Optimal Newton-Secant like methods without memory for solving nonlinear equations with its dynamics
aa r X i v : . [ m a t h . NA ] O c t Noname manuscript No. (will be inserted by the editor)
Optimal Newton-Secant like methods without memoryfor solving nonlinear equations with its dynamics
Mehdi Salimi a · Taher Lotfi b · SomayehSharifi c · Stefan Siegmund a Received: date / Accepted: date
Abstract
We construct two optimal Newton-Secant like iterative methods forsolving non-linear equations. The proposed classes have convergence order fourand eight and cost only three and four function evaluations per iteration, re-spectively. These methods support the Kung and Traub conjecture and possessa high computational efficiency. The new methods are illustrated by numer-ical experiments and a comparison with some existing optimal methods. Weconclude with an investigation of the basins of attraction of the solutions inthe complex plane.
Keywords
Multi-point iterative methods; Newton-Secant method; Kungand Traub’s conjecture.
A main tool for solving nonlinear problems is the approximation of simpleroots x ∗ of a nonlinear equation f ( x ∗ ) = 0 with a scalar function f : D ⊂ R → R which is defined on an open interval D (see e.g. [28,30,31,39] andthe references therein). The secant method is a simple root-finding algorithmwhich can be traced back to a historic precursor called “rule of double falseposition” [29]. A modern way to view the secant method would be to replacethe derivative in the Newton-Raphson method x n +1 = x n − f ( x n ) f ′ ( x n ) by a finite-difference approximation. The Newton-Raphson method is one of the mostwidely used algorithms for finding roots. It is of second order and requirestwo evaluations for each iteration step, one evaluation of f and one of f ′ . Corresponding author: [email protected] a Department of Mathematics, Technische Universit¨at Dresden, 01062 Dresden, Germany b Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran c Young Researchers and Elite Club, Hamedan Branch, Islamic Azad University, Hamedan,Iran Mehdi Salimi a et al. Newton-Raphson iteration is an example of a one-point iteration, i.e. in eachiteration step the evaluations are taken at one point. Multiple-point methodsevaluate at several points in each iteration step and in principle allow for ahigher convergence order with a lower number of function evaluations. Kungand Traub [20] conjectured that no multi-point method without memory with k evaluations could have a convergence order larger than 2 k − . A multi-pointmethod with convergence order 2 k − is called optimal.In this paper we construct two new optimal multi-point methods. Wepresent a two-point iteration with convergence order four which requires twoevaluations of f and one evaluation of f ′ and a three-point iteration withconvergence order eight which requires three evaluations of f and one evalua-tion of f ′ . Both methods combine the Newton and Secant methods and utilizethe idea of weight functions to obtain optimality in the sense of Kung andTraub. For an alternative construction of an optimal three-point method withconvergence order eight which also uses carefully chosen weight functions, see[23].For well known two-point methods without memory one can consult e.g.Jarrat [18], King [19] and Ostrowski [28]. Bi et al. [8] developed an optimalthree-point iterative method with convergence order eight. Wang and Liu usedweight functions to construct optimal three-point methods [21] and [41] andoptimal convergence order eight was achieved by Geum and Kim [15] and[16] utilizing parametric weight functions. Based on rational interpolation andweight functions, Sharma et al. introduced two three-point methods [33,34],see also Cordero et al. [12]-[14] and Soleymani et al. [35], Babajee et al. [7],Thukral and Petkovic [38] and for recent studies the interested reader is re-ferred to Chun and Lee [10] and Petkovic et al. [30] and Neta [24] has demon-strated methods of eight and sixteen order of convergence. Alberto et al. [1]have analyzed a different anomalies in a Jarrat family of iterative root-findingmethods. In [9] Chun et al. introduced weight functions with a parameter intoan iteration process to increase the order of the convergence and enhance thebehavior of the iteration process. In [22] Lotfi and Salimi pointed to seriouserrors that presented in the paper entitled ”A family of optimal iterative meth-ods with fifth and tenth order convergence for solving nonlinear equations” aswell.The paper is organized as follows: Section 2 is devoted to the constructionand convergence analysis of a new two-point method with optimal convergenceorder four and a new three-point method with optimal convergence order eight.Computational aspects, comparisons and dynamic behavior with other meth-ods are illustrated in Section 3. itle Suppressed Due to Excessive Length 3 y n = x n − f ( x n ) f ′ ( x n ) ,x n +1 = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) , ( n = 0 , , . . . ) , (1)where x is an initial approximation of x ∗ . The convergence order of (1) isthree and with three evaluations it is not optimal. We intend to increase theorder of convergence and extend (1) by an additional step y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) ,x n +1 = z n − f ( z n ) f ′ ( x n ) . (2)Method (2) uses four function evaluations with order of convergence four.Therefore, this method is not optimal. In order to decrease the number offunction evaluations, we approximate f ( z n ) by an expression based on f ( x n ), f ( y n ) and f ′ ( x n ). Taylor expansion of f at x n yields f ( z n ) = f ( x n ) + f ′ ( x n )( z n − x n ) + 12 f ′′ ( x n )( z n − x n ) + O (cid:0) ( z n − x n ) (cid:1) , (3)and similarly we have f ( y n ) = f ( x n ) + f ′ ( x n )( y n − x n ) + 12 f ′′ ( x n )( y n − x n ) + O (cid:0) ( y n − x n ) (cid:1) . (4)Using Newton’s method and (4), we obtain12 f ′′ ( x n ) ≈ f ( y n )( f ′ ( x n )) f ( x n ) . (5)According to (2), we have z n − x n = − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) . (6)Substituting (5) and (6) into (3), we obtain f ( z n ) ≈ f ( x n ) − f ( x n ) f ( x n ) − f ( y n ) + f ( y n ) f ( x n )( f ( x n ) − f ( y n )) . (7) Mehdi Salimi a et al. Substituting (7) into (2), yields y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) ,x n +1 = z n − h − f ( x n ) f ( x n ) − f ( y n ) (cid:16) f ( y n ) f ( x n ) − f ( y n ) (cid:17)i f ( x n ) f ′ ( x n ) . (8)Although we reduced the number of function evaluations compared to (2), theconvergence order of (8) is not yet four. In order to increase it, we consider anappropriate weight function, namely φ ( t n ), as follows: y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) ,x n +1 = z n − h − f ( x n ) f ( x n ) − f ( y n ) (cid:16) f ( y n ) f ( x n ) − f ( y n ) (cid:17)i f ( x n ) f ′ ( x n ) φ ( t n ) , (9)where t n = f ( y n ) f ( x n ) . In the following theorem, we provide sufficient conditionson the weight function φ ( t n ) which imply that method (9) has convergenceorder four. Theorem 1
Let D ⊆ R be an open interval, f : D → R four times continu-ously differentiable and let x ∗ ∈ D be a simple zero of f . If the initial point x is sufficiently close to x ∗ , then the method defined by (9) converges to x ∗ withorder at least four if the weight function φ : R → R is two times continuouslydifferentiable and satisfies the conditions φ (0) = 0 , φ ′ (0) = −
12 and | φ ′′ (0) | < ∞ . Proof
Let e n := x n − x ∗ , e n,y := y n − x ∗ , e n,z := z n − x ∗ and c n := f ( n ) ( x ∗ ) n ! f ′ ( x ∗ ) for n ∈ N . Using the fact that f ( x ∗ ) = 0, Taylor expansion of f at x ∗ yields f ( x n ) = f ′ ( x ∗ ) (cid:0) e n + c e n + c e n + c e n (cid:1) + O ( e n ) (10)and f ′ ( x n ) = f ′ ( x ∗ ) (cid:0) c e n + 3 c e n + 4 c e n (cid:1) + O ( e n ) . (11)Therefore f ( x n ) f ′ ( x n ) = e n − c e n + (cid:0) c − c (cid:1) e n + O ( e n ) , and hence e n,y = y n − x ∗ = c e n + O ( e n ) . For f ( y n ) we also have f ( y n ) = f ′ ( x ∗ ) (cid:0) c e n + ( − c + 2 c ) e n + (5 c − c c + 3 c ) e n (cid:1) + O ( e n ) , (12) itle Suppressed Due to Excessive Length 5 therefore, by substituting (10), (11) and (12) into (2), we get e n,z = z n − x ∗ = c e n + O ( e n ) . From (10) and (12), we obtain t n = f ( y n ) f ( x n ) = c e n + ( − c + 2 c ) e n + (8 c − c c + 3 c ) e n + O ( e n ) . (13)Expanding φ at 0, yields φ ( t n ) = φ (0) + φ ′ (0) t n + 12 φ ′′ (0) t n + O ( t n ) . (14)Substituting (10)-(14) into (9), we obtain e n +1 = x n +1 − x ∗ = R e n + R e n + R e n + O ( e n ) , where R = 2 c φ (0) ,R = c (cid:0) φ ′ (0) (cid:1) ,R = − c c + c (cid:0) + φ ′′ (0) (cid:1) . (15)By setting R = R = 0, the convergence order becomes four. Obviously φ (0) = 0 ⇒ R = 0 ,φ ′ (0) = − ⇒ R = 0 , | φ ′′ (0) | < ∞ ⇒ R = 0 . (16)Consequently, the error equation becomes e n +1 = R e n + O ( e n ) , which finishes the proof of the theorem.2.2 Optimal three-point methodIn this section we construct a new optimal three-point method based on thetwo-point method (9). We extend method (9) by a Newton step and get y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) ,v n = z n − h − f ( x n ) f ( x n ) − f ( y n ) (cid:16) f ( y n ) f ( x n ) − f ( y n ) (cid:17)i f ( x n ) f ′ ( x n ) φ ( t n ) ,x n +1 = v n − f ( v n ) f ′ ( v n ) , (17)where φ ( t n ) is a weight function as in Theorem 1.Method (17) evaluates functions for five times with order of convergenceeight, so the method is not optimal. In order to reduce the number of function Mehdi Salimi a et al. evaluation, we approximate f ′ ( v n ) by an expression which is based on f ( x n ), f ( y n ), f ( v n ), and f ′ ( x n ), namely its linear approximation f ′ ( v n ) ≈ f ′ ( x n ) + f ′ ( z n ) − f ′ ( x n ) z n − x n ( v n − x n ) . (18)We approximate f ′ ( z n ) by expressions which were calculated above. The Tay-lor expansion of f at y n yields f ( z n ) = f ( y n ) + f ′ ( y n )( z n − y n ) + 12 f ′′ ( y n )( z n − y n ) + O (cid:0) ( z n − y n ) (cid:1) , (19)and f ′ ( z n ) = f ′ ( y n ) + f ′′ ( y n )( z n − y n ) + O (cid:0) ( z n − y n ) (cid:1) . (20)According to (19), we have f ′ ( y n ) ≈ f ( z n ) − f ( y n ) z n − y n − f ′′ ( y n )( z n − y n ) . (21)On the other hand, we have f ′′ ( y n ) ≈ f [ z n , x n , x n ] = 2 (cid:16) f [ z n , x n ] − f ′ ( x n ) (cid:17) z n − x n , (22)where f [ z n , x n ] = f ( z n ) − f ( x n ) z n − x n . Substituting (21) and (22) into (20), we obtain f ′ ( z n ) ≈ f [ z n , y n ] + (cid:0) f [ z n , x n ] − f ′ ( x n ) (cid:1) z n − y n z n − x n , (23)where f [ z n , y n ] = f ( z n ) − f ( y n ) z n − y n . In a next step we replace f ( z n ) by an approx-imation to reduce the number of function evaluations. Taylor expansion of f at x n yields f ( z n ) = f ( x n ) + f ′ ( x n )( z n − x n ) + f ′′ ( x n )( z n − x n ) + f ′′′ ( x n )( z n − x n ) + O (cid:0) ( z n − x n ) (cid:1) , (24)and similarly we have f ( v n ) = f ( x n ) + f ′ ( x n )( v n − x n ) + f ′′ ( x n )( v n − x n ) + f ′′′ ( x n )( v n − x n ) + O (cid:0) ( v n − x n ) (cid:1) . (25)From (25), we calculate16 f ′′′ ( x n ) ≈ f ( v n ) − f ( x n ) v n − x n − f ′ ( x n ) − f ( y n ) (cid:16) f ′ ( x n ) (cid:17) f ( x n ) ( v n − x n ) v n − x n ) . (26) itle Suppressed Due to Excessive Length 7 Plugging (5) and (26) into (24), we obtain f ( z n ) ≈ f ( x n ) + f ′ ( x n )( z n − x n ) + f ( y n ) (cid:16) f ′ ( x n ) (cid:17) f ( x n ) ( z n − x n ) + " f [ v n , x n ] − f ′ ( x n ) − f ( y n ) (cid:16) f ′ ( x n ) (cid:17) f ( x n ) ( v n − x n ) ( z n − x n ) ( v n − x n ) . (27)Then, by replacing (23) into (18), we get f ′ ( v n ) ≈ f ′ ( x n ) + f [ z n , y n ] + (cid:16) f [ z n , x n ] − f ′ ( x n ) (cid:17) z n − y n z n − x n − f ′ ( x n ) z n − x n ( v n − x n ) , (28)where we can plug (27) instead of f ( z n ) in (28) as well. The following schemeevaluates functions for four times y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) ,v n = z n − h − f ( x n ) f ( x n ) − f ( y n ) (cid:16) f ( y n ) f ( x n ) − f ( y n ) (cid:17)i f ( x n ) f ′ ( x n ) φ ( t n ) ,x n +1 = v n − f ( v n ) (cid:18) f ′ ( x n ) + f [ z n ,y n ]+ (cid:16) f [ z n ,x n ] − f ′ ( x n ) (cid:17) zn − ynzn − xn − f ′ ( x n ) z n − x n ( v n − x n ) (cid:19) − , (29)where f ( z n ) is evaluated from (27) and t n = f ( y n ) f ( x n ) .Method (29) is not still optimal. Therefore we introduce a second weightfunction as follows: y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) ,v n = z n − h − f ( x n ) f ( x n ) − f ( y n ) (cid:16) f ( y n ) f ( x n ) − f ( y n ) (cid:17)i f ( x n ) f ′ ( x n ) φ ( t n ) ,x n +1 = v n − f ( v n ) (cid:18) f ′ ( x n ) + f [ z n ,y n ]+ (cid:16) f [ z n ,x n ] − f ′ ( x n ) (cid:17) zn − ynzn − xn − f ′ ( x n ) z n − x n ( v n − x n ) (cid:19) − ψ ( s n ) , (30)where f ( z n ) is evaluated from (27) and t n = f ( y n ) f ( x n ) and s n = f ( v n ) f ( x n ) .In the following theorem we prove that method (30) is of convergence ordereight if the weight functions φ ( t n ) and ψ ( s n ) satisfy the stated conditions inthe following theorem. Theorem 2
Let D ⊆ R be an open interval, f : D → R eight times continu-ously differentiable and let x ∗ ∈ D be a simple zero of f . If the initial point x is sufficiently close to x ∗ , then the method defined by (30) converges to x ∗ withorder at least eight if the weight function φ : R → R is two times continuously Mehdi Salimi a et al. differentiable, ψ : R → R is continuously differentiable and they satisfy theconditions of Theorem 1 and moreover φ ′′ (0) = − , ψ (0) = 1 and ψ ′ (0) = 1 . Proof
Let e n := x n − x ∗ , e n,y := y n − x ∗ , e n,z := z n − x ∗ , c n := f ( n ) ( x ∗ ) n ! f ′ ( x ∗ ) for n ∈ N . Using the fact that f ( x ∗ ) = 0, Taylor expansion of f at x ∗ yields f ( x n ) = f ′ ( x ∗ )( e n + c e n + c e n + c e n + c e n + c e n + c e n + c e n )+ O ( e n ) , (31)and f ′ ( x n ) = f ′ ( x ∗ )(1 + 2 c e n + 3 c e n + . . . + 9 c e n ) + O ( e n ) . (32)According to Theorem 1, we get e n,y = y n − x ∗ = c e n + ( − c + 2 c ) e n + (4 c − c c + 3 c ) e n + O ( e n ) , and e n,v = v n − x ∗ = (cid:0)(cid:0)
52 + φ ′′ (0) (cid:1) c − c c (cid:1) e n + O ( e n ) . By using Taylor’s theorem for f ( y n ) and f ( v n ) at x ∗ , we have f ( y n ) = f ′ ( x ∗ ) (cid:2) c e n − c − c ) e n + (5 c − c c + 3 c ) e n − c − c c + 3 c + 5 c c − c ) e n + (cid:0) c − c c + 37 c c + 34 c c − c c − c c + 5 c (cid:1) e n (cid:3) + O ( e n ) , (33)and f ( v n ) = f ′ ( x ∗ ) (cid:2) c c e n + (cid:0) c + 8 c c − c (cid:1) e n + (cid:0) − . c + 54 c c + 6 c c + 3 c c − c c − c c (cid:1) e n (1243 c − c c + 332 c c + 8 c (77 c + 2 c )+8(2 c − c − c c ) + 16 c (4 c c − c )) e n (cid:3) + O ( e n ) . (34)Also f ( z n ) = f ′ ( x ∗ ) (cid:2) c e n + 3 c ( − c + c ) e n + (6 c − c c + 2 c + 4 c c ) e n +( − c + 33 c c − c c − c c + 5 c c + 5 c c ) e n (3 c − c c + 48 c c + c (64 c − c )+( − c + 3 c + 6 c c ) + c ( − c c )) e n (cid:3) + O ( e n ) . (35)Moreover, for f ′ ( v n ), we also have f ′ ( v n ) = f ′ ( x ∗ ) (cid:2) − c c e n + (cid:0) c c − c − c c (cid:1) e n + (cid:0) c − c c + 104 c c + 60 c c − c c − c c (cid:1) e n − (243 c − c c − c + 39 c c + 6 c + 12 c c + c (165 c + 4 c ) + c ( − c c + 2 c )) e n (cid:3) + O ( e n ) . (36) itle Suppressed Due to Excessive Length 9 From (31) and (34), we calculate s n = f ( v n ) f ( x n ) = ( − c c ) e n + (cid:0) . c + 3 c c − c − c c (cid:1) e n + (cid:0) − c + 51 c c + 5 c c − c c + 9 c c − c c (cid:1) e n + O ( e n ) . (37)Expanding ψ at 0, yields ψ ( s n ) = ψ (0) + ψ ′ (0) s n + 12 ψ ′′ (0) s n + O ( s n ) . (38)By substituting (31)-(38) into (30), we obtain e n +1 = x n +1 − x ∗ = R e n + R e n + R e n + R e n + R e n + O ( e n ) , where R = − c ( − ψ (0)) (cid:0)(cid:0) φ ′′ (0) (cid:1) c − c (cid:1) ,R = 0 ,R = 0 ,R = − c (cid:0) − c + c (cid:0) φ ′′ (0) (cid:1)(cid:1)(cid:0) − c (cid:0) − ψ ′ (0) (cid:1) + c (cid:0) φ ′′ (0) (cid:1) ψ ′ (0) (cid:1) ,R = c c (cid:0) c + 4 c c − c (cid:1) . (39)To ensure convergence order eight for the three-point method (30), it is nec-essary to have R i = 0, ( i = 4 , , , ψ (0) = 1 ⇒ R = 0 ,ψ ′ (0) = 1 , φ ′′ (0) = − ⇒ R = 0 . (40)It is clear that R = 0, thus the error equation becomes e n +1 = R e n + O ( e n ) , and method (30) has convergence order eight, which proves the theorem.In what follows, we give some concrete explicit representations of (30) bychoosing different weight functions satisfying the provided condition for theweight functions φ ( t n ) and ψ ( t n ) in Theorems 1 and 2. Method 1.
Choose the weight functions φ ( t n ) and ψ ( s n ) as follows: φ ( t n ) = − t n − t n and ψ ( s n ) = 1 + 2 s n s n , (41) a et al. where t n = f ( y n ) f ( x n ) and s n = f ( v n ) f ( x n ) . The functions φ ( t n ) and ψ ( s n ) in (41) satisfythe assumptions of Theorem 2 denoted by SLSS, so y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) ,v n = z n − h − f ( x n ) f ( x n ) − f ( y n ) (cid:16) f ( y n ) f ( x n ) − f ( y n ) (cid:17)i (cid:18) − f ( y n )2 f ( x n ) − (cid:16) f ( y n ) f ( x n ) (cid:17) (cid:19) f ( x n ) f ′ ( x n ) ,x n +1 = v n − f ( v n ) (cid:16) f ( x n )+2 f ( v n ) f ( x n )+ f ( v n ) (cid:17) × (cid:18) f ′ ( x n ) + f [ z n ,y n ]+ (cid:16) f [ z n ,x n ] − f ′ ( x n ) (cid:17) zn − ynzn − xn − f ′ ( x n ) z n − x n ( v n − x n ) (cid:19) − , (42)where f ( z n ) is evaluated by (27). Method 2.
Choose the weight functions φ ( t n ) and ψ ( s n ) as follows: φ ( t n ) = t n + 9 t n t n − ψ ( s n ) = 11 − s n , (43)where t n = f ( y n ) f ( x n ) and s n = f ( v n ) f ( x n ) . The functions φ ( t n ) and ψ ( s n ) in (43) satisfythe assumptions of Theorem 2 and we get y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) ,v n = z n − h − f ( x n ) f ( x n ) − f ( y n ) (cid:16) f ( y n ) f ( x n ) − f ( y n ) (cid:17)i (cid:16) f ( y n ) f ( x n ) + f ( y n )5 f ( y n ) − f ( x n ) (cid:17) f ( x n ) f ′ ( x n ) ,x n +1 = v n − f ( v n ) (cid:16) f ( x n ) f ( x n ) − f ( v n ) (cid:17) × (cid:18) f ′ ( x n ) + f [ z n ,y n ]+ (cid:16) f [ z n ,x n ] − f ′ ( x n ) (cid:17) zn − ynzn − xn − f ′ ( x n ) z n − x n ( v n − x n ) (cid:19) − , (44)where f ( z n ) is evaluated by (27). Method 3.
Choose the weight functions φ ( t n ) and ψ ( s n ) as follows: φ ( t n ) = t n t n − ψ ( s n ) = 1 + 2 s n s n , (45)where t n = f ( y n ) f ( x n ) and s n = f ( v n ) f ( x n ) . The functions φ ( t n ) and ψ ( s n ) in (45) satisfythe assumptions of Theorem 2 and we get y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) ,v n = z n − h − f ( x n ) f ( x n ) − f ( y n ) (cid:16) f ( y n ) f ( x n ) − f ( y n ) (cid:17)i (cid:16) f ( y n )5 f ( y n ) − f ( x n ) (cid:17) f ( x n ) f ′ ( x n ) ,x n +1 = v n − f ( v n ) (cid:16) f ( v n )2 f ( x n )+5 f ( v n ) (cid:17) × (cid:18) f ′ ( x n ) + f [ z n ,y n ]+ (cid:16) f [ z n ,x n ] − f ′ ( x n ) (cid:17) zn − ynzn − xn − f ′ ( x n ) z n − x n ( v n − x n ) (cid:19) − , (46) itle Suppressed Due to Excessive Length 11 where f ( z n ) is evaluated by (27). Method 4.
Choose the weight functions φ ( t n ) and ψ ( s n ) as follows: φ ( t n ) = − t n + t n t n t n and ψ ( s n ) = (1 + s n ) sn +12 sn +1 , (47)where t n = f ( y n ) f ( x n ) and s n = f ( v n ) f ( x n ) . The functions φ ( t n ) and ψ ( s n ) in (47) satisfythe assumptions of Theorem 2 and we get y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n )( f ( x n ) − f ( y n )) f ′ ( x n ) ,v n = z n − h − f ( x n ) f ( x n ) − f ( y n ) (cid:16) f ( y n ) f ( x n ) − f ( y n ) (cid:17)i × (cid:16) − f ( y n )4 f ( x n ) − f ( y n )4 f ( x n ) + f ( y n ) f ( x n )+ f ( y n ) (cid:17) f ( x n ) f ′ ( x n ) ,x n +1 = v n − f ( v n ) (cid:16) f ( v n ) f ( x n ) (cid:17) f ( vn )+ f ( xn )2 f ( vn )+ f ( xn ) ! × (cid:18) f ′ ( x n ) + f [ z n ,y n ]+ (cid:16) f [ z n ,x n ] − f ′ ( x n ) (cid:17) zn − ynzn − xn − f ′ ( x n ) z n − x n ( v n − x n ) (cid:19) − , (48)where f ( z n ) is evaluated by (27).In the next section we apply the new methods (42), (44), (46) and (48)to several benchmark examples and compare them with existing three-pointmethods which have the same order of convergence and the same computa-tional efficiency index equal to θ √ r = 1 .
682 for the convergence order r = 8which is optimal for θ = 4 function evaluations per iteration [28,39]. W. Bi, H. Ren and Q. Wu method.
The method by Bi et al. [8] denotedby BRW is y n = x n − f ( x n ) f ′ ( x n ) ,z n = y n − f ( y n ) f ′ ( x n ) · f ( x n )+ βf ( y n ) f ( x n )+( β − f ( y n ) ,x n +1 = z n − f ( z n ) f [ z n ,y n ]+ f [ z n ,x n ,x n ]( z n − y n ) H ( t n ) , (49)with weight function H ( t n ) = 1(1 − αt n ) , α = 1 , (50) a et al. and t n = f ( z n ) f ( x n ) and β = − . Wang and Liu method.
The method by Wang and Liu [41] denoted byWL is y n = x n − f ( x n ) f ′ ( x n ) ,z n = x n − f ( x n ) f ′ ( x n ) G ( t n ) ,x n +1 = z n − f ( z n ) f ′ ( x n ) ( H ( t n ) + V ( t n ) W ( s n )) , (51)with weight functions G ( t n ) = 1 − t n − t n , H ( t n ) = 5 − t n + t n − t n , V ( t n ) = 1 + 4 t n , W ( s n ) = s n , (52)and t n = f ( y n ) f ( x n ) and s n = f ( z n ) f ( y n ) . Sharma and Sharma method.
The Sharma and Sharma method [33] de-noted by SS is y n = x n − f ( x n ) f ′ ( x n ) ,z n = y n − f ( y n ) f ′ ( x n ) · f ( x n ) f ( x n ) − f ( y n ) ,x n +1 = z n − f [ x n ,y n ] f ( z n ) f [ x n ,z n ] f [ y n ,z n ] W ( t n ) , (53)where weight functions are W ( t n ) = 1 + t n αt n , α = 1 , (54)and t n = f ( z n ) f ( x n ) . Babajee et al. method.
The method by Babajee et al., see [7], denotedby BCST, is y n = x n − f ( x n ) f ′ ( x n ) · (cid:18) (cid:16) f ( x n ) f ′ ( x n ) (cid:17) (cid:19) ,z n = y n − f ( y n ) f ′ ( x n ) · (cid:16) − f ( y n ) f ( x n ) (cid:17) − ,x n +1 = z n − f ( z n ) f ′ ( x n ) · ( f ( yn ) f ( xn ) ) +5 ( f ( yn ) f ( xn ) ) + f ( zn ) f ( yn ) ( − f ( yn ) f ( xn ) − f ( zn ) f ( xn ) ) . (55) Cordero et al. method.
The method by Cordero et al., see [14], denoted byCFGT, is y n = x n − f ( x n ) f ′ ( x n ) ,z n = y n − f ( x n ) f ( x n ) − f ( x n ) f ( y n ) − f ( x n ) f ( y n ) − f ( y n ) · f ( y n ) f ′ ( x n ) ,x n +1 = z n − f ( x n )+3 f ( z n ) f ( x n )+ f ( z n ) · f ( z n ) f [ z n ,y n ]+ f [ z n ,x n ,x n ]( z n − y n ) , (56) itle Suppressed Due to Excessive Length 13 with the divided differences f [ z n , y n ] = f ( z n ) − f ( y n ) z n − y n , f [ z n , x n , x n ] = f [ z n ,x n ] − f ′ ( x n ) z n − x n . Cordero et al. method.
The method by Cordero et al., see [13], denoted byCTV, is y n = x n − f ( x n ) f ′ ( x n ) ,z n = y n − f ( y n ) f ′ ( x n ) · f ( x n ) f ( x n ) − f ( y n ) ,x n +1 = v n − f ( z n ) f ′ ( x n ) · γ ( v n − z n ) β ( v n − z n )+ β ( y n − x n )+ β ( z n − x n ) , (57)where v n = z n − f ( z n ) f ′ ( x n ) · (cid:18) f ( x n ) − f ( y n ) f ( x n ) − f ( y n ) + 12 f ( z n ) f ( y n ) − f ( z n ) (cid:19) , and γ, β , β , β ∈ R such that γ = 3( β + β ) = 0. Thukral and Petkovic method.
The method by Thukral and Petkovic.,see [38], denoted by TP, is y n = x n − f ( x n ) f ′ ( x n ) ,z n = y n − f ( y n ) f ′ ( x n ) · f ( x n )+ βf ( y n ) f ( x n )+( β − f ( y n ) , ( α, β ∈ R ) x n +1 = z n − f ( z n ) f ′ ( x n ) · (cid:16) H ( t n ) + f ( z n ) f ( y n ) − αf ( z n ) + f ( z n ) f ( x n ) (cid:17) , (58)with weight functions H ( t n ) = 5 − β − (2 − β + 2 β ) t n + (1 + 4 β ) t n − β − (12 − β + 2 β ) t n , (59)where t n = f ( y n ) f ( x n ) . Chun and Lee method.
The method by Chun and Lee., see [10], denotedby CL, is y n = x n − f ( x n ) f ′ ( x n ) ,z n = y n − f ( y n ) f ′ ( x n ) · ( − f ( yn ) f ( xn ) ) ,x n +1 = z n − f ( z n ) f ′ ( x n ) · − H ( t n ) − J ( s n ) − P ( u n )) , (60)with weight functions H ( t n ) = − β − γ + t n + t n − t n , J ( s n ) = β + s n , P ( u n ) = γ + u n , (61) a et al. where t n = f ( y n ) f ( x n ) , s n = f ( z n ) f ( x n ) , u n = f ( z n ) f ( y n ) and β, γ ∈ R .The three-point method (30), more precisely, the explicitly proposed methods(42), (44), (46) and (48), are now tested on a number of nonlinear equations.To obtain a high accuracy and avoid the loss of significant digits, we employedmulti-precision arithmetic with 1800 significant decimal digits in the program-ming package of Mathematica 8. In order to compare them with the methods(49), (51), (53), (55), (56), (57), (58) and (60) we choose the initial value x using the Mathematica command
FindRoot [17, pp. 158–160] and computethe error, the computational order of convergence, (COC) by the approximateformula [42] COC ≈ ln | ( x n +1 − x ∗ ) / ( x n − x ∗ ) | ln | ( x n − x ∗ ) / ( x n − − x ∗ ) | . and the approximated computational order of convergence, (ACOC) by theformula [11] ACOC ≈ ln | ( x n +1 − x n ) / ( x n − x n − ) | ln | ( x n − x n − ) / ( x n − − x n − ) | . It is worth noting although the former formula, COC, has been used in therecent years, nevertheless, the later, ACOC, is more practical. Here we havecollect and use both of them for checking the accuracy of the considered meth-ods. Moreover, we should note that the results for these formula are generallydifferent from the exact convergence order of the method. The reason is thatin the error equations of the methods, we have some coefficients that dependon c k , and these c k s may vanish or vary for different kinds of examples. Seethe out puts in the Tables 1 and 2. We should be careful about these events.Indeed, it does not contradicts our discussed theory since all of the formulasare provided approximately and behave asymptotically. Table 1: f ( x ) = sin( x ) − x , x ∗ = 0 , x = 0 . | x − x ∗ | | x − x ∗ | | x − x ∗ | COC ACOC(42) 0 . e −
14 0 . e −
157 0 . e − . . . e −
14 0 . e −
157 0 . e − . . . e −
14 0 . e −
157 0 . e − . . . e −
14 0 . e −
157 0 . e − . . Table 2: f ( x ) = tan − ( x ) , x ∗ = 0 , x = 0 . | x − x ∗ | | x − x ∗ | | x − x ∗ | COC ACOC(42) 0 . e −
12 0 . e −
134 0 . e − . . . e −
12 0 . e −
134 0 . e − . . . e −
12 0 . e −
134 0 . e − . . . e −
12 0 . e −
134 0 . e − . . itle Suppressed Due to Excessive Length 15 In Table 1 and 2 our new three-point methods (42), (44), (46) and (48)with weight functions (41), (43), (45) and (47) are tested on two nonlinearequations. a et al. Table 3: f ( x ) = e sin( x ) − − x , x ∗ = 0 , x = 0 . | x − x ∗ | | x − x ∗ | | x − x ∗ | COC ACOC(42) 0 . e − . e −
83 0 . e −
748 9 . . . e − . e −
85 0 . e −
766 9 . . . e −
10 0 . e −
93 0 . e −
842 9 . . . e − . e −
81 0 . e −
734 9 . . . e − . e −
89 0 . e −
808 9 . . . e − . e −
68 0 . e −
552 8 . . . e − . e −
74 0 . e −
600 8 . . . e − . e −
74 0 . e −
598 8 . . . e − . e −
89 0 . e −
808 9 . . . e − . e −
73 0 . e −
598 8 . . . e − . e −
63 0 . e −
505 8 . . . e − . e −
74 0 . e −
599 8 . . Table 4: f ( x ) = ln(1 − x + x ) + 4 sin(1 − x ) , x ∗ = 1 , x = 1 . | x − x ∗ | | x − x ∗ | | x − x ∗ | COC ACOC(42) 0 . e −
12 0 . e −
117 0 . e − . . . e −
12 0 . e −
116 0 . e − . . . e −
12 0 . e −
114 0 . e − . . . e −
12 0 . e −
119 0 . e − . . . e −
12 0 . e −
114 0 . e − . . . e −
11 0 . e −
96 0 . e −
773 8 . . . e −
11 0 . e −
98 0 . e −
760 8 . . . e −
11 0 . e −
95 0 . e −
797 8 . . . e −
12 0 . e −
114 0 . e − . . . e −
11 0 . e −
98 0 . e −
788 8 . . . e −
11 0 . e −
92 0 . e −
739 8 . . . e −
11 0 . e −
97 0 . e −
782 8 . . itle Suppressed Due to Excessive Length 17 G : C → C be a rational map on the complex plane. For z ∈ C , wedefine its orbit as the set orb ( z ) = { z, G ( z ) , G ( z ) , . . . } . A point z ∈ C iscalled periodic point with minimal period m if G m ( z ) = z , where m is thesmallest integer with this property. A periodic point with minimal period 1is called fixed point. Moreover, a point z is called attracting if | G ′ ( z ) | < | G ′ ( z ) | >
1, and neutral otherwise. The Julia set of a nonlinearmap G ( z ), denoted by J ( G ), is the closure of the set of its repelling periodicpoints. The complement of J ( G ) is the Fatou set F ( G ), where the basin ofattraction of the different roots lie [7], [14].For the dynamical point of view, in fact, we take a 256 ×
256 grid of thesquare [ − , × [ − , ∈ C and assign a color to each point z ∈ D accordingto the simple root to which the corresponding orbit of the iterative methodstarting from z converges, and we mark the point as black if the orbit doesnot converge to a root, in the sense that after at most 100 iterations it hasa distance to any of the roots, which is larger than 10 − . In this way, wedistinguish the attraction basins by their color for different methods.We have tested several different examples, and the results on the perfor-mance of the tested methods were similar. Therefore we merely report thegeneral observation here for f ( z ) = z − /z . A visual inspection of the simu-lations indicates that for some examples the SLSS method (see Fig. 1) seems toproduce a larger basin of attraction than the BCST, SS, CTV, TP, CL, BRW,WL methods (see Figs. 2-5 and Figs. 7-9), but it seems to be smaller thanthat of the CFGT method (see Fig. 6). We stop here for a moment. Althoughwe were able to ignore the method CFGT, however, we should note that itis a very good example to discuss some aspects of our algorithms. It is well-known that any good algorithm should study these three concepts: accuracy,efficiency, and stability. All the work in this study have the same efficiency,four functional evaluations per iterate. On the other hand, comparing CFGTand method (44) reveal another fact: while a method may have a slightly bet-ter accuracy, see Table 4 and compare numerical results for methods (44) and(56), the other method may have produce a little better stability. Therefore,we cannot conclude which one is better in action. One has better accuracy,and the other has better stability. On the whole, finding such examples couldmake deeper understanding of devising new algorithms and it can be left forfuture works. Note that some points belong to no basin of attraction; these a et al. are starting points for which the methods do not converge, denoted by blackpoints. These exceptional points constitute the Julia set of methods, so namedin honor of G. Julia, a French mathematician who published an importantmemoir on this subject in 1918. Here, we would like to tell a little more aboutthese black points. We have said that these point do not converge to the roots.This statement is true only for the given number of iterations, say 100 here.If we increase the number of iteration, they might converge to a root, and thebasins or Fatou set might be larger. itle Suppressed Due to Excessive Length 19 Test problem f ( z ) = z − z Fig. 1
SLSS
Fig. 2
BRW
Fig. 3 WL Fig. 4 SS Fig. 5
BCST
Fig. 6
CFGT
Fig. 7
CTV
Fig. 8 TP Fig. 9
CL0 Mehdi Salimi a et al. Two new optimal classes of two-point and three-point methods without mem-ory have been developed which use only three and four function evaluationsper iteration, respectively. Both methods are based on the Newton and Secantmethods. A numerical comparison with other well-known optimal multi-pointmethods shows that our new classes are a valuable alternative to existing opti-mal multi-point methods. In addition, a numerical investigation of the basinsof attraction of the solutions illustrate that the stability region of our methodit typically larger than that of other methods. Indeed, among the eight com-pared methods, only one shows a larger stability region than our proposedmethods.
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