b = 1 , a = − 1 , c = 1 ) | (38), ( p = 0 ) | 3 | 0.3378e−668 | 8.00 | |

Soleymani [ | (32), ( a = b = 1 , c = − 1 ) | (38), ( p = 0 ) | 3 | 0.2023e−673 | 8.00 |

Thukral [ | (32), ( a = b = 1 , c = − 1 ) | (38), ( p = 1 ) | 3 | 0.1239e−672 | 8.00 |

M 1 , M 2 , M 3 [ | (45), ( η = 1 1 + γ ϕ n ) | (38), ( p = 1 ) | 3 | 0.4813e−670 | 8.00 |

P1 Thukral [ | (45), ( β = 0 ) | (13) | 3 | 0.1271e−667 | 8.00 |

P2 Thukral [ | (45), ( β = d ˜ n 1 + γ ϕ n ) | (46) | 3 | 0.3112e−669 | 8.00 |

Sharma et al. [ | (45), ( η = 0 ) | (38), ( p = 0 ) | 3 | 0.7836e−671 | 8.00 |

methods | τ ˜ n = H ( θ n ) | n | | x * − x n | | COC |
---|---|---|---|---|

choices of parameters | f 1 ( x ) , x 0 = 6 | |||

Lotfi [ | c = 1 , d = − 1 1 + γ φ n , b = 0 , ω = d ˜ n 2 | 3 | 0.2785e−672 | 8.00 |

King’s type [ | c = ω = 1 , d = β − 1 − d ˜ n , b = 2 − β 1 + γ φ n , ( β = 2 ) | 3 | 0.1004e−674 | 8.00 |

Zheng [ | c = 1 , d = − d ^ n , b = ω = 0 | 3 | 0.9462e−674 | 8.00 |

Sharma [ | c = 1 , d = − 1 1 + γ φ n , b = ω = 0 | 3 | 0.4414e−673 | 8.00 |

Chebyshew-Halley [ | c = 1 , d = − ( 2 α + 1 1 + γ φ n ) , b = 2 α 1 + γ φ n , ω = H ( θ n ) | 3 | 0.7466e−671 | 8.00 |

theory of convergence discussed in previous section. In addition to the comparison of new methods with other methods we include some special cases of proposed family (4) in

As τ ˜ n in

α ˜ n | f ( x ) | f 2 ( x ) | f 3 ( x ) | Methods | ||
---|---|---|---|---|---|---|

x 0 | −0.6 | −1.5 | ||||

a , b , c | n | | f ( x n ) | | n | | f ( x n ) | | ||

(32) | a = 1 | |||||

b = 1 | 4 | 1.60 (−691) | 4 | 1.37 (−349) | ||

c = − 1 | ||||||

a = 1 | ||||||

b = 1 | 5 | 2.44 (−395) | 4 | 2.20 (−260) | ||

c = − 1 | ||||||

a = 0 | ||||||

b = 1 | - | 5 | 6.80 (−1233) | (4) | ||

c = 0 | ||||||

a = 0 | ||||||

b = 1 | 4 | 9.48 (−541) | 4 | 9.00 (−300) | ||

c = 0 | ||||||

a = − 1 | ||||||

b = 1 | - | 5 | 8.55 (−988) | |||

c = 1 | ||||||

a = − 1 | ||||||

b = 1 | 4 | 7.55 (−316) | 5 | 2.25 (−1777) | ||

c = 1 | ||||||

a = 1 | ||||||

b = 1 | 4 | 2.72 (−484) | - | - | Soleymani [ | |

c = − 1 | ||||||

a = 1 | ||||||

b = 1 | 4 | 1.75 (−449) | 4 | 7.99 (−702) | Thukral [ | |

c = − 1 | ||||||

(45) | η = 1 1 + γ ϕ n | 4 | 1.75 (−449) | 4 | 9.49 (−267) | M 1 , M 3 [ |

β = 0 | 5 | 1.05 (−875) | 5 | 4.59 (−1301) | P1 Thukral [ | |

β = d ˜ n 1 + γ ϕ n | 5 | 1.09 (−707) | 5 | 1.14 (−1709) | P2 Thukral [ | |

η = 0 | 5 | 2.98 (−1069) | 4 | 2.33 (−373) | Sharma et al. [ |

Furthermore, when the iteration diverges for the considered initial guess x 0 , we denote it by “−”.

From

The result of

methods | α ˜ n | τ ˜ n | n | COC |
---|---|---|---|---|

(4) | (32), ( a = b = 1 , c = − 1 ) | (38), ( p = − 1 ) | 6 | 8.00 |

(32), ( b = 1 , a = 1 , c = − 1 ) | (13) | 5 | 8.00 | |

(32), ( b = 1 , a = c = 0 ) | (13) | 6 | 8.00 | |

(32), ( b = 1 , a = c = 0 ) | (38), ( p = 0 ) | 6 | 8.00 | |

(32), ( b = 1 , a = − 1 , c = 1 ) | (13) | 5 | 8.00 | |

(32), ( b = 1 , a = − 1 , c = 1 ) | (38), ( p = 0 ) | 10 | 8.00 | |

Soleymani [ | (32), ( a = b = 1 , c = − 1 ) | (38), ( p = 0 ) | 6 | 8.00 |

Thukral [ | (32), ( a = b = 1 , c = − 1 ) | (38), ( p = 1 ) | 8 | 8.00 |

P1 Thukral [ | (45), ( β = 0 ) | (13) | 14 | 8.00 |

P2 Thukral [ | (45), ( β = d ˜ n 1 + γ ϕ n ) | (46) | 8 | 8.00 |

Sharma et al. [ | (45), ( η = 0 ) | (38), ( p = 0 ) | 21 | 8.00 |

used lesser than other existing methods under condition | x n − x * | < 10 − 250 . However, the dynamic behavior of iterations may depend on the choices of parameters and problems under consideration. In sum, numerical results show that new iterative methods can be significant by its high precision and practical use.

We derive the necessary and sufficient conditions for derivative-free three-point iterations with the optimal order. The use of these conditions allows us to derive the families of optimal derivative-free iterations. We propose the families of optimal derivative-free iterations (4) with τ ˜ n given by (16) and α ˜ n given by (17), (29), (32), and (45). Our families include many existing iterations as particular cases, as well as new effective iterations. We reveal redundant terms in well-known methods given in [

The authors wish to thank the editor and anonymous referees for their valuable suggestions on the first version of this paper. This work was supported by the Foundation of Science and Technology of Mongolian under grant SST_18/2018.

The authors declare no conflicts of interest regarding the publication of this paper.

Zhanlav, T., Otgondorj, Kh. and Mijiddorj, R.-O. (2020) Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations. American Journal of Computational Mathematics, 10, 100-117. https://doi.org/10.4236/ajcm.2020.101007