cuasi-steffensen 3

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Research ArticleAn Efficient Family of Traub-Steffensen-Type Methods forSolving Systems of Nonlinear Equations

Janak Raj Sharma1 and Puneet Gupta2

Department o Mathematics, Sant Longowal Institute o Engineering and echnology, Longowal, Punjab , India Department o Mathematics, Government Ranbir College, Sangrur, Punjab , India

Correspondence should be addressed to Janak Raj Sharma; [email protected]

Received February ; Accepted June ; Published July

Academic Editor: Zhangxin Chen

Copyright J. R. Sharma and P. Gupta. Tis is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Based on raub-Steffensen method, we present a derivative ree three-step amily o sixth-order methods or solving systemso nonlinear equations. Te local convergence order o the amily is determined using rst-order divided difference operatoror unctions o several variables and direct computation by aylors expansion. Computational efficiency is discussed, and acomparison between the efficiencies o the proposed techniques with the existing ones is made. Numerical tests are perormedto compare the methods o the proposed amily with the existing methods and to conrm the theoretical results. It is shown thatthe new amily is especially efficient in solving large systems.

1. Introduction

Te problem o nding solution o the system o nonlinearequations() = 0, where : , is an openconvex domain in, by iterative methods is an importantand challenging task in numerical analysis and many appliedscientic branches. One o the basic procedures or solvingnonlinear equations is the quadratically convergent Newtonmethod (see [,]):

(+1) = () ()1 () , = 0,1,2 , . . . , ()where[()]1 is the inverse o the rst Frechet derivative()o the unction().

In many practical situations, it is preerable to avoid thecalculation o derivative()o the unction(). In suchsituations, it is preerable to use only the computed values o() and to approximate() by employing the values o() at suitable points. For example, a basic derivative reeiterative method is the raub-Steffensen method [], whichalso converges quadratically and ollows the scheme

(+1)

= 1,2

()

= ()

()

, ()

; 1

()

, ()

where[(), (); ]1 is the inverse o the rst-order divideddifference[(), (); ]o and() = () + (());isan arbitrary nonzero constant. Troughout this paper,,is used to denote theth iteration unction o convergenceorder. For = 1, the scheme () reduces to the well-knownSteffensen method [].

In recent years, many derivative ree higher-order meth-ods o great efficiency are developed or solving scalarequation() = 0; see [] and the reerences therein. Forsystems o nonlinear equations, however, the construction oefficient higher-order derivative ree methods is a difficulttask and thereore not many such methods can be ound inthe literature. Recently, based on Steffensens scheme, thatis, when = 1 in (), a amily o seventh-order methodshas been proposed in []. Some important members o thisamily, as shown in [], are given as ollows:

() = 1,2() ,() = 1,4(), ()

= ()

()

, ()

; + ()

, ()

;

Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2014, Article ID 152187, 11 pageshttp://dx.doi.org/10.1155/2014/152187


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Advances in Numerical Analysis

(), ();1 () ,(+1) = 1,7(), (), ()

= () (), (); + (), ();

(), ();1

() ,()

() = 1,2() ,() = 2,4(), ()

= () (), (); 1 (), (); (), ();

+ (), ();(), (); 1() ,(+1) = 2,7(), (), ()

= () (), (); + (), (); (), ();1 () .

()

Periteration, both methods use ourunctions, ve rst-orderdivided differences, and three matrix inversions. Te notableeature o these algorithms is their simple design whichmakes them easily implemented to solve systems o nonlinear

equations. Here, the ourth-order method1,4((), ()) isthe generalization o the method proposed by Ren et al. in[] and2,4((), ())is the generalization o the method byLiu et al. [].

In this paper, ouraim is to develop derivative ree iterativemethods that maysatisy thebasic requirements o generatingquality numerical algorithms, that is, the algorithms with(i) high convergence speed, (ii) minimum computationalcost, and (iii) simple design. In this way, we here propose aderivative ree amily o sixth-order methods. Te scheme iscomposed o three steps o which the rst two steps consisto any derivative ree ourth-order method with the baseas the raub-Steffensen iteration () whereas the third step

is weighted raub-Steffensen iteration. Te algorithm o thepresent contribution is as simple as the methods () and() but with an additional advantage that it possesses highcomputational efficiency, especially when applied or solvinglarge systems o equations.

Te rest o the paper is summarized as ollows. Te sixth-order scheme with its convergence analysis is presented inSection . InSection , the computational efficiency o newmethods is discussed and is compared with the methodswhich lie in the same category. Various numerical examplesare considered inSection to show the consistent conver-gence behavior o the methods and to veriy the theoreticalresults.Section contains the concluding remarks.

2. The Method and Its Convergence

Based on the above considerations o a quality numericalalgorithm, we begin with the ollowing three-step iterationscheme:

()

= 1,2()

,() = 4(), () ,(+1) = 6(), (), ()

= () + (), (); 1

(), (); + (), (); (), (); 1() ,

()

where4((), ())denotes any derivative ree ourth-orderscheme and,,are some parameters to be determined.

In order to nd the convergence order o scheme (),we rst dene divided difference operator or multivariableunction(see []). Te divided difference operator oisa mapping[, ; ] : ()dened by

[ + ,; ]= 10( + ) , , . ()

Expanding(+)in aylor series at the pointand thenintegrating, we have

[ + ,; ]= 10( + )

= ()+12() +16() 2 + 3 ,

()

where = (, , . . . ,), .Let() = () . Assuming that = [()]1 exists

and then developing

(()

)and its rst three derivatives in a

neighborhood o, we have () = () () + 2()2 + 3()3

+ 4()4 + 5()5 + ()6 ,()

() = () + 22() + 33()2+ 44()3

+ 55()4 + ()5 ,()


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() = () 22+ 63() + 124()2

+ 205()3 + ()4 , ()

() = () 63+ 244() + 605()2

+ ()3 , ()

where = (1/!)()() (, )and(()) = ((),(), . . . , ()), () . are symmetric operators that areused later on.

We can now analyze behavior o the scheme () throughthe ollowing theorem.

Teorem . Let the unction : be sufficientlydifferentiable in an open neighborhood o its zero and

4

(()

, ()

)is a ourth-order iteration unction which satises

() = () = 0()4 + ()5 , ()

where0 4(, ) and() = () . I an initialapproximation (0) is sufficiently close to, then the local ordero convergence o method () is at least provided = 3, = 1, and = 1.Proo. Let()= () = () + (()) . Ten, using

(), it ollows that

()= + () () + ()2()2

+ () 3()3 + ()4 . ()

Employing () or + = (), = (), and = () ()and then using ()(), we write

()

, ()

; =

() + 2()+ ()+ 3()2 + ()2 + ()()+ 4()3 + ()3 + ()2()

+ ()()2+ ()4 .

()

Expanding in the ormal power developments, the inverse opreceding operator can be written as

(), (); 1

= 2()+ () + 22 3 ()2 + ()2+ 222 3 ()() 32 32 23+ 4 ()3 + ()3 332 232 223+ 4

()2

() + ()()2

+ ()4.

()

Using ()and () to the required terms in the rst step o (),we nd that

()= () = 2()() 22 3

()2() + ()()2 + ()4 .()

Equation () or + = (), = (), and = () ()yields

(), (); = () + 2()+ ()+ 3()() + ()2

+ 4()

3

+ ()

4

.

()

Similarly, substituting+ = (), = (),and = ()()in (), we obtain

(), (); = () + 2()+ ()+ 3()()+ ()2+ 4()w3 + ()4 .

()


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Ten, using (), (), and () to the required terms, we ndthat

+ (), (); 1 (), (); + (), ();

=( + + ) 2

()

+ ()

+ 22 3()2 + ()2+( + ) 22 3 ()() +( + )2()+ ()3 .

()

Postmultiplying () by[(), (); ]1 and simpliying

= + ()

, ()

; 1

(), (); + (), (); (), (); 1

= ( + + ) ( + 2 + )2()( + + 2) 2()+ ( + 3 + ) 22( + 2 + ) 3 ()2

+ ( + + 3) 2

2( + + 2) 3 ()

2

+( + 2 ( + )) 222 3 ()()+( + )2()+ ()3,

()

aylor series o(())aboutyields

() = () ()+ 2()2 + ()3 . ()

Ten, using () and () in the third step o (), we obtainthe error equation as

(+1)= () () ()+ ()2= ( + + 1) ()+( + 2 + )2()()+( + + 2)2()()

( + 3 + ) 22

( + 2 + ) 3

()

2

()

( + + 3)22( + + 2)3 ()()2 ( + 2 ( + )) 222 3 ()()()

( + ) 2()()+ ()7

.()

In order to nd,, and, it will be sufficient to equate theactors + + 1, + 2 + , and + + 2to0and thensolving the resulting system o equations

+ + = 1, + 2 + = 0, + + 2 = 0, ()we obtain = 3, = 1, and = 1.

Tus, or this set o values, the above error equationreduces to

(+1) = 22()+ ()2()+ 22()() 3()()() + ()7

= 0 + ()622 3+2 ()222 ()6 + ()7 ,

()

which shows the sixth order o convergence and hence theresult ollows.

Finally, the sixth-order amily o methods is expressed by

() = 1,2() ,

() = 4(), () ,(+1) = 6(), (), ()

= () M (), (), ()() ,

()

wherein

M (), (), () = 3 (), (); 1 (), (); + (), ();(), (); 1.

()

Tus, the scheme () denes a new three-step amily oderivative ree sixth-order methods with the rst two steps asany ourth-order scheme whose base is the raub-Steffensenmethod (). Some simple members o this amily are asollows.

Method-I. Te rst method, which is denoted by1,6, isgivenby

() = 1,2() ,() = 1,4(), () ,

(+1)

= ()

M

()

, ()

, ()

()

,

()


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where1,4((), ()) is the ourth-order method as givenin the ormula (). It is clear that this ormula uses ourunctions, three rst-order divided differences, and twomatrix inversions per iteration.

Method-II. Te second method, that we denote by2,6, isgiven as

() = 1,2() ,() = 2,4(), () ,(+1) = () M (), (), ()() ,

()

where2,4((), ()) is the ourth-order method shown in(). Tis method also requires the same evaluations as in theabove method.

3. Computational Efficiency

Here, we estimate the computational efficiency o the pro-posed methods and compare it with the existing methods.o do this, we will make use o efficiency index, accordingto which the efficiency o an iterative method is given by

= 1/, where is the order o convergence andis the computational cost per iteration. For a system ononlinear equations inunknowns, the computational costper iteration is given by (see [])

, , = 0() + (, ) . ()Here,0() denotes the number o evaluations o scalarunctions used in the evaluation o

and

[,;], and

(,)denotes the number o products needed per iteration. Tedivided difference[,; ] o is an matrix withelements given as (see [,])

,;= 1, . . . , 1, , +1, . . . , 1, . . . , 1, , +1, . . . , + 1, . . . , 1, , +1, . . . , 1, . . . , 1, , +1, . . . ,

2

1

, 1 , .

()

In order to express the value o(,,) in terms o products,a ratio > 0 between products and evaluations o scalarunctions and a ratio 1between products and quotientsare required.

o compute in any iterative unction, we evaluate scalar unctions(1, 2, . . . , ), and i we compute adivided difference[,; ], then we evaluate2( 1)scalarunctions, where() and() are computed separately.We must add2 quotients rom any divided difference,2products or multiplication o a matrix with a vector or oa matrix by a scalar, and products or multiplication oa vector by a scalar. In order to compute an inverse linear

operator, we solve a linear system, where we have( 1)(2 1)/6 products and( 1)/2 quotients in the LUdecomposition and( 1)products andquotients in theresolution o two triangular linear systems.

Te computational efficiency o the present sixth-ordermethods1,6and2,6is compared with the existing ourth-order methods1,4 and2,4 and with the seventh-ordermethods1,7 and2,7. In addition, we also compare thepresent methods with each other. Let us denote efficiencyindices o, by, and computational cost by,. Ten,taking into account the above and previous considerations,we have

1,4= 62 3 +322 + 3 5 + 3 (4 + 1) ,1,4= 41/1,4 ,

()

2,4= 62 3 +322 + 9 8 + 3 (4 + 2) ,

2,4= 41/2,4 , ()

1,6= 62 2 +322 + 12 8 + 3 (4 + 3) ,1,6= 61/1,6 ,

()

2,6= 62 2 +322 + 18 11 + 12 ( + 1) ,2,6= 61/2,6 ,

()

1,7= 102

6 +222

+ 3 5 + (13 + 3) ,1,7= 71/1,7 ,()

2,7= 102 6 +222 + 7 7 + (13 + 5) ,2,7= 71/2,7 .

()

.. Comparison between Efficiencies. o compare the compu-tational efficiencies o the iterative methods, say, against,, we consider the ratio

,;,= log,log, =,log ,log . ()

It is clear that i,;,> 1, the iterative method,is moreefficient than,.1,4versus1,6Case. For this case, the ratio () is given by

1,6;1,4= log 6log 422 + 18 + 12 + 3 9 + 3 522 + 18 + 12 + 12 6 + 9 8,

()


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which shows that 1,6;1,4> 1 or > 0, 1, and 9. Tus,we have1,6> 1,4or > 0, 1, and 9.2,4 versus1,6 Case. In this case, the ratio () takes theollowing orm

1,6;2,4= log

6log 422

+ 18 + 12 + 9 9 + 6 822 + 18 + 12 + 12 6 + 9 8 . ()In this case, it is easy to prove that1,6;2,4> 1or > 0, 1,and 2, which implies that1,6> 2,4.1,4versus2,6Case.Te ratio () yields

2,6;1,4= log 6log 4 22 + 18 + 12 + 3 9 + 3 522 + 18 + 12 + 18 6 + 12 11 ,

()

which shows that2,6;1,4> 1or > 0, 1, and 19.Tus, we conclude that

2,6

> 1,4 or

> 0,

1, and

19.2,4versus2,6Case.In this case, the ratio () is given by2,6;2,4= log 6log 4

22 + 18 + 12 + 9 9 + 6 822 + 18 + 12 + 18 6 + 12 11 .

()

With the same range o,as in the previous case and 6,the ratio2,6;2,4> 1, which implies that2,6> 2,4.1,6 versus2,6 Case. In this case, it is enough to comparethe corresponding values o1,6and2,6rom () and ().Tus, we nd that1,6> 2,6or > 0, 1, and 2.1,6versus1,7Case. In this case, the ratio () is given by1,6;1,7= log 6log 7

322 + 20 + 13 + 3 12 + 3 5222 + 18 + 12 + 12 6 + 9 8 .

()

It is easy to show that1,6;1,7> 1or > 0, 1, and 4,which implies that1,6> 1,7or this range o values o theparameters(,, ).1,6versus2,7Case. Te ratio () is given by1,6;2,7= log 6log 7 32

2 + 20 + 13 + 7 12 + 5 7222 + 18 + 12 + 12 6 + 9 8 .()

With the same range o, as in the previous cases and 2,we have1,6;2,7> 1, which implies that1,6> 2,7.2,6versus1,7Case.Te ratio () yields2,6;1,7= log 6log 7

322 + 20 + 13 + 3 12 + 3 5222 + 18 + 12 + 18 6 + 12 11,

()

which shows that2,6;1,7> 1or > 0, 1, and 11, soit ollows that2,6> 1,7.2,6versus2,7Case.For this case, the ratio () is given by

2,6;2,7

= log 6log 7

322 + 20 + 13 + 7 12 + 5 7222 + 18 + 12 + 18 6 + 12 11 .()

In this case, also it is not difficult to prove that2,6;2,7> 1or > 0, 1, and 5, which implies that2,6> 2,7.We summarizethe above results in theollowing theorem.

Teorem . For > 0and 1, we have the ollowing:(i)1,6> 1,4or 9;

(ii)2,6> 1,4or 19;(iii)2,6> 2,4or 6;(iv)

1,6

> 1,7or

4;

(v)2,6> 1,7or 11;(vi)2,6> 2,7or 5;

(vii){1,6> 2,4, 1,6> 2,6, 1,6> 2,7}or 2.Otherwise, the comparison depends on,, and.4. Numerical Results

In this section, some numerical problems are consideredto illustrate the convergence behavior and computationalefficiency o the proposed methods. Te perormance is com-pared with the existing methods

1,4,

2,4,

1,7, and

2,7. All

computations areperormed usingthe programming packageMathematica [] using multiple-precision arithmetic with digits. For every method, we analyze the number oiterations() needed to converge to the solution such that(+1) () + (()) < 10200. In numerical results,we also include the CPU time utilized in the execution oprogram which is computed by the MathematicacommandimeUsed[]. In order to veriy the theoretical order oconvergence, we calculate the computational order o conver-gence()using the ormula

= log () / (1)

log

(1)

/ (2)

, ()

(see []) taking into consideration the last three approxima-tions in the iterative process.

o connect the analysis o computational efficiency withnumerical examples, the denition o the computational cost() is applied, according to which an estimation o theactor is claimed. For this, we express the cost o the evaluationo the elementary unctions in terms o products, whichdepends on the computer, the sofware, and the arithmeticsused (see [,]). Inable , the elapsed CPU time (mea-sured in milliseconds) in the computation o elementaryunctions and an estimation o the cost o the elementaryunctions in product units are displayed. Te programs are


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: CPU time and estimation o computational cost o the elementary unctions, where =3 1and =5.Functions / ln sin cos cos1 tan1 CPU time . . . . . . . . . .

Cost . .

perormed in the processor, Intel (R) Core (M) i-MCPU @ . GHz (-bit Machine) Microsof Windows Home Basic , and are compiled byMathematica . usingmultiple-precisionarithmetic.Itcan be observed rom able that, or this hardware and the sofware, the computationalcost o quotient with respect to product is = 2.8.

Te present methods1,6and2,6are tested by using thevalues0.01,0.01, and0.5or the parameter. Te ollowingproblems are chosen or numerical tests.

Problem . Considering the system o two equations,

21

2

+ 1 = 0,1cos 22 = 0. ()

In this problem,(,) = (2,52) are the values used in()() or calculating computational costs and efficiencyindices o the considered methods. Te initial approximation

chosen is(0) = {0.25, 0.5} and the solution is = {0, 1}.Problem . Consider the system o three equations:

101+sin 1+ 2 1 = 0,82cos2 3 2 1 = 0,

123+sin3 1 = 0,

()

with initial value(0) = {0.8,0.5, 0.125} towards the solution ={0.0689783491726666 . . . , 0.2464424186091830 . . . ,

0.0769289119875370...}.()

For this problem,(,) = (3, 103.33)are used in ()() tocalculate computational costs and efficiency indices.

Problem . Next, consider the ollowing boundary valueproblem (see []):

+ 3 = 0, (0)= 0, (1)= 1. ()Assume the ollowing partitioning o the interval[0, 1]:

0= 0 < 1< 2< < 1< = 1,+1= + , =1.

()

Let us dene0= (0) = 0, 1= (1) , . . . , 1=(1), = () = 1. I we discretize the problem byusing the numerical ormula or second derivative,

=1 2+ +1

2

, = 1, 2,3, . . . , 1, ()

we obtain a system o 1 nonlinear equations in 1variables:

1 2+ +1+ 23= 0, = 1, 2, 3, . . . , 1.()

In particular, we solve this problem or = 5so that = 4by selecting(0) = {0.5, 0.5, 0.5, 0.5} as the initial value. Tesolution o this problem is

={0.21054188948074775...,0.42071046387616439...,0.62790045371805633...,0.82518822786851363...}

()

and concrete values o the parameters are(, ) = (4,4).Problem . Consider the system o feen equations (see[]):

15=1, =

= 0, 1 15. ()In this problem, the concrete values o the parameters(, )are

(15, 81). Te initial approximation assumed is

(0)

={1 ,1 , . . . , 1 } and the solution o this problem is ={0.066812203179582582...,0.066812203179582582...,

. . . ,0.066812203179582582...}.

()

Problem . Consider the system o fy equations:

2 +1 1 = 0, ( = 1, 2, . . . , 49)

2

501 1 = 0. ()

In this problem,(,) = (50,2) are the values used in()() or calculating computational costs and efficiency

indices. Te initial approximation assumed is(0) = {1.5,1.5,1.5,...,1.5} or obtaining the solution = {1,1 ,1 , . . . , 1 }.Problem . Lastly, consider the nonlinear and nondifferen-tiable integral equation o mixed Hammerstein type (see[]):

()= 1 +1

21

0 (, ) | ()|+ ()2, ()


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: Comparison o the perormance o methods.

Methods , , e-timeProblem

1,4

7 4.000013 992.4 1.0013979 0.245182,4 7 4.000004 1004.0 1.0013817 0.247541,6( = .01) 5 5.999999 1117.6 1.0016045 0.197091,6( = .01) 5 5.999999 1117.6 1.0016045 0.198541,6( = .5) 5 6.000000 1117.6 1.0016045 0.197732,6( = .01) 5 6.000000 1129.2 1.0015880 0.214272,6( = .01) 5 6.000000 1129.2 1.0015880 0.215642,6( = .5) 5 6.000001 1129.2 1.0015880 0.218451,7 5 7.120115 1546.2 1.0012593 0.248912,7 5 7.078494 1557.8 1.0012499 0.25246Problem

1,4 5 4.000000 4781.05 1.0002899 1.31891

2,4

5 4.000000 4804.45 1.0002886 1.323731,6( = .01) 4 6.000000 5131.84 1.0003492 1.107551,6( = .01) 4 6.000000 5131.84 1.0003492 1.110751,6( = .5) 4 5.999930 5131.84 1.0003492 1.117522,6( = .01) 4 6.000000 5155.24 1.0003476 1.111912,6( = .01) 4 6.000000 5155.24 1.0003476 1.116182,6( = .5) 4 5.999906 5155.24 1.0003476 1.114551,7 4 7.000000 7649.20 1.0002544 1.694732,7 4 7.000000 7672.60 1.0002537 1.70182Problem

1,4 5 4.000123 578.4 1.0023996 0.179032,4 5 4.000035 617.6 1.0022472 0.18164

1,6

( = .01) 4 5.999970 660.8 1.0027152 0.134731,6( = .01) 4 5.999968 660.8 1.0027152 0.129001,6( = .5) 4 5.999887 660.8 1.0027152 0.126272,6( = .01) 4 5.999957 700.0 1.0025629 0.141822,6( = .01) 4 5.999955 700.0 1.0025629 0.140822,6( = .5) 4 5.999870 700.0 1.0025629 0.144641,7 4 6.999331 930.0 1.0020946 0.193272,7 4 6.999216 969.2 1.0020098 0.19891Problem

1,4 5 4.000000 110717.0 1.00001252 6.676812,4 5 4.000000 111194.0 1.00001246 7.691751,6( = .01) 4 6.000000 112676.0 1.00001590 6.16291

1,6

( = .01) 4 6.000000 112676.0 1.00001590 5.741891,6( = .5) 4 6.000000 112676.0 1.00001590 6.034522,6( = .01) 4 6.000000 113153.0 1.00001584 6.539692,6( = .01) 4 6.000000 113153.0 1.00001584 6.625722,6( = .5) 4 6.000000 113153.0 1.00001584 6.723121,7 4 7.000000 182793.0 1.00001065 7.292282,7 4 7.000000 183270.0 1.00001062 7.95691Problem

1,4 7 4.000000 143590.0 1.0000097 8.124362,4 7 4.000000 148680.0 1.0000093 8.322871,6( = .01) 5 6.000000 151420.0 1.0000118 6.08475


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: Continued.

Methods , , e-time1,6( = .01) 5 6.000000 151420.0 1.0000118 5.991521,6( = .5) 5 6.000000 151420.0 1.0000118 5.897342,6( = .01) 5 6.000000 156510.0 1.0000114 6.618332,6( = .01) 5 6.000000 156510.0 1.0000114 6.458342,6( = .5) 5 6.000000 156510.0 1.0000114 6.665831,7 5 7.000000 223735.0 1.0000087 7.845672,7 5 7.000000 228825.0 1.0000085 7.89437

Problem

1,4 5 4.000000 5091.200 1.0002723 2.029452,4 5 4.000000 5233.600 1.0002649 2.135731,6( = .01) 4 6.000000 5408.000 1.0003314 1.963381,6( = .01) 4 6.000000 5408.000 1.0003314 1.832271,6( = .5) 4 6.000000 5408.000 1.0003314 1.822362,6( = .01) 4 6.000000 5550.400 1.0003229 1.796732,6( = .01) 4 6.000000 5550.400 1.0003229 1.80255

2,6( = .5) 4 6.000000 5550.400 1.0003229 1.906091,7 4 8.000000 8298.400 1.0002345 2.342912,7 4 8.000000 8440.800 1.0002306 2.48182

where [0, 1];, [0,1]and the kernelis given asollows:

(, )= (1 ) , , (1 ) , . ()

We transorm the above equation into a nite-dimensionalproblem by using Gauss-Legendre quadrature ormula givenas

10 ()

=1

, ()

where the abscissasand the weightsare determined or = 8by Gauss-Legendre quadrature ormula. Denoting theapproximation o

(

)by

( = 1, 2, . . . , 8), we obtain the

system o nonlinear equations:

2 2 8=1

+ 2 = 0, ()

where

=

1 i ,

1

i

< .

()

In this problem, (,) = (8,10) are the values usedin ()() or calculating computational costs and effi-

ciency indices. Te initial approximation assumed is(0) ={1,1,1, . . . ,1} or obtaining the solution

={1.0115010875012980 . . . , 1.0546781130247093 . . . ,1.1098925075633296 . . . , 1.1481439774759322, . . . ,1.1481439774759322 . . . , 1.1098925075633296 . . . ,1.0546781130247093...,1.0115010875012980... }.

()

able shows the numerical results obtained or theconsidered problems by various methods. Displayed in thistable are the number o iterations(), the computationalorder o convergence(), the computational costs(,)interms o products, the computational efficiencies

(,

), and

the mean elapsed CPU time (e-time). Computational costand efficiency are calculated according to the correspondingexpressions given by ()() by using the values o parame-ters and as calculated in each problem, while taking = 2.8in each case. Te mean elapsed CPU time is calculated bytaking the mean o perormances o the program, where

we use(+1) () + (()) < 10200 as the stoppingcriterion in single perormance o the program.

From the numerical results, we can observe that likethe existing methods the present methods show consistentconvergence behavior. It is seen that some methods donot preserve the theoretical order o convergence, especiallywhen applied or solving some typical type o nonlinear


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systems. Tis can be observed in the last problem o non-differentiable mixed Hammerstein integral equation wherethe seventh-order methods1,7 and2,7 yield the eighthorder o convergence. However, or the present methods,the computational order o convergence overwhelminglysupports the theoretical order o convergence.Comparison o

the numerical values o computational efficiencies exhibitedin the second last column oable veries the theoreticalresults oTeorem . As we know, the computational effi-ciency is proportional to the reciprocal value o the total CPUtime necessary to complete running iterative process. Tismeans that the method with high efficiency utilizes less CPUtime than the method with low efficiency. Te truthulnesso this act can be judged rom the numerical values ocomputational efficiency and elapsed CPU time displayedin the last two columns oable , which are in completeagreement according to the notion.

5. Concluding Remarks

In the oregoing study, we have proposed iterative methodswith the sixth order o convergence or solving systems ononlinear equations. Te schemes are totally derivative reeand thereore particularly suited to those problems in whichderivatives require lengthy computation. A development orst-order divided difference operator or unctions o several

variables and direct computation by aylors expansion areused to prove the local convergence order o new methods.Comparison o efficiencies o the new schemes with the exist-ing schemes is shown. It is observed that the present methodshave an edge over similar existing methods, especially whenapplied or solving large systems o equations. Six numerical

examples have been presented and the relevant perormancesare compared with the existing methods. Computationalresults have conrmed the robust and efficient character othe proposed techniques. Similar numerical experimenta-tions have been carried out or a number o problems andresults are ound to be on a par with those presented here.

We conclude the paper with the remark that in manynumerical applications multiprecision in computations isrequired.Te results o numericalexperiments justiy that thehigh-orderefficient methodsassociated with a multiprecisionarithmetic oating point are very useul, because they yielda clear reduction in the number o iterations to achieve therequired solution.

Conflict of Interests

Te authors declare that there is no conict o interestsregarding the publication o this paper.

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