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Research ArticlePotra-Ptaacutek Iterative Method with Memory
Taher Lotfi1 Stanford Shateyi2 and Sommayeh Hadadi1
1 Department of Mathematics Hamedan Branch Islamic Azad University Hamedan 65138 Iran2Department of Mathematics University of Venda Private Bag X5050 Thohoyandou 0950 South Africa
Correspondence should be addressed to Stanford Shateyi stanfordshateyiunivenacza
Received 8 September 2013 Accepted 5 November 2013 Published 22 January 2014
Academic Editors I Straskraba and C Zhu
Copyright copy 2014 Taher Lotfi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The problem is to extend the method proposed by Soleymani et al (2012) to a method with memory Following this aim a freeparameter is calculated using Newtonrsquos interpolatory polynomial of the third degree So the R-order of convergence is increasedfrom 4 to 6 without any new function evaluations Numerically the extended method is examined along with comparison to someexisting methods with the similar properties
1 Introduction
Root finding is a great task in mathematics both historicallyand practically It has attracted attention of great mathemati-cians like Gauss and Newton It has real major applicationsand because of these real features it is still alive as a researchfield
Kung and Traubrsquos conjecture is the basic fact to constructoptimal multipoint methods without memory [1] On theother hand multipoint methods with memory can increaseefficiency index of an optimalmethodwithoutmemory with-out consuming any new functional evaluations and merelyusing accelerator parameter(s) This great power of meth-ods with memory has not been well considered until veryrecently So we have been motivated to extend modifiedPotra-Ptak [2] to its with memory method
Traub in his book [3] introduced methods with andwithoutmemory for the first timeMoreover he constructed aSteffensen-typemethodwithmemory using secant approachIn fact he increased the order of convergence of the Stef-fensenmethod [4] from 2 to 241This is the firstmethodwithmemory based on our best knowledge In other words Traubchanged Steffensenrsquos method slightly as follows (see [3 pages185ndash187])
1199090 11990801205740are given suitably
119909119899+1
= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
0 = 120574119899isin 119877 119899 = 0 1 2
1198731 (
119909) = 119891 (119909119899) + (119909 minus 119909
119899) 119891 [119909
119899 119908119899]
120574119899+1
= minus
1
1198731015840
1(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(1)
The parameter 120574119899is called self-accelerator and method (1)
has convergence order of 241 It is still possible to increasethe convergence order using better self-accelerator parameterbased on better Newton interpolation Free derivative can beconsidered as another virtue of (1)
We use the symbols rarr 119874 and sim according to the fol-lowing conventions [3] If lim
119909119899rarrinfin
119892(119909119899) = 119862 we write
119892(119909119899) rarr 119862 or 119892 rarr 119862 If lim
119909rarr119886119892(119909) = 119862 we write
119892(119909) rarr 119862 or 119892 rarr 119862 If 119891119892 rarr 119862 where 119862 is a nonzeroconstant we write 119891 = 119874(119892) or 119891 sim 119862119892 Let 119891(119909) be afunction defined on an interval 119868 where 119868 is the smallestinterval containing 119896 + 1 distinct nodes 119909
1 1199092 119909
119896 The
divided difference 119891[1199090 1199091 119909
119896] with 119896th-order is defined
as follows 119891[1199090] = 119891(119909
0)
119891 [1199090] =
119891 [1199091] minus 119891 [119909
0]
1199091minus 1199090
119891 [1199090 1199091 119909
119896]
=
119891 [1199091 1199092 119909
119896] minus 119891 [119909
0 1199091 119909119896minus1
]
119909119896minus 1199090
(2)
Hindawi Publishing CorporationISRN Mathematical AnalysisVolume 2014 Article ID 697642 6 pageshttpdxdoiorg1011552014697642
2 ISRNMathematical Analysis
Moreover we recall the definition of efficiency index (EI) as119864 = 119901
1119899 where 119901 is the order of convergence and 119899 is thetotal number of function evaluations per iteration
This paper is organized as follows Section 2 reviewsmod-ified Potra-Ptakrsquos method and we try to remodify it slightlytoo Error equation for our modification is provided InSection 3 development to with memory is carried out alongwith the discussion of its 119877-order Numerical examinationsand comparisons are presented in the last section
2 Remodified Optimal Derivative-FreePotra-Ptaacutekrsquos Method
In this section our primal goal is to modify Soleymani et almethod slightly so that its error equation can provide betterform in the case with memory In fact we prove that ourmodified method can generate order of convergence of 6while theirs has order of convergence of 52 in the case of withmemory
Derivative-free iterative methods for solving nonlinearequation 119891(119909) = 0 are important in the sense that in manypractical situation it is preferable to avoid calculation ofderivative of 119891 One such scheme is
119909119896+1
= 119909119896minus
120574119891(119909119896)2
119891 (119909119896+ 120574119891 (119909
119896)) minus 119891 (119909
119896)
119896 = 0 1 2 120574 isin 119877 minus 0
(3)
which is obtained from Newtonrsquos method
119909119896+1
= 119909119896minus
119891 (119909119896)
1198911015840(119909119896)
119896 = 0 1 2 (4)
by approximating the derivative 1198911015840(119909119896) by the quotient
(119891(119909119896+ 120574119891(119909
119896)) minus 119891(119909
119896))120574119891(119909
119896) Scheme (3) defines a one-
parameter (120574) family of methods and has the same order andefficiency index as that of Newtonrsquos method [3 4]
Recently based on scheme (3) Soleymani et al [2]have extended the idea of this family and presented Potra-Ptakrsquos derivative free families of two-point methods withoutmemory as follows
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119908119899= 119909119899+ 120574119891 (119909
119899) 120574 isin 119877 minus 0
119909119899+1
= 119909119899minus
119891 (119909119899) + 119891 (119910
119899)
119891 [119909119899 119908119899]
minus (
2119891 (119909119899) + 119886119891 (119910
119899)
119891 [119909119899 119908119899]
(
119891 (119910119899)
119891 (119909119899)
)
2
)
times (1 minus
120574119891 [119909119899 119908119899]
2 + 2120574119891 [119909119899 119908119899]
)
(5)
Moreover they have proved
Theorem 1 (see [2]) Let 120572 be a simple root of the sufficientlydifferentiable function 119891 in an open interval 119863 If 119909
0is
sufficiently close to120572 then (5) is of local forth order and satisfiesthe error equation below
119890119899+1
= (1 + 1205741198911015840(120572))119860
11198904
119899+ 119874 (119890
5
119899) (6)
where 119890119899= 119909119899minus 120572 119886 isin 119877 119888
119896= 119891(119896)
(120572)(1198961198911015840(120572)) 119896 = 2 3
and
1198601= minus
1198882
2
[2 (1198883+ 11988831198911015840(120572) 120574)
+ 1198882
2(119886 (1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
minus2 (5 + 1198911015840(120572) 120574 (5 + 119891
1015840(120572) 120574)))]
(7)
As you can see the order of convergence is 4 It is clear thaterror equation (6) has linear factor (1 +119891
1015840(120572)120574) it is better to
correct approach (5) in such a way that its error equation hasthe quadratic factor (1 + 119891
1015840(120572)120574)2 So as we can prove later
this factor increases convergence order up to 6 To this endit is just enough to correct second step in (5) as follows
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119908119899= 119909119899+ 120574119891 (119909
119899) 120574 isin 119877 minus 0
119909119899+1
= 119909119899minus
119891 (119909119899) + 119891 (119910
119899)
119891 [119909119899 119908119899]
minus (
2119891 (119909119899) + 119886119891 (119910
119899)
119891 [119910119899 119908119899]
(
119891(119910119899)
119891 (119909119899)
)
2
)
times (1 minus
120574119891 [119909119899 119908119899]
2 + 2120574119891 [119909119899 119908119899]
)
(8)
Hence method without memory (8) is still optimal andin the following theorem we establish its error equation
Theorem 2 Let 120572 be a simple root of the sufficiently differ-entiable function 119891 in an open interval 119863 If 119909
0is sufficiently
close to 120572 then (8) is of local forth order and satisfies the errorequation below
119890119899+1
= (1 + 1205741198911015840(120572))
2
11986021198904
119899+ 119874 (119890
5
119899) (9)
where 119890119899= 119909119899minus120572 119886 isin 119877 119888
119896= 119891(119896)
(120572)(1198961198911015840(120572)) 119896 = 2 3
and
1198602= minus
1
2
1198882((2 (minus3 + 119886) + (minus2 + 119886) 119891
1015840(120572) 120574) 119888
2
2+ 21198883)
(10)
ISRNMathematical Analysis 3
Proof We provide the Taylor expansion of any term involvedin (8) By Taylor expanding around the simple root in the nthiterate we have
119891 (119909119899)
119891 [119909119899 119908119899]
= 1198901
119899minus 1198882(1 + 119891
1015840(120572) 120574) 119890
2
119899
+ (minus1198883(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
+1198882
2(2 + 119891
1015840(120572) 120574 (2 + 119891
1015840(120572) 120574))) 119890
3
119899
+ 119874 (1198904
119899)
(11)
By considering this relation and the first step of (8) we obtain
119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
= 120572 + 1198882(1 + 119891
1015840(120572) 120574) 119890
2
119899
minus (minus1198883(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
+1198882
2(2 + 119891
1015840(120572) 120574 (2 + 119891
1015840(120572) 120574))) 119890
3
119899
+ 119874 (1198904
119899)
(12)
At this time we should expand 119891(119910119899) around the root by
taking into consideration (12) Accordingly we have
119891 (119910119899)
= 1198882(1 + 119891
1015840(120572) 120574) 119891
1015840(120572) 1198902
119899
minus (minus1198883(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
+1198882
2(2 + 119891
1015840(120572) 120574 (2 + 119891
1015840(120572) 120574))) 119891
1015840(120572) 1198903
119899+ 1198911015840(120572)
times (1198884(1 + 119891
1015840(120572) 120574) (3 + 119891
1015840(120572) 120574 (3 + 119891
1015840(120572) 120574))
+ 1198883
2(5 + 119891
1015840(120572) 120574 (7 + 119891
1015840(120572) 120574 (4 + 119891
1015840(120572) 120574))) minus 119888
21198883
times (7 + 1198911015840(120572) 120574 (10 + 119891
1015840(120572) 120574 (7 + 2119891
1015840(120572) 120574)))) 119890
4
119899
+ 119874 (1198905
119899)
(13)
Additionally we obtain
119909119899minus
119891 (119909119899) + 119891 (119910
119899)
119891 [119909119899 119908119899]
= 120572 + 1198882
2(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574) 119890
3
119899+ 119874 (119890
4
119899)
(14)
Similarly the same Taylor expansion results in
(
2119891 (119909119899) + 119886119891 (119910
119899)
119891 [119910119899 119908119899]
(
119891 (119910119899)
119891 (119909119899)
)
2
)(1 minus
120574119891 [119909119899 119908119899]
2 + 2120574119891 [119909119899 119908119899]
)
= 1198882
2(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574) 119890
3
119899
+
1
2
1198882(1 + 119891
1015840(120572) 120574)
2
times ((2 (minus3 + 119886) + (minus2 + 119886) 1198911015840(120572) 120574) 119888
2
2+ 21198883) 1198904
119899
+ 119874 (1198905
119899)
(15)
Using (12)ndash(15) in the last step of (8) provides finally
119909119899+1
minus 120572
= 119909119899minus
119891 (119909119899) + 119891 (119910
119899)
119891 [119909119899 119908119899]
minus (
2119891 (119909119899) + 119886119891 (119910
119899)
119891 [119910119899 119908119899]
(
119891 (119910119899)
119891 (119909119899)
)
2
)
times (1 minus
120574119891 [119909119899 119908119899]
2 + 2120574119891 [119909119899 119908119899]
) minus 120572
= minus
1
2
1198882(1 + 119891
1015840(120572) 120574)
2
times ((2 (minus3 + 119886) + (minus2 + 119886) 1198911015840(120572) 120574) 119888
2
2+ 21198883) 1198904
119899
+ 119874 (1198905
119899)
(16)
which shows that (8) is a derivative-free family of two-stepmethods with optimal convergence rate of 4 This completesthe proof
3 Development and Construction withMemory Family
This section concerns with extension of (8) to a method withmemory since its error equation contains the parameter 120574
which can be approximated in such a way that increase thelocal order of convergence So we set 120574 = 120574
119896as the iteration
proceeds by the formula 120574119896
= minus1119891
1015840
(120572) for 119896 = 1 2 where 119891
1015840
(120572) is an approximation of 1198911015840(120572) We have a methodthrough the following forms of 120574
119896
120574119896= minus
1
119891
1015840
(120572)
= minus
1
1198731015840
3(119909119896)
(17)
4 ISRNMathematical Analysis
where11987310158403(119905) = 119873
3(119905 119909119896 119910119896minus1
119908119896minus1
119909119896minus1
) is Newtonrsquos interpo-latory polynomial of third degree set through four availableapproximations 119909
119896 119910119896minus1
119908119896minus1
and 119909119896minus1
and
1198731015840
3(119909119896) = [
119889
119889119905
1198733 (
119905)]
119905=119909119896
= [
119889
119889119905
[119891 (119909119896) + 119891 [119909
119896 119910119896minus1
] (119905 minus 119909119896)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
] (119905 minus 119909119896) (119905 minus 119910
119896minus1)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
119909119896minus1
]
times (119905 minus 119909119896) (119905 minus 119910
119896minus1) (119905 minus 119908
119896minus1)]]
119905=119909119896
= 119891 [119909119896 119910119896minus1
] + 119891 [119909119896 119910119896minus1
119908119896minus1
] (119909119896minus 119910119896minus1
)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
119909119896minus1
] (119909119896minus 119910119896minus1
) (119909119896minus 119908119896minus1
)
(18)
By using Taylorrsquos expansion of 119891(119909) around the root 120572 wehave
119891 (119909) = 1198911015840(120572) (119890 + 119888
21198902+ 11988831198903+ 11988841198904+ 11988851198905+ sdot sdot sdot ) (19)
where 119890 = 119909 minus 120572 By using (18) and (19) we calculate
1198731015840
3(119909119896) = 1198911015840(120572) [1 + 2119888
2119890119896+ 311988831198902
119896
+ 1198884(119890119896minus1
1198902
119896+ 1198902
119896minus11199101198902
119896
+ 1198902
119896minus11199081198902
119896minus 119890119896minus1
119890119896minus1119910
119890119896
minus 119890119896minus1119910
119890119896minus1119908
119890119896minus 119890119896minus1
119890119896minus1119908
119890119896
+119890119896minus1
119890119896minus1119910
119890119896minus1119908
+ 31198903
119896) + sdot sdot sdot ]
= 1198911015840(120572) [1 + 119888
4119890119896minus1
119890119896minus1119910
119890119896minus1119908
+ 119874 (119890119896)]
(20)
According to this and (17) we find
1 + 1205741198911015840(120572) sim 119888
4119890119896minus1
119890119896minus1119910
119890119896minus1119908
(21)
For general case one can consult [3]In order to obtain the order of convergence of the
family of two-point methods with memory (8) where 120574119896is
calculated using the formula (17) we will use the conceptof the 119877-order of convergence [3] Now we can state thefollowing convergence theorem
Theorem 3 If an initial approximation 1199090is sufficiently close
to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative
scheme (8) is recursively calculated by the forms given in (17)then the 119877-order of convergence is at least 6
Proof Let 119909119896 be a sequence of approximations generated
by an iterative method with memory (IM) If this sequence
converges to the zero 120572 of119891with the119877-order (ger) of IM thenwe write
119890119896+1
sim 119863119896119903
119890119903
119896 119890
119896= 119909119896minus 120572 (22)
where 119863119896119903
tends to the asymptotic error constant 119863119903of IM
when 119896 rarr infin Thus
119890119896+1
sim 119863119896119903
(119863119896minus1119903
119890119903
119896minus1)119903= 119863119896119903
119863119903
119896minus11199031198901199032
119896minus1 (23)
Let 119890119896minus1119910
= 119910119896minus1
minus 120572 119890119896minus1119908
= 119908119896minus1
minus 120572 then we have
119890119896119908
sim (1 + 1205741198961198911015840(120572)) 119890119896
+ 119874 (1198902
119896) (24)
119890119896119910
sim 1198882(1 + 120574
1198961198911015840(120572)) 119890
2
119896+ 119874 (119890
3
119896) (25)
119890119896+1
sim 1198602(1 + 120574
1198961198911015840(120572))
2
1198904
119896+ 119874 (119890
5
119896) (26)
where 1198602= minus(12)119888
2[(2(minus3 + 119886) + (minus2 + 119886)119891
1015840(120572)120574)119888
2
2+ 21198883]
In the sequel we obtain the 119877-order of convergence of family(8) for approach (17) applied to the calculation of 120574
119896
Assume that the iterative sequences 119910119896and 119909119896have the119877-
orders 119901 and 119903 respectively then bearing in mind (22) weobtain
119890119896119910
sim 119863119896119901
119890119901
119896sim 119863119896119901
(119863119896minus1119903
119890119903
119896minus1)119901sim 119863119896119901
119863119901
119896minus1119903119890119903119901
119896minus1 (27)
and then we obtain
119890119896119910
sim 1198882(1 + 120574
1198961198911015840(120572)) 119890
2
119896sim 1198882(1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
) 1198902
119896
sim 11988821198884(119890119896minus1
) (119863119896minus1119901
119890119901
119896minus1) (119863119896minus1119904
119890119904
119896minus1) (119863119896minus1119903
119890119903
119896minus1)2
sim 11988821198884119863119896minus1119901
119863119896minus1119904
1198632
119896minus11199031198902119903+119904+119901+1
119896minus1
(28)
Assume that the iterative sequence119908119896has the 119877-order 119904 then
bearing in mind (22) we obtain
119890119896119908
sim 119863119896119904
119890119904
119896sim 119863119896119904
(119863119896minus1119903
119890119903
119896minus1)119904sim 119863119896119904
119863119904
119896minus1119903119890119903119904
119896minus1 (29)
and then we obtain
119890119896119908
sim (1 + 1205741198961198911015840(120572)) 119890119896
sim (1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
) 119890119896
sim 1198884119890119896minus1
(119863119896minus1119901
119890119901
119896minus1) (119863119896minus1119904
119890119904
119896minus1) (119863119896minus1119903
119890119903
119896minus1)
sim 1198884119863119896minus1119901
119863119896minus1119904
119863119896minus1119903
119890119903+119904+119901+1
119896minus1
(30)
119890119896+1
sim 1198861198964
(1 + 1205741198961198911015840(120572))
2
1198904
119896sim 1198861198964
(1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
)
2
1198904
119896
sim 1198861198964
1198884(119890119896minus1
)2(119863119896minus1119901
119890119901
119896minus1)
2
(119863119896minus1119904
119890119904
119896minus1)2(119863119896minus1119903
119890119903
119896minus1)4
sim 1198861198964
11988841198632
119896minus11199011198632
119896minus11199041198634
119896minus11199031198904119903+2119904+2119901+2
119896minus1
(31)
Combining the exponents of 119890119896minus1
on the right-hand sidesof (27)-(28) (29)-(30) and (23)ndash(31) we form the nonlinearsystem of three equations in 119901 119904 and 119903
119903119901 minus 2119903 minus (119901 + 119904) minus 1 = 0
119903119904 minus 119903 minus (119901 + 119904) minus 1 = 0
1199032minus 4119903 minus 2 (119901 + 119904) minus 2 = 0
(32)
ISRNMathematical Analysis 5
Table 1 1198911(119909) = log(1 + 119909
2) + 1198901199092minus3119909 sin119909 120572 = 0 119909
0= 035 120574
0= 001
Methods |1199091minus 120572| |119909
2minus 120572| |119909
3minus 120572| 119903
119888(20)
Potra-Ptak without memory 6040 (minus3) 5672 (minus10) 8280 (minus37) 3822Equations (8) and (17) with memory 6050 (minus3) 8723 (minus15) 6425 (minus84) 5841
Kung and Traub [1] without memory 5545 (minus3) 4864 (minus9) 3045 (minus33) 3999
Equation (34) with memory 5546 (minus3) 4708 (minus17) 1374 (minus96) 5654
Zheng et al [5] without memory 3541 (minus2) 2974 (minus6) 2127 (minus22) 3930Equation (35) with memory 5647 (minus3) 8272 (minus18) 1022 (minus101) 5658
Table 2 1198912(119909) = prod
5
119894=1(119909 minus 119894) 120572 = 2 119909
0= 16 120574
0= 001
Methods |1199091minus 120572| |119909
2minus 120572| |119909
3minus 120572| 119903
119888(20)
Potra-Ptak without memory 5522 (minus3) 1063 (minus10) 7391 (minus41) 3910Equations (8) and (17) with memory 5623 (minus3) 3822 (minus16) 1170 (minus93) 5888
Kung and Traub [1] (without memory) 6125 (minus3) 2402 (minus9) 5453 (minus35) 4004
Equation (34) with memory 6125 (minus3) 5888 (minus15) 1651 (minus86) 5955
Zheng at al [5] without memory 6021 (minus3) 1525 (minus9) 6077 (minus36) 4003Equation (35) with memory 6021 (minus3) 3251 (minus15) 1274 (minus88) 5985
Nontrivial solution of this system is 119904 = 2 119901 = 3 and119903 = 6 and we conclude that the lower bound of the 119877-orderof the method with memory is 6
Similarly one can prove the following
Theorem 4 If an initial approximation 1199090is sufficiently close
to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative
scheme (5) is recursively calculated by the forms given in (17)then the 119877-order of convergence is at least 52
4 Numerical Examples
To examine practical aspects of the proposedmodified Potra-Ptakrsquos without and with memory we implement it here inaction In other words we demonstrate the convergencebehavior of the method with memory (8) where 120574
119896is
calculated by (17) For comparison purposes we pick upKung and Traub [1] and Zheng et al [5] with and withoutmemories We use these notations The errors |119909
119896minus 120572|
denote approximations to the sought zeros 119860(minusℎ) stands for119860 times 10
minusℎ Moreover 119903119888indicates computational order of
convergence and is computed [2]
119903119888=
log (1003816100381610038161003816119891 (119909119896) 119891 (119909
119896minus1)1003816100381610038161003816)
log (1003816100381610038161003816119891 (119909119896minus1
) 119891 (119909119896minus2
)1003816100381610038161003816)
(33)
The software Mathematica 8 with 1000 arbitrary precisionarithmetic has been used in our computations The resultsalongside the test functions are given in Tables 1 and 2while 120574 = 120574
0= 001 [3] From Tables 1 and 2 we can
conclude that our methods work numerically well and aresuccessfully competingwith the existingmethods Indeed thelast columns of these tables show that both numerical andtheoretical aspects support each other
For comparison purposes we consider the followingmethods
Two-Point Method by Kung and Traub [1]
1199090 11990801205740are given suitably
119910119899= 119909119899minus
119891 (119909119899)
119891 [119908119899 119909119899]
119899 = 0 1 2
119909119899+1
= 119910119899minus
119891 (119910119899) 119891 (119908
119899)
[119891 (119908119899) minus 119891 (119910
119899)] 119891 [119909
119899 119910119899]
119896 = 0 1
120574119899+1
= minus
1
1198731015840
3(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(34)
Two-Point Method by Zheng et al [5]
1199090 11990801205740are given suitably
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119899 = 0 1 2
119909119899+1
= 119910119899minus
119891 (119910119899)
119891 [119910119899 119908119899] + 119891 [119910
119899 119909119899 119908119899] (119910119899minus 119909119899)
120574119899+1
= minus
1
1198731015840
3(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(35)
From Tables 1 and 2 it can be seen that our modifiedmethod without memory works truly moreover its withmemory competes the existing methods To sum up Potraand Ptak [6] constructed two-pointmethodwithoutmemory
6 ISRNMathematical Analysis
with convergence order of 3 it is not optimal in the sense ofKung and Traub Cordero et al [7] could make it optimalIn other words they introduce optimal two- and three-pointmethods with order of convergence of 4 and 8 respectivelyThough their methods are optimal they are not derivative-free Freshly Soleymani et al [2] have drawn two pointmethods without memory from Potra and ptak methodOne is derivative-free and the other is not In addition theirderivative method results two steps method by Cordero et al[7] for 119886 = 0 (See (5)) In this work we modified theirderivative-free method at first Then we generalized it tomethod with memory with efficiency index 119864(119901 119899) = 4
13=
18 see more about efficiency index in [7] Therefore atwo-step method with memory can obtain performanceeven better than four-step methods without memory withefficiency index 16
15= 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the University of Venda andthe Islamic Azad University Hamedan Branch
References
[1] H T Kung and J F Traub ldquoOptimal order of one-point andmultipoint iterationrdquo Journal of the Association for ComputingMachinery vol 21 pp 643ndash651 1974
[2] F Soleymani R Sharma X Li and E Tohidi ldquoAn optimizedderivative-free form of the Potra-Ptak methodrdquo Mathematicaland Computer Modelling vol 56 no 5-6 pp 97ndash104 2012
[3] J F Traub Iterative Methods for the Solution of EquationsPrentice Hall New York NY USA 1964
[4] J F Steffensen ldquoRemarks on iterationrdquo Scandinavian ActuarialJournal vol 1933 no 1 pp 64ndash72 1933
[5] Q Zheng J Li and F Huang ldquoAn optimal Steffensen-typefamily for solving nonlinear equationsrdquo Applied Mathematicsand Computation vol 217 no 23 pp 9592ndash9597 2011
[6] F A Potra and V Ptak ldquoNondiscrete introduction and iterativeprocessesrdquo in Research Notes in Mathematics vol 103 PitmanBoston Mass USA 1984
[7] A Cordero J L Hueso E Martınez and J R Torregrosa ldquoNewmodifications of Potra-Ptakrsquos method with optimal fourth andeighth orders of convergencerdquo Journal of Computational andApplied Mathematics vol 234 no 10 pp 2969ndash2976 2010
Submit your manuscripts athttpwwwhindawicom
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 ISRNMathematical Analysis
Moreover we recall the definition of efficiency index (EI) as119864 = 119901
1119899 where 119901 is the order of convergence and 119899 is thetotal number of function evaluations per iteration
This paper is organized as follows Section 2 reviewsmod-ified Potra-Ptakrsquos method and we try to remodify it slightlytoo Error equation for our modification is provided InSection 3 development to with memory is carried out alongwith the discussion of its 119877-order Numerical examinationsand comparisons are presented in the last section
2 Remodified Optimal Derivative-FreePotra-Ptaacutekrsquos Method
In this section our primal goal is to modify Soleymani et almethod slightly so that its error equation can provide betterform in the case with memory In fact we prove that ourmodified method can generate order of convergence of 6while theirs has order of convergence of 52 in the case of withmemory
Derivative-free iterative methods for solving nonlinearequation 119891(119909) = 0 are important in the sense that in manypractical situation it is preferable to avoid calculation ofderivative of 119891 One such scheme is
119909119896+1
= 119909119896minus
120574119891(119909119896)2
119891 (119909119896+ 120574119891 (119909
119896)) minus 119891 (119909
119896)
119896 = 0 1 2 120574 isin 119877 minus 0
(3)
which is obtained from Newtonrsquos method
119909119896+1
= 119909119896minus
119891 (119909119896)
1198911015840(119909119896)
119896 = 0 1 2 (4)
by approximating the derivative 1198911015840(119909119896) by the quotient
(119891(119909119896+ 120574119891(119909
119896)) minus 119891(119909
119896))120574119891(119909
119896) Scheme (3) defines a one-
parameter (120574) family of methods and has the same order andefficiency index as that of Newtonrsquos method [3 4]
Recently based on scheme (3) Soleymani et al [2]have extended the idea of this family and presented Potra-Ptakrsquos derivative free families of two-point methods withoutmemory as follows
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119908119899= 119909119899+ 120574119891 (119909
119899) 120574 isin 119877 minus 0
119909119899+1
= 119909119899minus
119891 (119909119899) + 119891 (119910
119899)
119891 [119909119899 119908119899]
minus (
2119891 (119909119899) + 119886119891 (119910
119899)
119891 [119909119899 119908119899]
(
119891 (119910119899)
119891 (119909119899)
)
2
)
times (1 minus
120574119891 [119909119899 119908119899]
2 + 2120574119891 [119909119899 119908119899]
)
(5)
Moreover they have proved
Theorem 1 (see [2]) Let 120572 be a simple root of the sufficientlydifferentiable function 119891 in an open interval 119863 If 119909
0is
sufficiently close to120572 then (5) is of local forth order and satisfiesthe error equation below
119890119899+1
= (1 + 1205741198911015840(120572))119860
11198904
119899+ 119874 (119890
5
119899) (6)
where 119890119899= 119909119899minus 120572 119886 isin 119877 119888
119896= 119891(119896)
(120572)(1198961198911015840(120572)) 119896 = 2 3
and
1198601= minus
1198882
2
[2 (1198883+ 11988831198911015840(120572) 120574)
+ 1198882
2(119886 (1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
minus2 (5 + 1198911015840(120572) 120574 (5 + 119891
1015840(120572) 120574)))]
(7)
As you can see the order of convergence is 4 It is clear thaterror equation (6) has linear factor (1 +119891
1015840(120572)120574) it is better to
correct approach (5) in such a way that its error equation hasthe quadratic factor (1 + 119891
1015840(120572)120574)2 So as we can prove later
this factor increases convergence order up to 6 To this endit is just enough to correct second step in (5) as follows
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119908119899= 119909119899+ 120574119891 (119909
119899) 120574 isin 119877 minus 0
119909119899+1
= 119909119899minus
119891 (119909119899) + 119891 (119910
119899)
119891 [119909119899 119908119899]
minus (
2119891 (119909119899) + 119886119891 (119910
119899)
119891 [119910119899 119908119899]
(
119891(119910119899)
119891 (119909119899)
)
2
)
times (1 minus
120574119891 [119909119899 119908119899]
2 + 2120574119891 [119909119899 119908119899]
)
(8)
Hence method without memory (8) is still optimal andin the following theorem we establish its error equation
Theorem 2 Let 120572 be a simple root of the sufficiently differ-entiable function 119891 in an open interval 119863 If 119909
0is sufficiently
close to 120572 then (8) is of local forth order and satisfies the errorequation below
119890119899+1
= (1 + 1205741198911015840(120572))
2
11986021198904
119899+ 119874 (119890
5
119899) (9)
where 119890119899= 119909119899minus120572 119886 isin 119877 119888
119896= 119891(119896)
(120572)(1198961198911015840(120572)) 119896 = 2 3
and
1198602= minus
1
2
1198882((2 (minus3 + 119886) + (minus2 + 119886) 119891
1015840(120572) 120574) 119888
2
2+ 21198883)
(10)
ISRNMathematical Analysis 3
Proof We provide the Taylor expansion of any term involvedin (8) By Taylor expanding around the simple root in the nthiterate we have
119891 (119909119899)
119891 [119909119899 119908119899]
= 1198901
119899minus 1198882(1 + 119891
1015840(120572) 120574) 119890
2
119899
+ (minus1198883(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
+1198882
2(2 + 119891
1015840(120572) 120574 (2 + 119891
1015840(120572) 120574))) 119890
3
119899
+ 119874 (1198904
119899)
(11)
By considering this relation and the first step of (8) we obtain
119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
= 120572 + 1198882(1 + 119891
1015840(120572) 120574) 119890
2
119899
minus (minus1198883(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
+1198882
2(2 + 119891
1015840(120572) 120574 (2 + 119891
1015840(120572) 120574))) 119890
3
119899
+ 119874 (1198904
119899)
(12)
At this time we should expand 119891(119910119899) around the root by
taking into consideration (12) Accordingly we have
119891 (119910119899)
= 1198882(1 + 119891
1015840(120572) 120574) 119891
1015840(120572) 1198902
119899
minus (minus1198883(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
+1198882
2(2 + 119891
1015840(120572) 120574 (2 + 119891
1015840(120572) 120574))) 119891
1015840(120572) 1198903
119899+ 1198911015840(120572)
times (1198884(1 + 119891
1015840(120572) 120574) (3 + 119891
1015840(120572) 120574 (3 + 119891
1015840(120572) 120574))
+ 1198883
2(5 + 119891
1015840(120572) 120574 (7 + 119891
1015840(120572) 120574 (4 + 119891
1015840(120572) 120574))) minus 119888
21198883
times (7 + 1198911015840(120572) 120574 (10 + 119891
1015840(120572) 120574 (7 + 2119891
1015840(120572) 120574)))) 119890
4
119899
+ 119874 (1198905
119899)
(13)
Additionally we obtain
119909119899minus
119891 (119909119899) + 119891 (119910
119899)
119891 [119909119899 119908119899]
= 120572 + 1198882
2(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574) 119890
3
119899+ 119874 (119890
4
119899)
(14)
Similarly the same Taylor expansion results in
(
2119891 (119909119899) + 119886119891 (119910
119899)
119891 [119910119899 119908119899]
(
119891 (119910119899)
119891 (119909119899)
)
2
)(1 minus
120574119891 [119909119899 119908119899]
2 + 2120574119891 [119909119899 119908119899]
)
= 1198882
2(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574) 119890
3
119899
+
1
2
1198882(1 + 119891
1015840(120572) 120574)
2
times ((2 (minus3 + 119886) + (minus2 + 119886) 1198911015840(120572) 120574) 119888
2
2+ 21198883) 1198904
119899
+ 119874 (1198905
119899)
(15)
Using (12)ndash(15) in the last step of (8) provides finally
119909119899+1
minus 120572
= 119909119899minus
119891 (119909119899) + 119891 (119910
119899)
119891 [119909119899 119908119899]
minus (
2119891 (119909119899) + 119886119891 (119910
119899)
119891 [119910119899 119908119899]
(
119891 (119910119899)
119891 (119909119899)
)
2
)
times (1 minus
120574119891 [119909119899 119908119899]
2 + 2120574119891 [119909119899 119908119899]
) minus 120572
= minus
1
2
1198882(1 + 119891
1015840(120572) 120574)
2
times ((2 (minus3 + 119886) + (minus2 + 119886) 1198911015840(120572) 120574) 119888
2
2+ 21198883) 1198904
119899
+ 119874 (1198905
119899)
(16)
which shows that (8) is a derivative-free family of two-stepmethods with optimal convergence rate of 4 This completesthe proof
3 Development and Construction withMemory Family
This section concerns with extension of (8) to a method withmemory since its error equation contains the parameter 120574
which can be approximated in such a way that increase thelocal order of convergence So we set 120574 = 120574
119896as the iteration
proceeds by the formula 120574119896
= minus1119891
1015840
(120572) for 119896 = 1 2 where 119891
1015840
(120572) is an approximation of 1198911015840(120572) We have a methodthrough the following forms of 120574
119896
120574119896= minus
1
119891
1015840
(120572)
= minus
1
1198731015840
3(119909119896)
(17)
4 ISRNMathematical Analysis
where11987310158403(119905) = 119873
3(119905 119909119896 119910119896minus1
119908119896minus1
119909119896minus1
) is Newtonrsquos interpo-latory polynomial of third degree set through four availableapproximations 119909
119896 119910119896minus1
119908119896minus1
and 119909119896minus1
and
1198731015840
3(119909119896) = [
119889
119889119905
1198733 (
119905)]
119905=119909119896
= [
119889
119889119905
[119891 (119909119896) + 119891 [119909
119896 119910119896minus1
] (119905 minus 119909119896)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
] (119905 minus 119909119896) (119905 minus 119910
119896minus1)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
119909119896minus1
]
times (119905 minus 119909119896) (119905 minus 119910
119896minus1) (119905 minus 119908
119896minus1)]]
119905=119909119896
= 119891 [119909119896 119910119896minus1
] + 119891 [119909119896 119910119896minus1
119908119896minus1
] (119909119896minus 119910119896minus1
)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
119909119896minus1
] (119909119896minus 119910119896minus1
) (119909119896minus 119908119896minus1
)
(18)
By using Taylorrsquos expansion of 119891(119909) around the root 120572 wehave
119891 (119909) = 1198911015840(120572) (119890 + 119888
21198902+ 11988831198903+ 11988841198904+ 11988851198905+ sdot sdot sdot ) (19)
where 119890 = 119909 minus 120572 By using (18) and (19) we calculate
1198731015840
3(119909119896) = 1198911015840(120572) [1 + 2119888
2119890119896+ 311988831198902
119896
+ 1198884(119890119896minus1
1198902
119896+ 1198902
119896minus11199101198902
119896
+ 1198902
119896minus11199081198902
119896minus 119890119896minus1
119890119896minus1119910
119890119896
minus 119890119896minus1119910
119890119896minus1119908
119890119896minus 119890119896minus1
119890119896minus1119908
119890119896
+119890119896minus1
119890119896minus1119910
119890119896minus1119908
+ 31198903
119896) + sdot sdot sdot ]
= 1198911015840(120572) [1 + 119888
4119890119896minus1
119890119896minus1119910
119890119896minus1119908
+ 119874 (119890119896)]
(20)
According to this and (17) we find
1 + 1205741198911015840(120572) sim 119888
4119890119896minus1
119890119896minus1119910
119890119896minus1119908
(21)
For general case one can consult [3]In order to obtain the order of convergence of the
family of two-point methods with memory (8) where 120574119896is
calculated using the formula (17) we will use the conceptof the 119877-order of convergence [3] Now we can state thefollowing convergence theorem
Theorem 3 If an initial approximation 1199090is sufficiently close
to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative
scheme (8) is recursively calculated by the forms given in (17)then the 119877-order of convergence is at least 6
Proof Let 119909119896 be a sequence of approximations generated
by an iterative method with memory (IM) If this sequence
converges to the zero 120572 of119891with the119877-order (ger) of IM thenwe write
119890119896+1
sim 119863119896119903
119890119903
119896 119890
119896= 119909119896minus 120572 (22)
where 119863119896119903
tends to the asymptotic error constant 119863119903of IM
when 119896 rarr infin Thus
119890119896+1
sim 119863119896119903
(119863119896minus1119903
119890119903
119896minus1)119903= 119863119896119903
119863119903
119896minus11199031198901199032
119896minus1 (23)
Let 119890119896minus1119910
= 119910119896minus1
minus 120572 119890119896minus1119908
= 119908119896minus1
minus 120572 then we have
119890119896119908
sim (1 + 1205741198961198911015840(120572)) 119890119896
+ 119874 (1198902
119896) (24)
119890119896119910
sim 1198882(1 + 120574
1198961198911015840(120572)) 119890
2
119896+ 119874 (119890
3
119896) (25)
119890119896+1
sim 1198602(1 + 120574
1198961198911015840(120572))
2
1198904
119896+ 119874 (119890
5
119896) (26)
where 1198602= minus(12)119888
2[(2(minus3 + 119886) + (minus2 + 119886)119891
1015840(120572)120574)119888
2
2+ 21198883]
In the sequel we obtain the 119877-order of convergence of family(8) for approach (17) applied to the calculation of 120574
119896
Assume that the iterative sequences 119910119896and 119909119896have the119877-
orders 119901 and 119903 respectively then bearing in mind (22) weobtain
119890119896119910
sim 119863119896119901
119890119901
119896sim 119863119896119901
(119863119896minus1119903
119890119903
119896minus1)119901sim 119863119896119901
119863119901
119896minus1119903119890119903119901
119896minus1 (27)
and then we obtain
119890119896119910
sim 1198882(1 + 120574
1198961198911015840(120572)) 119890
2
119896sim 1198882(1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
) 1198902
119896
sim 11988821198884(119890119896minus1
) (119863119896minus1119901
119890119901
119896minus1) (119863119896minus1119904
119890119904
119896minus1) (119863119896minus1119903
119890119903
119896minus1)2
sim 11988821198884119863119896minus1119901
119863119896minus1119904
1198632
119896minus11199031198902119903+119904+119901+1
119896minus1
(28)
Assume that the iterative sequence119908119896has the 119877-order 119904 then
bearing in mind (22) we obtain
119890119896119908
sim 119863119896119904
119890119904
119896sim 119863119896119904
(119863119896minus1119903
119890119903
119896minus1)119904sim 119863119896119904
119863119904
119896minus1119903119890119903119904
119896minus1 (29)
and then we obtain
119890119896119908
sim (1 + 1205741198961198911015840(120572)) 119890119896
sim (1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
) 119890119896
sim 1198884119890119896minus1
(119863119896minus1119901
119890119901
119896minus1) (119863119896minus1119904
119890119904
119896minus1) (119863119896minus1119903
119890119903
119896minus1)
sim 1198884119863119896minus1119901
119863119896minus1119904
119863119896minus1119903
119890119903+119904+119901+1
119896minus1
(30)
119890119896+1
sim 1198861198964
(1 + 1205741198961198911015840(120572))
2
1198904
119896sim 1198861198964
(1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
)
2
1198904
119896
sim 1198861198964
1198884(119890119896minus1
)2(119863119896minus1119901
119890119901
119896minus1)
2
(119863119896minus1119904
119890119904
119896minus1)2(119863119896minus1119903
119890119903
119896minus1)4
sim 1198861198964
11988841198632
119896minus11199011198632
119896minus11199041198634
119896minus11199031198904119903+2119904+2119901+2
119896minus1
(31)
Combining the exponents of 119890119896minus1
on the right-hand sidesof (27)-(28) (29)-(30) and (23)ndash(31) we form the nonlinearsystem of three equations in 119901 119904 and 119903
119903119901 minus 2119903 minus (119901 + 119904) minus 1 = 0
119903119904 minus 119903 minus (119901 + 119904) minus 1 = 0
1199032minus 4119903 minus 2 (119901 + 119904) minus 2 = 0
(32)
ISRNMathematical Analysis 5
Table 1 1198911(119909) = log(1 + 119909
2) + 1198901199092minus3119909 sin119909 120572 = 0 119909
0= 035 120574
0= 001
Methods |1199091minus 120572| |119909
2minus 120572| |119909
3minus 120572| 119903
119888(20)
Potra-Ptak without memory 6040 (minus3) 5672 (minus10) 8280 (minus37) 3822Equations (8) and (17) with memory 6050 (minus3) 8723 (minus15) 6425 (minus84) 5841
Kung and Traub [1] without memory 5545 (minus3) 4864 (minus9) 3045 (minus33) 3999
Equation (34) with memory 5546 (minus3) 4708 (minus17) 1374 (minus96) 5654
Zheng et al [5] without memory 3541 (minus2) 2974 (minus6) 2127 (minus22) 3930Equation (35) with memory 5647 (minus3) 8272 (minus18) 1022 (minus101) 5658
Table 2 1198912(119909) = prod
5
119894=1(119909 minus 119894) 120572 = 2 119909
0= 16 120574
0= 001
Methods |1199091minus 120572| |119909
2minus 120572| |119909
3minus 120572| 119903
119888(20)
Potra-Ptak without memory 5522 (minus3) 1063 (minus10) 7391 (minus41) 3910Equations (8) and (17) with memory 5623 (minus3) 3822 (minus16) 1170 (minus93) 5888
Kung and Traub [1] (without memory) 6125 (minus3) 2402 (minus9) 5453 (minus35) 4004
Equation (34) with memory 6125 (minus3) 5888 (minus15) 1651 (minus86) 5955
Zheng at al [5] without memory 6021 (minus3) 1525 (minus9) 6077 (minus36) 4003Equation (35) with memory 6021 (minus3) 3251 (minus15) 1274 (minus88) 5985
Nontrivial solution of this system is 119904 = 2 119901 = 3 and119903 = 6 and we conclude that the lower bound of the 119877-orderof the method with memory is 6
Similarly one can prove the following
Theorem 4 If an initial approximation 1199090is sufficiently close
to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative
scheme (5) is recursively calculated by the forms given in (17)then the 119877-order of convergence is at least 52
4 Numerical Examples
To examine practical aspects of the proposedmodified Potra-Ptakrsquos without and with memory we implement it here inaction In other words we demonstrate the convergencebehavior of the method with memory (8) where 120574
119896is
calculated by (17) For comparison purposes we pick upKung and Traub [1] and Zheng et al [5] with and withoutmemories We use these notations The errors |119909
119896minus 120572|
denote approximations to the sought zeros 119860(minusℎ) stands for119860 times 10
minusℎ Moreover 119903119888indicates computational order of
convergence and is computed [2]
119903119888=
log (1003816100381610038161003816119891 (119909119896) 119891 (119909
119896minus1)1003816100381610038161003816)
log (1003816100381610038161003816119891 (119909119896minus1
) 119891 (119909119896minus2
)1003816100381610038161003816)
(33)
The software Mathematica 8 with 1000 arbitrary precisionarithmetic has been used in our computations The resultsalongside the test functions are given in Tables 1 and 2while 120574 = 120574
0= 001 [3] From Tables 1 and 2 we can
conclude that our methods work numerically well and aresuccessfully competingwith the existingmethods Indeed thelast columns of these tables show that both numerical andtheoretical aspects support each other
For comparison purposes we consider the followingmethods
Two-Point Method by Kung and Traub [1]
1199090 11990801205740are given suitably
119910119899= 119909119899minus
119891 (119909119899)
119891 [119908119899 119909119899]
119899 = 0 1 2
119909119899+1
= 119910119899minus
119891 (119910119899) 119891 (119908
119899)
[119891 (119908119899) minus 119891 (119910
119899)] 119891 [119909
119899 119910119899]
119896 = 0 1
120574119899+1
= minus
1
1198731015840
3(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(34)
Two-Point Method by Zheng et al [5]
1199090 11990801205740are given suitably
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119899 = 0 1 2
119909119899+1
= 119910119899minus
119891 (119910119899)
119891 [119910119899 119908119899] + 119891 [119910
119899 119909119899 119908119899] (119910119899minus 119909119899)
120574119899+1
= minus
1
1198731015840
3(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(35)
From Tables 1 and 2 it can be seen that our modifiedmethod without memory works truly moreover its withmemory competes the existing methods To sum up Potraand Ptak [6] constructed two-pointmethodwithoutmemory
6 ISRNMathematical Analysis
with convergence order of 3 it is not optimal in the sense ofKung and Traub Cordero et al [7] could make it optimalIn other words they introduce optimal two- and three-pointmethods with order of convergence of 4 and 8 respectivelyThough their methods are optimal they are not derivative-free Freshly Soleymani et al [2] have drawn two pointmethods without memory from Potra and ptak methodOne is derivative-free and the other is not In addition theirderivative method results two steps method by Cordero et al[7] for 119886 = 0 (See (5)) In this work we modified theirderivative-free method at first Then we generalized it tomethod with memory with efficiency index 119864(119901 119899) = 4
13=
18 see more about efficiency index in [7] Therefore atwo-step method with memory can obtain performanceeven better than four-step methods without memory withefficiency index 16
15= 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the University of Venda andthe Islamic Azad University Hamedan Branch
References
[1] H T Kung and J F Traub ldquoOptimal order of one-point andmultipoint iterationrdquo Journal of the Association for ComputingMachinery vol 21 pp 643ndash651 1974
[2] F Soleymani R Sharma X Li and E Tohidi ldquoAn optimizedderivative-free form of the Potra-Ptak methodrdquo Mathematicaland Computer Modelling vol 56 no 5-6 pp 97ndash104 2012
[3] J F Traub Iterative Methods for the Solution of EquationsPrentice Hall New York NY USA 1964
[4] J F Steffensen ldquoRemarks on iterationrdquo Scandinavian ActuarialJournal vol 1933 no 1 pp 64ndash72 1933
[5] Q Zheng J Li and F Huang ldquoAn optimal Steffensen-typefamily for solving nonlinear equationsrdquo Applied Mathematicsand Computation vol 217 no 23 pp 9592ndash9597 2011
[6] F A Potra and V Ptak ldquoNondiscrete introduction and iterativeprocessesrdquo in Research Notes in Mathematics vol 103 PitmanBoston Mass USA 1984
[7] A Cordero J L Hueso E Martınez and J R Torregrosa ldquoNewmodifications of Potra-Ptakrsquos method with optimal fourth andeighth orders of convergencerdquo Journal of Computational andApplied Mathematics vol 234 no 10 pp 2969ndash2976 2010
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRNMathematical Analysis 3
Proof We provide the Taylor expansion of any term involvedin (8) By Taylor expanding around the simple root in the nthiterate we have
119891 (119909119899)
119891 [119909119899 119908119899]
= 1198901
119899minus 1198882(1 + 119891
1015840(120572) 120574) 119890
2
119899
+ (minus1198883(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
+1198882
2(2 + 119891
1015840(120572) 120574 (2 + 119891
1015840(120572) 120574))) 119890
3
119899
+ 119874 (1198904
119899)
(11)
By considering this relation and the first step of (8) we obtain
119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
= 120572 + 1198882(1 + 119891
1015840(120572) 120574) 119890
2
119899
minus (minus1198883(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
+1198882
2(2 + 119891
1015840(120572) 120574 (2 + 119891
1015840(120572) 120574))) 119890
3
119899
+ 119874 (1198904
119899)
(12)
At this time we should expand 119891(119910119899) around the root by
taking into consideration (12) Accordingly we have
119891 (119910119899)
= 1198882(1 + 119891
1015840(120572) 120574) 119891
1015840(120572) 1198902
119899
minus (minus1198883(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574)
+1198882
2(2 + 119891
1015840(120572) 120574 (2 + 119891
1015840(120572) 120574))) 119891
1015840(120572) 1198903
119899+ 1198911015840(120572)
times (1198884(1 + 119891
1015840(120572) 120574) (3 + 119891
1015840(120572) 120574 (3 + 119891
1015840(120572) 120574))
+ 1198883
2(5 + 119891
1015840(120572) 120574 (7 + 119891
1015840(120572) 120574 (4 + 119891
1015840(120572) 120574))) minus 119888
21198883
times (7 + 1198911015840(120572) 120574 (10 + 119891
1015840(120572) 120574 (7 + 2119891
1015840(120572) 120574)))) 119890
4
119899
+ 119874 (1198905
119899)
(13)
Additionally we obtain
119909119899minus
119891 (119909119899) + 119891 (119910
119899)
119891 [119909119899 119908119899]
= 120572 + 1198882
2(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574) 119890
3
119899+ 119874 (119890
4
119899)
(14)
Similarly the same Taylor expansion results in
(
2119891 (119909119899) + 119886119891 (119910
119899)
119891 [119910119899 119908119899]
(
119891 (119910119899)
119891 (119909119899)
)
2
)(1 minus
120574119891 [119909119899 119908119899]
2 + 2120574119891 [119909119899 119908119899]
)
= 1198882
2(1 + 119891
1015840(120572) 120574) (2 + 119891
1015840(120572) 120574) 119890
3
119899
+
1
2
1198882(1 + 119891
1015840(120572) 120574)
2
times ((2 (minus3 + 119886) + (minus2 + 119886) 1198911015840(120572) 120574) 119888
2
2+ 21198883) 1198904
119899
+ 119874 (1198905
119899)
(15)
Using (12)ndash(15) in the last step of (8) provides finally
119909119899+1
minus 120572
= 119909119899minus
119891 (119909119899) + 119891 (119910
119899)
119891 [119909119899 119908119899]
minus (
2119891 (119909119899) + 119886119891 (119910
119899)
119891 [119910119899 119908119899]
(
119891 (119910119899)
119891 (119909119899)
)
2
)
times (1 minus
120574119891 [119909119899 119908119899]
2 + 2120574119891 [119909119899 119908119899]
) minus 120572
= minus
1
2
1198882(1 + 119891
1015840(120572) 120574)
2
times ((2 (minus3 + 119886) + (minus2 + 119886) 1198911015840(120572) 120574) 119888
2
2+ 21198883) 1198904
119899
+ 119874 (1198905
119899)
(16)
which shows that (8) is a derivative-free family of two-stepmethods with optimal convergence rate of 4 This completesthe proof
3 Development and Construction withMemory Family
This section concerns with extension of (8) to a method withmemory since its error equation contains the parameter 120574
which can be approximated in such a way that increase thelocal order of convergence So we set 120574 = 120574
119896as the iteration
proceeds by the formula 120574119896
= minus1119891
1015840
(120572) for 119896 = 1 2 where 119891
1015840
(120572) is an approximation of 1198911015840(120572) We have a methodthrough the following forms of 120574
119896
120574119896= minus
1
119891
1015840
(120572)
= minus
1
1198731015840
3(119909119896)
(17)
4 ISRNMathematical Analysis
where11987310158403(119905) = 119873
3(119905 119909119896 119910119896minus1
119908119896minus1
119909119896minus1
) is Newtonrsquos interpo-latory polynomial of third degree set through four availableapproximations 119909
119896 119910119896minus1
119908119896minus1
and 119909119896minus1
and
1198731015840
3(119909119896) = [
119889
119889119905
1198733 (
119905)]
119905=119909119896
= [
119889
119889119905
[119891 (119909119896) + 119891 [119909
119896 119910119896minus1
] (119905 minus 119909119896)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
] (119905 minus 119909119896) (119905 minus 119910
119896minus1)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
119909119896minus1
]
times (119905 minus 119909119896) (119905 minus 119910
119896minus1) (119905 minus 119908
119896minus1)]]
119905=119909119896
= 119891 [119909119896 119910119896minus1
] + 119891 [119909119896 119910119896minus1
119908119896minus1
] (119909119896minus 119910119896minus1
)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
119909119896minus1
] (119909119896minus 119910119896minus1
) (119909119896minus 119908119896minus1
)
(18)
By using Taylorrsquos expansion of 119891(119909) around the root 120572 wehave
119891 (119909) = 1198911015840(120572) (119890 + 119888
21198902+ 11988831198903+ 11988841198904+ 11988851198905+ sdot sdot sdot ) (19)
where 119890 = 119909 minus 120572 By using (18) and (19) we calculate
1198731015840
3(119909119896) = 1198911015840(120572) [1 + 2119888
2119890119896+ 311988831198902
119896
+ 1198884(119890119896minus1
1198902
119896+ 1198902
119896minus11199101198902
119896
+ 1198902
119896minus11199081198902
119896minus 119890119896minus1
119890119896minus1119910
119890119896
minus 119890119896minus1119910
119890119896minus1119908
119890119896minus 119890119896minus1
119890119896minus1119908
119890119896
+119890119896minus1
119890119896minus1119910
119890119896minus1119908
+ 31198903
119896) + sdot sdot sdot ]
= 1198911015840(120572) [1 + 119888
4119890119896minus1
119890119896minus1119910
119890119896minus1119908
+ 119874 (119890119896)]
(20)
According to this and (17) we find
1 + 1205741198911015840(120572) sim 119888
4119890119896minus1
119890119896minus1119910
119890119896minus1119908
(21)
For general case one can consult [3]In order to obtain the order of convergence of the
family of two-point methods with memory (8) where 120574119896is
calculated using the formula (17) we will use the conceptof the 119877-order of convergence [3] Now we can state thefollowing convergence theorem
Theorem 3 If an initial approximation 1199090is sufficiently close
to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative
scheme (8) is recursively calculated by the forms given in (17)then the 119877-order of convergence is at least 6
Proof Let 119909119896 be a sequence of approximations generated
by an iterative method with memory (IM) If this sequence
converges to the zero 120572 of119891with the119877-order (ger) of IM thenwe write
119890119896+1
sim 119863119896119903
119890119903
119896 119890
119896= 119909119896minus 120572 (22)
where 119863119896119903
tends to the asymptotic error constant 119863119903of IM
when 119896 rarr infin Thus
119890119896+1
sim 119863119896119903
(119863119896minus1119903
119890119903
119896minus1)119903= 119863119896119903
119863119903
119896minus11199031198901199032
119896minus1 (23)
Let 119890119896minus1119910
= 119910119896minus1
minus 120572 119890119896minus1119908
= 119908119896minus1
minus 120572 then we have
119890119896119908
sim (1 + 1205741198961198911015840(120572)) 119890119896
+ 119874 (1198902
119896) (24)
119890119896119910
sim 1198882(1 + 120574
1198961198911015840(120572)) 119890
2
119896+ 119874 (119890
3
119896) (25)
119890119896+1
sim 1198602(1 + 120574
1198961198911015840(120572))
2
1198904
119896+ 119874 (119890
5
119896) (26)
where 1198602= minus(12)119888
2[(2(minus3 + 119886) + (minus2 + 119886)119891
1015840(120572)120574)119888
2
2+ 21198883]
In the sequel we obtain the 119877-order of convergence of family(8) for approach (17) applied to the calculation of 120574
119896
Assume that the iterative sequences 119910119896and 119909119896have the119877-
orders 119901 and 119903 respectively then bearing in mind (22) weobtain
119890119896119910
sim 119863119896119901
119890119901
119896sim 119863119896119901
(119863119896minus1119903
119890119903
119896minus1)119901sim 119863119896119901
119863119901
119896minus1119903119890119903119901
119896minus1 (27)
and then we obtain
119890119896119910
sim 1198882(1 + 120574
1198961198911015840(120572)) 119890
2
119896sim 1198882(1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
) 1198902
119896
sim 11988821198884(119890119896minus1
) (119863119896minus1119901
119890119901
119896minus1) (119863119896minus1119904
119890119904
119896minus1) (119863119896minus1119903
119890119903
119896minus1)2
sim 11988821198884119863119896minus1119901
119863119896minus1119904
1198632
119896minus11199031198902119903+119904+119901+1
119896minus1
(28)
Assume that the iterative sequence119908119896has the 119877-order 119904 then
bearing in mind (22) we obtain
119890119896119908
sim 119863119896119904
119890119904
119896sim 119863119896119904
(119863119896minus1119903
119890119903
119896minus1)119904sim 119863119896119904
119863119904
119896minus1119903119890119903119904
119896minus1 (29)
and then we obtain
119890119896119908
sim (1 + 1205741198961198911015840(120572)) 119890119896
sim (1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
) 119890119896
sim 1198884119890119896minus1
(119863119896minus1119901
119890119901
119896minus1) (119863119896minus1119904
119890119904
119896minus1) (119863119896minus1119903
119890119903
119896minus1)
sim 1198884119863119896minus1119901
119863119896minus1119904
119863119896minus1119903
119890119903+119904+119901+1
119896minus1
(30)
119890119896+1
sim 1198861198964
(1 + 1205741198961198911015840(120572))
2
1198904
119896sim 1198861198964
(1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
)
2
1198904
119896
sim 1198861198964
1198884(119890119896minus1
)2(119863119896minus1119901
119890119901
119896minus1)
2
(119863119896minus1119904
119890119904
119896minus1)2(119863119896minus1119903
119890119903
119896minus1)4
sim 1198861198964
11988841198632
119896minus11199011198632
119896minus11199041198634
119896minus11199031198904119903+2119904+2119901+2
119896minus1
(31)
Combining the exponents of 119890119896minus1
on the right-hand sidesof (27)-(28) (29)-(30) and (23)ndash(31) we form the nonlinearsystem of three equations in 119901 119904 and 119903
119903119901 minus 2119903 minus (119901 + 119904) minus 1 = 0
119903119904 minus 119903 minus (119901 + 119904) minus 1 = 0
1199032minus 4119903 minus 2 (119901 + 119904) minus 2 = 0
(32)
ISRNMathematical Analysis 5
Table 1 1198911(119909) = log(1 + 119909
2) + 1198901199092minus3119909 sin119909 120572 = 0 119909
0= 035 120574
0= 001
Methods |1199091minus 120572| |119909
2minus 120572| |119909
3minus 120572| 119903
119888(20)
Potra-Ptak without memory 6040 (minus3) 5672 (minus10) 8280 (minus37) 3822Equations (8) and (17) with memory 6050 (minus3) 8723 (minus15) 6425 (minus84) 5841
Kung and Traub [1] without memory 5545 (minus3) 4864 (minus9) 3045 (minus33) 3999
Equation (34) with memory 5546 (minus3) 4708 (minus17) 1374 (minus96) 5654
Zheng et al [5] without memory 3541 (minus2) 2974 (minus6) 2127 (minus22) 3930Equation (35) with memory 5647 (minus3) 8272 (minus18) 1022 (minus101) 5658
Table 2 1198912(119909) = prod
5
119894=1(119909 minus 119894) 120572 = 2 119909
0= 16 120574
0= 001
Methods |1199091minus 120572| |119909
2minus 120572| |119909
3minus 120572| 119903
119888(20)
Potra-Ptak without memory 5522 (minus3) 1063 (minus10) 7391 (minus41) 3910Equations (8) and (17) with memory 5623 (minus3) 3822 (minus16) 1170 (minus93) 5888
Kung and Traub [1] (without memory) 6125 (minus3) 2402 (minus9) 5453 (minus35) 4004
Equation (34) with memory 6125 (minus3) 5888 (minus15) 1651 (minus86) 5955
Zheng at al [5] without memory 6021 (minus3) 1525 (minus9) 6077 (minus36) 4003Equation (35) with memory 6021 (minus3) 3251 (minus15) 1274 (minus88) 5985
Nontrivial solution of this system is 119904 = 2 119901 = 3 and119903 = 6 and we conclude that the lower bound of the 119877-orderof the method with memory is 6
Similarly one can prove the following
Theorem 4 If an initial approximation 1199090is sufficiently close
to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative
scheme (5) is recursively calculated by the forms given in (17)then the 119877-order of convergence is at least 52
4 Numerical Examples
To examine practical aspects of the proposedmodified Potra-Ptakrsquos without and with memory we implement it here inaction In other words we demonstrate the convergencebehavior of the method with memory (8) where 120574
119896is
calculated by (17) For comparison purposes we pick upKung and Traub [1] and Zheng et al [5] with and withoutmemories We use these notations The errors |119909
119896minus 120572|
denote approximations to the sought zeros 119860(minusℎ) stands for119860 times 10
minusℎ Moreover 119903119888indicates computational order of
convergence and is computed [2]
119903119888=
log (1003816100381610038161003816119891 (119909119896) 119891 (119909
119896minus1)1003816100381610038161003816)
log (1003816100381610038161003816119891 (119909119896minus1
) 119891 (119909119896minus2
)1003816100381610038161003816)
(33)
The software Mathematica 8 with 1000 arbitrary precisionarithmetic has been used in our computations The resultsalongside the test functions are given in Tables 1 and 2while 120574 = 120574
0= 001 [3] From Tables 1 and 2 we can
conclude that our methods work numerically well and aresuccessfully competingwith the existingmethods Indeed thelast columns of these tables show that both numerical andtheoretical aspects support each other
For comparison purposes we consider the followingmethods
Two-Point Method by Kung and Traub [1]
1199090 11990801205740are given suitably
119910119899= 119909119899minus
119891 (119909119899)
119891 [119908119899 119909119899]
119899 = 0 1 2
119909119899+1
= 119910119899minus
119891 (119910119899) 119891 (119908
119899)
[119891 (119908119899) minus 119891 (119910
119899)] 119891 [119909
119899 119910119899]
119896 = 0 1
120574119899+1
= minus
1
1198731015840
3(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(34)
Two-Point Method by Zheng et al [5]
1199090 11990801205740are given suitably
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119899 = 0 1 2
119909119899+1
= 119910119899minus
119891 (119910119899)
119891 [119910119899 119908119899] + 119891 [119910
119899 119909119899 119908119899] (119910119899minus 119909119899)
120574119899+1
= minus
1
1198731015840
3(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(35)
From Tables 1 and 2 it can be seen that our modifiedmethod without memory works truly moreover its withmemory competes the existing methods To sum up Potraand Ptak [6] constructed two-pointmethodwithoutmemory
6 ISRNMathematical Analysis
with convergence order of 3 it is not optimal in the sense ofKung and Traub Cordero et al [7] could make it optimalIn other words they introduce optimal two- and three-pointmethods with order of convergence of 4 and 8 respectivelyThough their methods are optimal they are not derivative-free Freshly Soleymani et al [2] have drawn two pointmethods without memory from Potra and ptak methodOne is derivative-free and the other is not In addition theirderivative method results two steps method by Cordero et al[7] for 119886 = 0 (See (5)) In this work we modified theirderivative-free method at first Then we generalized it tomethod with memory with efficiency index 119864(119901 119899) = 4
13=
18 see more about efficiency index in [7] Therefore atwo-step method with memory can obtain performanceeven better than four-step methods without memory withefficiency index 16
15= 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the University of Venda andthe Islamic Azad University Hamedan Branch
References
[1] H T Kung and J F Traub ldquoOptimal order of one-point andmultipoint iterationrdquo Journal of the Association for ComputingMachinery vol 21 pp 643ndash651 1974
[2] F Soleymani R Sharma X Li and E Tohidi ldquoAn optimizedderivative-free form of the Potra-Ptak methodrdquo Mathematicaland Computer Modelling vol 56 no 5-6 pp 97ndash104 2012
[3] J F Traub Iterative Methods for the Solution of EquationsPrentice Hall New York NY USA 1964
[4] J F Steffensen ldquoRemarks on iterationrdquo Scandinavian ActuarialJournal vol 1933 no 1 pp 64ndash72 1933
[5] Q Zheng J Li and F Huang ldquoAn optimal Steffensen-typefamily for solving nonlinear equationsrdquo Applied Mathematicsand Computation vol 217 no 23 pp 9592ndash9597 2011
[6] F A Potra and V Ptak ldquoNondiscrete introduction and iterativeprocessesrdquo in Research Notes in Mathematics vol 103 PitmanBoston Mass USA 1984
[7] A Cordero J L Hueso E Martınez and J R Torregrosa ldquoNewmodifications of Potra-Ptakrsquos method with optimal fourth andeighth orders of convergencerdquo Journal of Computational andApplied Mathematics vol 234 no 10 pp 2969ndash2976 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 ISRNMathematical Analysis
where11987310158403(119905) = 119873
3(119905 119909119896 119910119896minus1
119908119896minus1
119909119896minus1
) is Newtonrsquos interpo-latory polynomial of third degree set through four availableapproximations 119909
119896 119910119896minus1
119908119896minus1
and 119909119896minus1
and
1198731015840
3(119909119896) = [
119889
119889119905
1198733 (
119905)]
119905=119909119896
= [
119889
119889119905
[119891 (119909119896) + 119891 [119909
119896 119910119896minus1
] (119905 minus 119909119896)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
] (119905 minus 119909119896) (119905 minus 119910
119896minus1)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
119909119896minus1
]
times (119905 minus 119909119896) (119905 minus 119910
119896minus1) (119905 minus 119908
119896minus1)]]
119905=119909119896
= 119891 [119909119896 119910119896minus1
] + 119891 [119909119896 119910119896minus1
119908119896minus1
] (119909119896minus 119910119896minus1
)
+ 119891 [119909119896 119910119896minus1
119908119896minus1
119909119896minus1
] (119909119896minus 119910119896minus1
) (119909119896minus 119908119896minus1
)
(18)
By using Taylorrsquos expansion of 119891(119909) around the root 120572 wehave
119891 (119909) = 1198911015840(120572) (119890 + 119888
21198902+ 11988831198903+ 11988841198904+ 11988851198905+ sdot sdot sdot ) (19)
where 119890 = 119909 minus 120572 By using (18) and (19) we calculate
1198731015840
3(119909119896) = 1198911015840(120572) [1 + 2119888
2119890119896+ 311988831198902
119896
+ 1198884(119890119896minus1
1198902
119896+ 1198902
119896minus11199101198902
119896
+ 1198902
119896minus11199081198902
119896minus 119890119896minus1
119890119896minus1119910
119890119896
minus 119890119896minus1119910
119890119896minus1119908
119890119896minus 119890119896minus1
119890119896minus1119908
119890119896
+119890119896minus1
119890119896minus1119910
119890119896minus1119908
+ 31198903
119896) + sdot sdot sdot ]
= 1198911015840(120572) [1 + 119888
4119890119896minus1
119890119896minus1119910
119890119896minus1119908
+ 119874 (119890119896)]
(20)
According to this and (17) we find
1 + 1205741198911015840(120572) sim 119888
4119890119896minus1
119890119896minus1119910
119890119896minus1119908
(21)
For general case one can consult [3]In order to obtain the order of convergence of the
family of two-point methods with memory (8) where 120574119896is
calculated using the formula (17) we will use the conceptof the 119877-order of convergence [3] Now we can state thefollowing convergence theorem
Theorem 3 If an initial approximation 1199090is sufficiently close
to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative
scheme (8) is recursively calculated by the forms given in (17)then the 119877-order of convergence is at least 6
Proof Let 119909119896 be a sequence of approximations generated
by an iterative method with memory (IM) If this sequence
converges to the zero 120572 of119891with the119877-order (ger) of IM thenwe write
119890119896+1
sim 119863119896119903
119890119903
119896 119890
119896= 119909119896minus 120572 (22)
where 119863119896119903
tends to the asymptotic error constant 119863119903of IM
when 119896 rarr infin Thus
119890119896+1
sim 119863119896119903
(119863119896minus1119903
119890119903
119896minus1)119903= 119863119896119903
119863119903
119896minus11199031198901199032
119896minus1 (23)
Let 119890119896minus1119910
= 119910119896minus1
minus 120572 119890119896minus1119908
= 119908119896minus1
minus 120572 then we have
119890119896119908
sim (1 + 1205741198961198911015840(120572)) 119890119896
+ 119874 (1198902
119896) (24)
119890119896119910
sim 1198882(1 + 120574
1198961198911015840(120572)) 119890
2
119896+ 119874 (119890
3
119896) (25)
119890119896+1
sim 1198602(1 + 120574
1198961198911015840(120572))
2
1198904
119896+ 119874 (119890
5
119896) (26)
where 1198602= minus(12)119888
2[(2(minus3 + 119886) + (minus2 + 119886)119891
1015840(120572)120574)119888
2
2+ 21198883]
In the sequel we obtain the 119877-order of convergence of family(8) for approach (17) applied to the calculation of 120574
119896
Assume that the iterative sequences 119910119896and 119909119896have the119877-
orders 119901 and 119903 respectively then bearing in mind (22) weobtain
119890119896119910
sim 119863119896119901
119890119901
119896sim 119863119896119901
(119863119896minus1119903
119890119903
119896minus1)119901sim 119863119896119901
119863119901
119896minus1119903119890119903119901
119896minus1 (27)
and then we obtain
119890119896119910
sim 1198882(1 + 120574
1198961198911015840(120572)) 119890
2
119896sim 1198882(1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
) 1198902
119896
sim 11988821198884(119890119896minus1
) (119863119896minus1119901
119890119901
119896minus1) (119863119896minus1119904
119890119904
119896minus1) (119863119896minus1119903
119890119903
119896minus1)2
sim 11988821198884119863119896minus1119901
119863119896minus1119904
1198632
119896minus11199031198902119903+119904+119901+1
119896minus1
(28)
Assume that the iterative sequence119908119896has the 119877-order 119904 then
bearing in mind (22) we obtain
119890119896119908
sim 119863119896119904
119890119904
119896sim 119863119896119904
(119863119896minus1119903
119890119903
119896minus1)119904sim 119863119896119904
119863119904
119896minus1119903119890119903119904
119896minus1 (29)
and then we obtain
119890119896119908
sim (1 + 1205741198961198911015840(120572)) 119890119896
sim (1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
) 119890119896
sim 1198884119890119896minus1
(119863119896minus1119901
119890119901
119896minus1) (119863119896minus1119904
119890119904
119896minus1) (119863119896minus1119903
119890119903
119896minus1)
sim 1198884119863119896minus1119901
119863119896minus1119904
119863119896minus1119903
119890119903+119904+119901+1
119896minus1
(30)
119890119896+1
sim 1198861198964
(1 + 1205741198961198911015840(120572))
2
1198904
119896sim 1198861198964
(1198884119890119896minus1
119890119896minus1119910
119890119896minus1119908
)
2
1198904
119896
sim 1198861198964
1198884(119890119896minus1
)2(119863119896minus1119901
119890119901
119896minus1)
2
(119863119896minus1119904
119890119904
119896minus1)2(119863119896minus1119903
119890119903
119896minus1)4
sim 1198861198964
11988841198632
119896minus11199011198632
119896minus11199041198634
119896minus11199031198904119903+2119904+2119901+2
119896minus1
(31)
Combining the exponents of 119890119896minus1
on the right-hand sidesof (27)-(28) (29)-(30) and (23)ndash(31) we form the nonlinearsystem of three equations in 119901 119904 and 119903
119903119901 minus 2119903 minus (119901 + 119904) minus 1 = 0
119903119904 minus 119903 minus (119901 + 119904) minus 1 = 0
1199032minus 4119903 minus 2 (119901 + 119904) minus 2 = 0
(32)
ISRNMathematical Analysis 5
Table 1 1198911(119909) = log(1 + 119909
2) + 1198901199092minus3119909 sin119909 120572 = 0 119909
0= 035 120574
0= 001
Methods |1199091minus 120572| |119909
2minus 120572| |119909
3minus 120572| 119903
119888(20)
Potra-Ptak without memory 6040 (minus3) 5672 (minus10) 8280 (minus37) 3822Equations (8) and (17) with memory 6050 (minus3) 8723 (minus15) 6425 (minus84) 5841
Kung and Traub [1] without memory 5545 (minus3) 4864 (minus9) 3045 (minus33) 3999
Equation (34) with memory 5546 (minus3) 4708 (minus17) 1374 (minus96) 5654
Zheng et al [5] without memory 3541 (minus2) 2974 (minus6) 2127 (minus22) 3930Equation (35) with memory 5647 (minus3) 8272 (minus18) 1022 (minus101) 5658
Table 2 1198912(119909) = prod
5
119894=1(119909 minus 119894) 120572 = 2 119909
0= 16 120574
0= 001
Methods |1199091minus 120572| |119909
2minus 120572| |119909
3minus 120572| 119903
119888(20)
Potra-Ptak without memory 5522 (minus3) 1063 (minus10) 7391 (minus41) 3910Equations (8) and (17) with memory 5623 (minus3) 3822 (minus16) 1170 (minus93) 5888
Kung and Traub [1] (without memory) 6125 (minus3) 2402 (minus9) 5453 (minus35) 4004
Equation (34) with memory 6125 (minus3) 5888 (minus15) 1651 (minus86) 5955
Zheng at al [5] without memory 6021 (minus3) 1525 (minus9) 6077 (minus36) 4003Equation (35) with memory 6021 (minus3) 3251 (minus15) 1274 (minus88) 5985
Nontrivial solution of this system is 119904 = 2 119901 = 3 and119903 = 6 and we conclude that the lower bound of the 119877-orderof the method with memory is 6
Similarly one can prove the following
Theorem 4 If an initial approximation 1199090is sufficiently close
to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative
scheme (5) is recursively calculated by the forms given in (17)then the 119877-order of convergence is at least 52
4 Numerical Examples
To examine practical aspects of the proposedmodified Potra-Ptakrsquos without and with memory we implement it here inaction In other words we demonstrate the convergencebehavior of the method with memory (8) where 120574
119896is
calculated by (17) For comparison purposes we pick upKung and Traub [1] and Zheng et al [5] with and withoutmemories We use these notations The errors |119909
119896minus 120572|
denote approximations to the sought zeros 119860(minusℎ) stands for119860 times 10
minusℎ Moreover 119903119888indicates computational order of
convergence and is computed [2]
119903119888=
log (1003816100381610038161003816119891 (119909119896) 119891 (119909
119896minus1)1003816100381610038161003816)
log (1003816100381610038161003816119891 (119909119896minus1
) 119891 (119909119896minus2
)1003816100381610038161003816)
(33)
The software Mathematica 8 with 1000 arbitrary precisionarithmetic has been used in our computations The resultsalongside the test functions are given in Tables 1 and 2while 120574 = 120574
0= 001 [3] From Tables 1 and 2 we can
conclude that our methods work numerically well and aresuccessfully competingwith the existingmethods Indeed thelast columns of these tables show that both numerical andtheoretical aspects support each other
For comparison purposes we consider the followingmethods
Two-Point Method by Kung and Traub [1]
1199090 11990801205740are given suitably
119910119899= 119909119899minus
119891 (119909119899)
119891 [119908119899 119909119899]
119899 = 0 1 2
119909119899+1
= 119910119899minus
119891 (119910119899) 119891 (119908
119899)
[119891 (119908119899) minus 119891 (119910
119899)] 119891 [119909
119899 119910119899]
119896 = 0 1
120574119899+1
= minus
1
1198731015840
3(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(34)
Two-Point Method by Zheng et al [5]
1199090 11990801205740are given suitably
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119899 = 0 1 2
119909119899+1
= 119910119899minus
119891 (119910119899)
119891 [119910119899 119908119899] + 119891 [119910
119899 119909119899 119908119899] (119910119899minus 119909119899)
120574119899+1
= minus
1
1198731015840
3(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(35)
From Tables 1 and 2 it can be seen that our modifiedmethod without memory works truly moreover its withmemory competes the existing methods To sum up Potraand Ptak [6] constructed two-pointmethodwithoutmemory
6 ISRNMathematical Analysis
with convergence order of 3 it is not optimal in the sense ofKung and Traub Cordero et al [7] could make it optimalIn other words they introduce optimal two- and three-pointmethods with order of convergence of 4 and 8 respectivelyThough their methods are optimal they are not derivative-free Freshly Soleymani et al [2] have drawn two pointmethods without memory from Potra and ptak methodOne is derivative-free and the other is not In addition theirderivative method results two steps method by Cordero et al[7] for 119886 = 0 (See (5)) In this work we modified theirderivative-free method at first Then we generalized it tomethod with memory with efficiency index 119864(119901 119899) = 4
13=
18 see more about efficiency index in [7] Therefore atwo-step method with memory can obtain performanceeven better than four-step methods without memory withefficiency index 16
15= 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the University of Venda andthe Islamic Azad University Hamedan Branch
References
[1] H T Kung and J F Traub ldquoOptimal order of one-point andmultipoint iterationrdquo Journal of the Association for ComputingMachinery vol 21 pp 643ndash651 1974
[2] F Soleymani R Sharma X Li and E Tohidi ldquoAn optimizedderivative-free form of the Potra-Ptak methodrdquo Mathematicaland Computer Modelling vol 56 no 5-6 pp 97ndash104 2012
[3] J F Traub Iterative Methods for the Solution of EquationsPrentice Hall New York NY USA 1964
[4] J F Steffensen ldquoRemarks on iterationrdquo Scandinavian ActuarialJournal vol 1933 no 1 pp 64ndash72 1933
[5] Q Zheng J Li and F Huang ldquoAn optimal Steffensen-typefamily for solving nonlinear equationsrdquo Applied Mathematicsand Computation vol 217 no 23 pp 9592ndash9597 2011
[6] F A Potra and V Ptak ldquoNondiscrete introduction and iterativeprocessesrdquo in Research Notes in Mathematics vol 103 PitmanBoston Mass USA 1984
[7] A Cordero J L Hueso E Martınez and J R Torregrosa ldquoNewmodifications of Potra-Ptakrsquos method with optimal fourth andeighth orders of convergencerdquo Journal of Computational andApplied Mathematics vol 234 no 10 pp 2969ndash2976 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRNMathematical Analysis 5
Table 1 1198911(119909) = log(1 + 119909
2) + 1198901199092minus3119909 sin119909 120572 = 0 119909
0= 035 120574
0= 001
Methods |1199091minus 120572| |119909
2minus 120572| |119909
3minus 120572| 119903
119888(20)
Potra-Ptak without memory 6040 (minus3) 5672 (minus10) 8280 (minus37) 3822Equations (8) and (17) with memory 6050 (minus3) 8723 (minus15) 6425 (minus84) 5841
Kung and Traub [1] without memory 5545 (minus3) 4864 (minus9) 3045 (minus33) 3999
Equation (34) with memory 5546 (minus3) 4708 (minus17) 1374 (minus96) 5654
Zheng et al [5] without memory 3541 (minus2) 2974 (minus6) 2127 (minus22) 3930Equation (35) with memory 5647 (minus3) 8272 (minus18) 1022 (minus101) 5658
Table 2 1198912(119909) = prod
5
119894=1(119909 minus 119894) 120572 = 2 119909
0= 16 120574
0= 001
Methods |1199091minus 120572| |119909
2minus 120572| |119909
3minus 120572| 119903
119888(20)
Potra-Ptak without memory 5522 (minus3) 1063 (minus10) 7391 (minus41) 3910Equations (8) and (17) with memory 5623 (minus3) 3822 (minus16) 1170 (minus93) 5888
Kung and Traub [1] (without memory) 6125 (minus3) 2402 (minus9) 5453 (minus35) 4004
Equation (34) with memory 6125 (minus3) 5888 (minus15) 1651 (minus86) 5955
Zheng at al [5] without memory 6021 (minus3) 1525 (minus9) 6077 (minus36) 4003Equation (35) with memory 6021 (minus3) 3251 (minus15) 1274 (minus88) 5985
Nontrivial solution of this system is 119904 = 2 119901 = 3 and119903 = 6 and we conclude that the lower bound of the 119877-orderof the method with memory is 6
Similarly one can prove the following
Theorem 4 If an initial approximation 1199090is sufficiently close
to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative
scheme (5) is recursively calculated by the forms given in (17)then the 119877-order of convergence is at least 52
4 Numerical Examples
To examine practical aspects of the proposedmodified Potra-Ptakrsquos without and with memory we implement it here inaction In other words we demonstrate the convergencebehavior of the method with memory (8) where 120574
119896is
calculated by (17) For comparison purposes we pick upKung and Traub [1] and Zheng et al [5] with and withoutmemories We use these notations The errors |119909
119896minus 120572|
denote approximations to the sought zeros 119860(minusℎ) stands for119860 times 10
minusℎ Moreover 119903119888indicates computational order of
convergence and is computed [2]
119903119888=
log (1003816100381610038161003816119891 (119909119896) 119891 (119909
119896minus1)1003816100381610038161003816)
log (1003816100381610038161003816119891 (119909119896minus1
) 119891 (119909119896minus2
)1003816100381610038161003816)
(33)
The software Mathematica 8 with 1000 arbitrary precisionarithmetic has been used in our computations The resultsalongside the test functions are given in Tables 1 and 2while 120574 = 120574
0= 001 [3] From Tables 1 and 2 we can
conclude that our methods work numerically well and aresuccessfully competingwith the existingmethods Indeed thelast columns of these tables show that both numerical andtheoretical aspects support each other
For comparison purposes we consider the followingmethods
Two-Point Method by Kung and Traub [1]
1199090 11990801205740are given suitably
119910119899= 119909119899minus
119891 (119909119899)
119891 [119908119899 119909119899]
119899 = 0 1 2
119909119899+1
= 119910119899minus
119891 (119910119899) 119891 (119908
119899)
[119891 (119908119899) minus 119891 (119910
119899)] 119891 [119909
119899 119910119899]
119896 = 0 1
120574119899+1
= minus
1
1198731015840
3(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(34)
Two-Point Method by Zheng et al [5]
1199090 11990801205740are given suitably
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119899 = 0 1 2
119909119899+1
= 119910119899minus
119891 (119910119899)
119891 [119910119899 119908119899] + 119891 [119910
119899 119909119899 119908119899] (119910119899minus 119909119899)
120574119899+1
= minus
1
1198731015840
3(119909119899)
119908119899+1
= 119909119899+1
+ 120574119899+1
119891 (119909119899+1
)
(35)
From Tables 1 and 2 it can be seen that our modifiedmethod without memory works truly moreover its withmemory competes the existing methods To sum up Potraand Ptak [6] constructed two-pointmethodwithoutmemory
6 ISRNMathematical Analysis
with convergence order of 3 it is not optimal in the sense ofKung and Traub Cordero et al [7] could make it optimalIn other words they introduce optimal two- and three-pointmethods with order of convergence of 4 and 8 respectivelyThough their methods are optimal they are not derivative-free Freshly Soleymani et al [2] have drawn two pointmethods without memory from Potra and ptak methodOne is derivative-free and the other is not In addition theirderivative method results two steps method by Cordero et al[7] for 119886 = 0 (See (5)) In this work we modified theirderivative-free method at first Then we generalized it tomethod with memory with efficiency index 119864(119901 119899) = 4
13=
18 see more about efficiency index in [7] Therefore atwo-step method with memory can obtain performanceeven better than four-step methods without memory withefficiency index 16
15= 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the University of Venda andthe Islamic Azad University Hamedan Branch
References
[1] H T Kung and J F Traub ldquoOptimal order of one-point andmultipoint iterationrdquo Journal of the Association for ComputingMachinery vol 21 pp 643ndash651 1974
[2] F Soleymani R Sharma X Li and E Tohidi ldquoAn optimizedderivative-free form of the Potra-Ptak methodrdquo Mathematicaland Computer Modelling vol 56 no 5-6 pp 97ndash104 2012
[3] J F Traub Iterative Methods for the Solution of EquationsPrentice Hall New York NY USA 1964
[4] J F Steffensen ldquoRemarks on iterationrdquo Scandinavian ActuarialJournal vol 1933 no 1 pp 64ndash72 1933
[5] Q Zheng J Li and F Huang ldquoAn optimal Steffensen-typefamily for solving nonlinear equationsrdquo Applied Mathematicsand Computation vol 217 no 23 pp 9592ndash9597 2011
[6] F A Potra and V Ptak ldquoNondiscrete introduction and iterativeprocessesrdquo in Research Notes in Mathematics vol 103 PitmanBoston Mass USA 1984
[7] A Cordero J L Hueso E Martınez and J R Torregrosa ldquoNewmodifications of Potra-Ptakrsquos method with optimal fourth andeighth orders of convergencerdquo Journal of Computational andApplied Mathematics vol 234 no 10 pp 2969ndash2976 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 ISRNMathematical Analysis
with convergence order of 3 it is not optimal in the sense ofKung and Traub Cordero et al [7] could make it optimalIn other words they introduce optimal two- and three-pointmethods with order of convergence of 4 and 8 respectivelyThough their methods are optimal they are not derivative-free Freshly Soleymani et al [2] have drawn two pointmethods without memory from Potra and ptak methodOne is derivative-free and the other is not In addition theirderivative method results two steps method by Cordero et al[7] for 119886 = 0 (See (5)) In this work we modified theirderivative-free method at first Then we generalized it tomethod with memory with efficiency index 119864(119901 119899) = 4
13=
18 see more about efficiency index in [7] Therefore atwo-step method with memory can obtain performanceeven better than four-step methods without memory withefficiency index 16
15= 17
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the University of Venda andthe Islamic Azad University Hamedan Branch
References
[1] H T Kung and J F Traub ldquoOptimal order of one-point andmultipoint iterationrdquo Journal of the Association for ComputingMachinery vol 21 pp 643ndash651 1974
[2] F Soleymani R Sharma X Li and E Tohidi ldquoAn optimizedderivative-free form of the Potra-Ptak methodrdquo Mathematicaland Computer Modelling vol 56 no 5-6 pp 97ndash104 2012
[3] J F Traub Iterative Methods for the Solution of EquationsPrentice Hall New York NY USA 1964
[4] J F Steffensen ldquoRemarks on iterationrdquo Scandinavian ActuarialJournal vol 1933 no 1 pp 64ndash72 1933
[5] Q Zheng J Li and F Huang ldquoAn optimal Steffensen-typefamily for solving nonlinear equationsrdquo Applied Mathematicsand Computation vol 217 no 23 pp 9592ndash9597 2011
[6] F A Potra and V Ptak ldquoNondiscrete introduction and iterativeprocessesrdquo in Research Notes in Mathematics vol 103 PitmanBoston Mass USA 1984
[7] A Cordero J L Hueso E Martınez and J R Torregrosa ldquoNewmodifications of Potra-Ptakrsquos method with optimal fourth andeighth orders of convergencerdquo Journal of Computational andApplied Mathematics vol 234 no 10 pp 2969ndash2976 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
