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Research Article Effective Root-Finding Methods for Nonlinear Equations Based on Multiplicative Calculi Ali ÖzyapJcJ, 1 Zehra B. Sensoy, 1 and Tolgay Karanfiller 2 1 Faculty of Engineering, Cyprus International University, Nicosia, Mersin 10, Turkey 2 Faculty of Education, Cyprus International University, Nicosia, Mersin 10, Turkey Correspondence should be addressed to Ali ¨ Ozyapıcı; [email protected] Received 29 July 2016; Accepted 14 September 2016 Academic Editor: Liwei Zhang Copyright © 2016 Ali ¨ Ozyapıcı et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In recent studies, papers related to the multiplicative based numerical methods demonstrate applicability and efficiency of these methods. Numerical root-finding methods are essential for nonlinear equations and have a wide range of applications in science and engineering. erefore, the idea of root-finding methods based on multiplicative and Volterra calculi is self-evident. Newton- Raphson, Halley, Broyden, and perturbed root-finding methods are used in numerical analysis for approximating the roots of nonlinear equations. In this paper, Newton-Raphson methods and consequently perturbed root-finding methods are developed in the frameworks of multiplicative and Volterra calculi. e efficiency of these proposed root-finding methods is exposed by examples, and the results are compared with some ordinary methods. One of the striking results of the proposed method is that the rate of convergence for many problems are considerably larger than the original methods. 1. Introduction Ever since Grossman and Katz introduced multiplicative calculus in [1] in the last quarter of nineteenth century, the importance of multiplicative calculi is recently understood by the researchers from the different branches. Especially aſter the paper by Bashirov et al. [2], some important applications of multiplicative calculus in various applications have been introduced. Some of these are [3] in biomedical image analysis, [4] in complex analysis, [5] in growth phenomena, [6–9] in numerical analysis, [10] in actuarial science, finance, demography, and so forth, [11] in biology, and recently [12] in accounting. It is important to note that first serious application of multiplicative numerical methods was applied by Bilgehan in [13] for representing real time signals in signal processing. Additionally, Volterra (bigeometric by Grossman terminology) calculus also provides a wide range of applica- tions in science. We refer to [14–16] with references therein for further discussions of Volterra (bigeometric) calculus. In this section, we give an overview of some basic concepts of multiplicative and Volterra calculi. (1) Multiplicative Calculus Definition 1. Let () be a function, where dom(). If the limit () = lim ℎ→0 ( ( + ℎ) () ) 1/ℎ (1) exists, then is called multiplicative differentiable at . If is a positive function and the derivative of at exists, then th multiplicative derivative of exists and ∗() () = exp {(ln ∘) () ()} . (2) Some properties and basic theorems of multiplicative derivative can be found in [2, 8] and papers therein. Referring to Bashirov et al. [2], multiplicative Taylor theorems with one variable can be given, respectively. eorem 2 (multiplicative Taylor theorem with one variable). Let be an open interval and let :→ R be +1 times Hindawi Publishing Corporation Journal of Mathematics Volume 2016, Article ID 8174610, 7 pages http://dx.doi.org/10.1155/2016/8174610

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Page 1: Research Article Effective Root-Finding Methods for ...downloads.hindawi.com/journals/jmath/2016/8174610.pdfResearch Article Effective Root-Finding Methods for Nonlinear Equations

Research ArticleEffective Root-Finding Methods for NonlinearEquations Based on Multiplicative Calculi

Ali OumlzyapJcJ1 Zehra B Sensoy1 and Tolgay Karanfiller2

1Faculty of Engineering Cyprus International University Nicosia Mersin 10 Turkey2Faculty of Education Cyprus International University Nicosia Mersin 10 Turkey

Correspondence should be addressed to Ali Ozyapıcı aliozyapiciemuedutr

Received 29 July 2016 Accepted 14 September 2016

Academic Editor Liwei Zhang

Copyright copy 2016 Ali Ozyapıcı 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

In recent studies papers related to the multiplicative based numerical methods demonstrate applicability and efficiency of thesemethods Numerical root-finding methods are essential for nonlinear equations and have a wide range of applications in scienceand engineering Therefore the idea of root-finding methods based on multiplicative and Volterra calculi is self-evident Newton-Raphson Halley Broyden and perturbed root-finding methods are used in numerical analysis for approximating the roots ofnonlinear equations In this paper Newton-Raphson methods and consequently perturbed root-finding methods are developedin the frameworks of multiplicative and Volterra calculi The efficiency of these proposed root-finding methods is exposed byexamples and the results are compared with some ordinary methods One of the striking results of the proposed method is thatthe rate of convergence for many problems are considerably larger than the original methods

1 Introduction

Ever since Grossman and Katz introduced multiplicativecalculus in [1] in the last quarter of nineteenth century theimportance ofmultiplicative calculi is recently understood bythe researchers from the different branches Especially afterthe paper by Bashirov et al [2] some important applicationsof multiplicative calculus in various applications have beenintroduced Some of these are [3] in biomedical imageanalysis [4] in complex analysis [5] in growth phenomena[6ndash9] in numerical analysis [10] in actuarial science financedemography and so forth [11] in biology and recently [12]in accounting It is important to note that first seriousapplication of multiplicative numerical methods was appliedby Bilgehan in [13] for representing real time signals in signalprocessing Additionally Volterra (bigeometric by Grossmanterminology) calculus also provides a wide range of applica-tions in science We refer to [14ndash16] with references thereinfor further discussions of Volterra (bigeometric) calculus Inthis section we give an overview of some basic concepts ofmultiplicative and Volterra calculi

(1) Multiplicative Calculus

Definition 1 Let 119891(119909) be a function where 119909 isin dom(119891) Ifthe limit

119891lowast (119909) = limℎrarr0

(119891 (119909 + ℎ)

119891 (119909))1ℎ

(1)

exists then 119891 is called multiplicative differentiable at 119909If 119891 is a positive function and the derivative of 119891 at 119909

exists then 119899th multiplicative derivative of 119891 exists and

119891lowast(119899) (119909) = exp (ln ∘119891)(119899) (119909) (2)

Some properties and basic theorems of multiplicativederivative can be found in [2 8] and papers therein Referringto Bashirov et al [2] multiplicative Taylor theorems with onevariable can be given respectively

Theorem2 (multiplicative Taylor theoremwith one variable)Let 119860 be an open interval and let 119891 119860 rarr R be 119899 + 1 times

Hindawi Publishing CorporationJournal of MathematicsVolume 2016 Article ID 8174610 7 pageshttpdxdoiorg10115520168174610

2 Journal of Mathematics

lowast differentiable on 119860 Then for any 119909 119909 + ℎ isin 119860 there exists anumber 120579 isin (0 1) such that

119891 (119909 + ℎ) =119899

prod119898=0

(119891lowast(119898) (119909))ℎ119898

119898

sdot (119891lowast(119899+1) (119909 + 120579ℎ))ℎ119899+1

(119899+1)

(3)

Definition 3 Given 119909 isin R+ = (0infin) the multiplicativeabsolute value of 119909 is denoted by |119909|lowastsuch that

|119909|lowast =

119909 if 119909 ge 11

119909if 119909 lt 1

(4)

(2) Volterra (Bigeometric) Multiplicative Calculus

Definition 4 Let 119891 be a positive function over the openinterval (119886 119887) If the limit

119891120587 (119909) =119889120587119891 (119909)

119889119909= limℎrarr0

(119891 ((1 + ℎ) 119909)

119891 (119909))1ℎ

(5)

exists then 119891 is said to be Volterra type differentiable at 119909 isin(119886 119887)

In [7] the relation between these two multiplicativecalculi can be given as

119891120587 (119909) = (119891lowast

(119909))119909

(6)

119891120587120587 (119909) = (119891lowastlowast

(119909))1199092

(119891lowast)119909

(7)

119891120587(3) (119909) = (119891lowast(3)

(119909))1199093

(119891lowastlowast (119909))31199092

(119891lowast)119909

(8)

Furthermore necessary concepts onVolterra calculus caneasily be derived by using the above relations (6) and (8)together with multiplicative concepts

Taylor expansion for one variable cannot be obtainedeasily in Volterra calculus Few factors of the Volterra typeTaylor expansion are deduced respectively in [14] and [7] as

119891 (119909 + ℎ)

= 119891 (119909) 119891120587

(119909)ℎ119909minusℎ

2

21199092

+sdotsdotsdot 119891120587120587 (119909)ℎ2

21199092

minussdotsdotsdot sdot sdot sdot (9)

Recently the closed form of Volterra Taylor theorem (asBigeometric Taylor theorem) was presented in the paper [17]

(3) Zero-Finding Methods andTheir Applications The processof determining roots of nonlinear functions is involvedin many applications in various fields such as image andaudio processing mathematics fuzzy systems and fluidmechanics In fuzzy systems it is important to be able to solvenonlinear systems of equations In fluid mechanics root-finding methods arise in finding depth of water In the caseof image processing it is applied to detect the significant localchanges (zero crossings) of the intensity levels under thework

of edge detection It is known that accurate prompt resultsyield best selection of localization which produce smoothimage In the case of digital audio processing zero crossingpoints represent the sample at zero amplitude At any otherpoint the amplitude of the wave is rising towards its peak orsinking towards zero It is very important to determine theexact zero crossing in a shortest time interval especially inthe case of looping where the joins between the audio mustbe as smooth as possible

As mentioned above the zeros of functions (especiallynonlinear functions) are very significant in real applicationsas well as mathematical applications such as critical pointsof nonlinear functions Therefore using numerical methodsare essential in relation to these problems Newton-RaphsonChebyshevrsquos Halley Broyden and perturbed methods aresome of the important methods for approximating the zerosof functions The papers [18ndash20] and many papers thereinpresent recent developments and applications of the ordinaryroot-findingmethodsThemain problem of these methods isthat the initial quest of each method should be sufficientlyclose to the root Moreover the rate of convergence of thesemethods is different in many applications

It is also possible to create alternative methods basedon multiplicative calculi which provide larger intervals forinitial assumptions with more improved rate of convergenceFirstly the multiplicative root-finding methods have beendiscussed in master thesis [21] For many applications theuse of multiplicative based methods will reduce the amountof computational complexity and the time consumption

2 Newton-Raphson Methods viaMultiplicative Calculi

In this section multiplicative root-finding methods based onNewton-Raphsonmethod are alternatively introducedThesemethods will provide better performance formany problems

21 Multiplicative Newton-RaphsonMethod (MN) The start-ing point of multiplicative Newton formulae will be thecorresponding multiplicative Taylor theorems In order toobtain Volterra taylor theorem the relationships betweenmultiplicative andVolterra derivatives stated inRiza et al [7]were applied

Theorem 5 (multiplicative Newton-Raphson) Assume that119891 isin 1198622[119886 119887] and there exist a number 119898 isin [119886 119887] such that119891(119898) = 1 If119891lowast(119898) = 1 and 119892(119909) = 119909minus ln119891(119909) ln119891lowast(119909) thenthere exist a 120575 gt 0 such that the sequence 119898

119896infin119896=1

defined byiteration

119898119896= 119892 (119898

119896minus1) = 119898

119896minus1minus

ln119891 (119898119896minus1)

ln119891lowast (119898119896minus1)

forall119896 = 1 2 3

(10)

will converge to119898 for any initial value1198980isin [119898 minus 120575119898 + 120575]

Proof Since 119891(119898) = 1 we have 119892(119898) = 119898 To find the point119909 such that 119891(119909) = 1 is accomplished by finding a fixed

Journal of Mathematics 3

point of function 119892(119909) The multiplicative Taylor theorem(Theorem 2) of degree of 1 with its reminder term can begiven as

119891 (119909) asymp 119891 (1198980) 119891lowast (119898

0)(119909minus119898

0)

119891lowastlowast (119888)(119909minus119898

0)2

2 (11)

where 119888 isin [1198980 119909] Substituting 119909 = 119898 into (11) gives

119891 (1198980) 119891lowast (119898

0)(119898minus119898

0)

119891lowastlowast (119888)(119898minus119898

0)2

2 cong 1 (12)

whenever 119891(119898) = 1 If 1198980is close enough to 119898 we can

consider first two terms of (12) as

119891 (1198980) 119891lowast (119898

0)(119898minus119898

0)

asymp 1 (13)

Thus

119898 = 1198980minus (

ln119891 (1198980)

ln119891lowast (1198980)) (14)

Letting119898119896rarr 119898 and119898

119896minus1rarr 1198980yields (10)

22 Volterra (Bigeometric) Newton-Raphson Methods (AVM)The following theorem states the iteration of root-findingalgorithms with the convergent criteria in the framework ofVolterra calculus

Theorem 6 (Volterra Newton-Raphson) Assume that 119891 isin

1198622[119892] and there exist a number V isin [119886 119887] such that119891(V) = 1 If119891120587(V) = 1 and 119892(119909) = 119909 minus 119909 ln119891(119909) ln119891120587(119909) then there exista 120575 gt 0 such that the sequence V

119896infin119896=1

defined by iteration

V119896= 119892 (V

119896minus1) = V119896minus1minus V119896minus1

ln119891 (V119896minus1)

ln119891120587 (V119896minus1)

for 119896 = 1 2

(15)

will converge to V for any initial value V0isin [V minus 120575 V + 120575]

Proof Insight can be easily given of the proof of multiplica-tive Newton-Raphson method

It is very important to note that methods (10) and (15)are identical because first two terms of multiplicative andVolterra Taylor theorems in the derivation ofNewtonmethodyield same iterations

Alternatively the following theorem can be considered asa modification of Volterra Newton-Raphson method

Theorem 7 (alternative Volterra Newton-Raphson) Assumethat 119891 isin 1198622[119892] and there exist a number V isin [119886 119887] such that119891(V) = 1 If 119891120587(V) = 1 119892(119909) = 119909 minus 119909 ln119891(119909) ln119891120587(119909)and ℎ(119909) = (1199092)(32 minus ln119891(119909) ln119891120587(119909) + (12)(1 minusln119891(119909) ln119891120587(119909))2) then there exists 120575 gt 0 such that thesequence V

119896infin119896=1

defined by iteration

V119896= ℎ (V

119896minus1) =

V119896minus1

2(3

2minus

ln119891 (V119896minus1)

ln119891120587 (V119896minus1)

+1

2(1 minus

ln119891 (V119896minus1)

ln119891120587 (V119896minus1))

2

) for 119896 = 1 2

(16)

will converge to V for any initial value V0isin [V minus 120575 V + 120575]

Proof Insight can be easily given of the proof of multiplica-tive Newton-Raphson method

3 Perturbed Root-Finding Methods viaMultiplicative Calculi

In this section multiplicative perturbed root-finding meth-ods are derived based on corresponding Taylor theoremsThe effort to derive these methods will provide betterapproximation with less computational time and complexityThe number of perturbed terms in corresponding ordinaryperturbed methods (see [20]) should be increased to havethe same approximations as multiplicative perturbed meth-ods for many nonlinear equations The ordinary perturbedmethod (OP) with two terms can be given as (see [20 22])

119909119899+1= 119909119899minus [

119891 (119909119899)

1198911015840 (119909119899)+1

2

11989110158401015840 (119909119899) (119891 (119909

119899))2

(1198911015840 (119909119899))3

] (17)

The multiplicative perturbed methods with two terms arealternatively derived Numerical results tabulated in nextsection demonstrate efficiency of the proposed method forthe solutions of nonlinear equations

31 Multiplicative Perturbed Method We will start withthe multiplicative Taylor theorem to derive correspondingmultiplicative perturbedmethodThe first three factors in themultiplicative Taylor theorem can be given as

119891 (119909 + ℎ) asymp 119891 (119909) 119891lowast

(119909)ℎ 119891lowastlowast (119909)

ℎ2

2 (18)

Assume that there exists at least a value 119909 such that thefunction 119891(119909) = 1 Representing the root value withperturbation terms as 119909 = 119909

0+ 1205761199091+ 12059821199092and substituting

them into (18) give

119891 (1199090+ 1205761199091+ 12059821199092) asymp 119891 (119909

0) [119891lowast (119909

0)]1205761199091

sdot (119891lowast (1199090))1205982

1199092 (119891lowastlowast (119909

0))1205762

1199092

12

(119891lowastlowast (1199090))1205763

11990911199092

sdot (119891lowastlowast (1199090))1205764

1199092

22

(19)

Omitting the last two terms in (19) and setting 1 give

119891 (1199090) 119891lowast (119909

0)1205761199091 = 1

(119891lowast (1199090))1205982

1199092 (119891lowastlowast (119909

0))1205762

1199092

12

= 1

(20)

The solutions of the equations in (20) are respectively

1205761199091= minus

ln119891 (1199090)

ln119891lowast (1199090)

12059821199092=minus (12) (ln119891 (119909

0) ln119891lowast (119909

0))2 ln119891lowastlowast (119909

0)

ln119891lowast (1199090)

(21)

4 Journal of Mathematics

Let the initial estimation of 119909 where 119891(119909) = 1 to be denotedas 1199090 According to the equations in (21) the second iterative

value 1199091can be computed as

1199091= 1199090minus

ln119891 (1199090)

ln119891lowast (1199090)minus1

2

ln119891lowastlowast (1199090) (ln119891 (119909

0))2

(ln119891lowast (1199090))3

(22)

Let 1199091= 119909119899+1

and 1199090= 119909119899in (22) Then

119909119899+1= 119909119899minus

ln119891 (119909119899)

ln119891lowast (119909119899)minus1

2

ln119891lowastlowast (119909119899) (ln119891 (119909

119899))2

(ln119891lowast (119909119899))3

(23)

Equation (23) expresses multiplicative perturbed iteration(MP)

32 Volterra (Bigeometric) Perturbed Method (VP) We willderive Volterra perturbed method in the framework ofVolterra calculus The starting point is (9) in the form

119891 (119909 + ℎ) asymp 119891 (119909) 119891120587

(119909)ℎ119909 119891120587120587 (119909)

ℎ2

21199092

(24)

Assume that there exists at least a value 119909 such that thefunction119891(119909) = 1 Let the root value with perturbation termsbe represented as 119909 = 119909

0+ 1205761199091+ 12059821199092and substituting it in

(24) gives

119891 (1199090+ 1205761199091+ 12059821199092) asymp 119891 (119909

0) 119891120587 (119909

0)12057611990911199090

sdot 119891120587 (1199090)1205982

11990921199090 119891120587120587 (119909

0)1205982

1199092

12(1199090)2

sdot 119891120587120587 (1199090)1205763

11990911199092(1199090)2

119891120587120587 (1199090)1205764

1199092

22(1199090)2

(25)

Omitting the last two terms in (25) and setting it to 1 we have

1205761199091= 1199090[minus

ln119891 (1199090)

ln119891120587 (1199090)]

12059821199092= minus1199090[ln119891120587120587 (119909

0) (ln119891 (119909

0))2

2 (ln119891120587 (1199090))3

]

(26)

If the initial estimation of 119891(119909) = 1 is 1199090 the second iterative

value 1199091can be computed by using (26) as

1199091

= 1199090[1 minus

ln119891 (1199090)

ln119891120587 (1199090)minus1

2

ln119891120587120587 (1199090) (ln119891 (119909

0))2

(ln119891120587 (1199090))3

] (27)

Let 1199091= 119909119899+1

and 1199090= 119909119899in (27) Then we obtain

119909119899+1

= 119909119899[1 minus

ln119891 (119909119899)

ln119891120587 (119909119899)minus1

2

ln119891120587120587 (119909119899) (ln119891 (119909

119899))2

(ln119891120587 (119909119899))3

] (28)

Equation (28) expresses Volterra perturbed iteration (VP)

4 Convergence Criterion of Proposed Methods

The selection of initial value is also very important for theconvergence of the multiplicative root-finding algorithms Inthis section the criterion of the convergence of the proposedmethods will be given which will also lead to a selection ofthe starting point for any given problem Before giving theconvergence of the multiplicative Newton-Raphson methodsdue to the initial value the fixed point theorems shouldbe derived in the multiplicative sense Additionally multi-plicative Rolle and Mean Value theorems discussed in [2]should be considered Moreover Volterra Rolle and VolterraMeanValue theorems which were firstly given in [16] shouldexplicitly be derived

Theorem 8 Suppose that 119892lowast is defined over (119886 119887) and let apositive constant 119896 lt 119890 (27182818 )with |119892lowast(119909)119892(119909)|lowast le 119896 lt119890 for all 119909 isin (119886 119887)Then 119892 has unique fixed point 119865 in [119886 119887]

Proof Let 1198651and 1198652be two fixed pointsThen by multiplica-

tive Mean Value theorem there exist 119888 isin (119886 119887) such that

(119890119892(119888))lowast

= (119890119892(1198652)

119890119892(1198651))

1(1198652minus1198651)

= 119890ln(11989011986521198901198651)1(1198652minus1198651)

= 119890 (29)

which contradicts the statementTherefore119892(119909) has a uniquefixed point 119865 in [119886 119887]

By the assumption in multiplicative Newton-Raphsonmethod 119891(119898) = 1 and thus 119892lowast(119898) = 1 The above theoremproduces a sufficient condition for initial value 119898

0to yield a

sequence 119898119896 for 119896 = 0 1 2 for the root of 119891(119898) = 1 so

that1198980isin (119898 minus 120575119898 + 120575) and 120575 can be selected such that

(119890119892(119909))lowast

lt 119890 (30)

for all 119909 isin (119898 minus 120575119898 + 120575)Hence the condition in (30) is convergence criteria of the

initial value of the proposed multiplicative methodAnalogously it is possible to derive similar condition for

the corresponding Volterra method

Theorem 9 (Volterra Rollersquos theorem) Assume that 119891 isin119862[119886 119887] and 119891 is positive on [119886 119887] Let 119891120587(119909) exist for all 119909 isin(119886 119887) If 119891(119886) = 119891(119887) = 1 then exist119888 isin (119886 119887) such that

119891120587 (119888) = 1 (31)

Proof It can easily be given by proof of multiplicative Rollersquostheorem

Theorem 10 (Volterra Mean Value theorem) Assume that119891 isin 119862[119886 119887] and 119891 is positive on [119886 119887] Let 119891120587(119909) exist for all119909 isin (119886 119887)Then exist119888 isin (119886 119887) such that

119891120587 (119888) = [119891 (119887)

119891 (119886)]1(ln 119887minusln 119886)

(32)

Proof Let us consider the function

119865 (119905) = 119891 (119886) [119891 (119887)

119891 (119886)](ln 119905minusln 119886)(ln 119887minusln 119886)

(33)

Journal of Mathematics 5

Then 119865(119886) = 119891(119886) and 119865(119887) = 119891(119887) Let 119866(119905) = 119865(119905)119891(119905)so that 119866(119886) = 119866(119887) = 1 By Volterra Rollersquos theorem exist119888 isin(119886 119887) such that

119866120587 (119888) =119865120587 (119905)

119891120587 (119905)= 1 (34)

Hence 119865120587(119905) = 119891120587(119905) = 119890119905[(ln119891(119887)minusln119891(119886))119905(ln 119887minusln 119886)] whichgives formula (32)

Theorem 11 Suppose that 119892120587 is defined over (119886 119887) and let apositive constant 119870 le 119890 with |[119892(119909)120587]|lowast le 119870 le 119890119909 for all 119909 isin(119886 119887) where 119890 = 27182818 Then 119892 has unique fixed point119865 in [119886 119887]

Proof It can be easily shown by using Mean Value theoremin Volterra calculus

Sufficient condition for initial value V0to yield a conver-

gence sequence V119896 for 119896 = 0 1 2 for the root of 119891(V) = 1

is that V0isin (119898 minus 120575119898 + 120575) and 120575 can be selected such that

[119892 (119909)120587] lt 119890119909 (35)

for all 119909 isin (119898 minus 120575119898 + 120575)Hence condition in (35) is convergence criteria of the

initial value of the proposed Volterra method

5 Some Numerical Results ofProposed Methods

In this section some examples will be considered to reveal theapplicability of the introduced methods Numerical resultsare reported which indicate that the proposed methods mayallow a considerable saving in both the number of step sizesand reduction of computational cost Besides the examplesconsisting of different type of functions reveal the advantagesof proposed methods compared to the ordinary methods

51 Comparisons of Multiplicative and Ordinary Methods Itis important to note that finding a zero of the function ℎ(119909)at 119909 = 119903 is accomplished by finding the multiplicative root119903 such that 119891(119903) = ℎ(119903) + 1 = 1 The numerical results ofsome nonlinear equations using proposed and ordinary root-finding methods are listed in Tables 1 and 2

Displayed in Table 1 is the number of function evaluationsrequired such that |1 minus 119891(119909

119899)119891(119909

119890)| lt 10minus15 The functions

in Table 1 with their roots 119909119890 respectively are

1198911(119909) = 10119890

minus1199092

minus 1 119909119890= 16796306104284499

1198912(119909) = sin119909 minus 1

2119909 119909

119890= 18954942670339809

1198913(119909) = tan119909 minus tanh119909

119909119890= 39266023120479185

Table 1 OP represents ordinary perturbation iterations MP rep-resents multiplicative perturbation iterations and VP representsVolterra perturbation iterations 119909

0represent initial value NC states

that the given method does not converge for the given zero offunction 119909

119890

119891 (119909119899) 119909

0OP MP VP

1198911

25 NC 4 4

055 NC 6 6

1198912

07 NC 5 NC2 3 3 4

1198913

45 5 4 3

35 3 3 3

1198914

25 8 4 6

075 NC 3 4

1198915

265 NC 6 6

225 4 4 4

1198916

175 5 3 4

25 NC 4 6

Table 2 NM represents ordinary Newton-Raphson iterations MNrepresents multiplicative Newton-Raphson iterations and AVNrepresents alternative Volterra Newton-Raphson iterations 119909

0rep-

resent initial value NC states that the given method does notconverge for the given zero of function 119909

119890

119891(119909119899) 119909

0NM MN AVN

1198911

25 NC 5 5

075 NC 8 11

1198912

16 5 5 5

2 4 4 4

1198913

45 7 5 5

35 6 4 5

1198914

25 12 4 5

075 11 4 5

1198915

3 7 6 6

225 4 3 4

1198916

175 7 6 6

25 NC 6 5

1198914(119909) = 119909119890

1199092

minus sin2119909 + 3 cos119909 minus 4

119909119890= 10651360157761873

1198915(119909) = 119890

minus119909 + cos119909 119909119890= 17461395304080125

1198916(119909) = 119909 minus ln (1199092 minus 3) minus 2

119909119890= 17461395304080125

(36)

According to the obtained results multiplicative root-finding algorithms can be used effectively and efficientlyin real applications mentioned in section one Moreoverthese methods yield better approximations for the nonlinearequations especially when the equations involve exponentiallogarithmic and hyperbolic function On the other hand the

6 Journal of Mathematics

Table 3The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (38) 119909119890= 05576473009191445

119891(119909119899) 119909

0OP MP VP NM MN AVN

175 NC 4 4 NC 6 6

125 NC 4 7 14 5 7

Table 4The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (41) 119909119890= 21954

119891(119909119899) 119909

0OP MP VP NM MN AVN

19 NC 6 NC 7 5 5

16 3 3 4 3 3 4

ordinary methods can give more accurate results especiallyfor polynomial equations Thus the method should beselected according to the functions appeared in the equations

52 Realistic Applications In this subsection two exampleswill be considered to demonstrate possible impacts of theintroduced methods to science and engineering

Example 12 The process in which chemicals interact to formnew chemicals with different compositions is called chemicalreactionsThis process is the results of chemical properties ofthe element or compound causing changes in compositionThese chemical changes are chemistsrsquo main purpose It isimportant to mention a remark from [23] where ldquothe twoquestions that must be answered for a chemically reactingsystem are (1) what changes are expected to occur and (2)how fast will they occurrdquo This is an indication that math-ematical representations are very important for chemicalinvestigations Consequently a chemical reaction is mainlyrepresented by a chemical equation which represents thechange from reactants to products This process generallyinvolves nonlinear functions so it is compulsory to useand apply numerical approaches Exemplarily suppose thata chemical reaction is made and the concentration of aparticular ion at the time 119905 is given by a nonlinear function

119891 (119909) = 5119890minus3119909 + 119890minus5119909 (37)

If we are interested in when this concentration will be one-half of its value at initial time 0 we need to solve this problemnumerically If 119891(0) = 2 as an initial assumption this will beequivalent to finding a root of nonlinear equation

5119890minus3119909 + 119890minus5119909 minus 1 = 0 (38)

Usually chemists tend to use ordinary numerical methodsto estimate the root of (38) However the multiplicativebase methods whose effectiveness was proved in problemsthat involve exponential functions should not be disregardedin many applications The superiority of the multiplicativenumerical methods can be easily observed for this problemby Table 3

Example 13 The Rayleigh function which corresponds toRayleigh distribution

119891 (119909) =119909

120590119890minus1199092

21205902

for arbitrary 119886 119887 119888 isin 119877 (39)

plays an important role in magnetic resonance imaging(MRI) and probability theory It is a striking example in orderto show the efficiency of the multiplicative and Volterra root-finding methods Assume that the density function of thecontinuous random variable119883 on the interval [0infin) is givenby

119891 (119909) = 119909119890minus1199092

2 where 120590 = 1 (40)

It may be interesting to consider the root of the equation

120572 minus 119891 (119909) = 120572 minus 119909119890minus1199092

2 = 0 0 le 120572 le 1 (41)

which estimates the point119909 according to the given probability120572 of an event in119883 We attempt to use root-finding algorithmsfor (41) for 120572 = 019720178928946303 which will be given byTable 4

6 Conclusion

In this study the multiplicative and Volterra based root-finding methods are presented These methods were testedfor some nontrivial problems and compared with the originalroot-finding method The results show that in certain prob-lems the multiplicative andor Volterra methods give moreaccurate results compared to the original root-finding meth-ods Especially the examples showed that the nature of theunderlying calculus plays an important role in approximatingthe zeros of the function

The selection of the initial value is very important for theconvergence of the iteration Two theorems in Section 4 indi-cate the conditions for convergence related to multiplicativeand Volterra methods respectively Section 4 also highlightsoptimal selection of the initial value Evidently in certain sit-uations the multiplicative andor Volterra methods convergefaster compared to the ordinary methods Particularly it canbe easily observed that multiplicative and Volterra Newton-Raphson methods are more accurate than ordinary Newton-Raphson method in many applications Therefore these

Journal of Mathematics 7

methods based on multiplicative calculi have proven theirimportance in the process of numerical approximations ofnonlinear equations Also the numerical results obtained inthe paper encourage the usage of multiplicative and Volterramethods for solving nonlinear equations

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Grossman and R Katz Non-Newtonian Calculus Lee PressPigeon Cove Mass USA 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] L Florack and H V Assen ldquoMultiplicative calculus in biomedi-cal image analysisrdquo Journal ofMathematical Imaging andVisionvol 42 no 1 pp 64ndash75 2012

[4] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[5] D A Filip and C Piatecki ldquoA non-Newtonian examinationof the theory of exogenous economic growthrdquo CNCSISmdashUEFISCSU (project number PNII IDEI 23662008) and Lab-oratoire dEconomie dOrleans (LEO) 2010

[6] E Misirli and Y Gurefe ldquoMultiplicative adams bashforthmdashmoulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011

[7] M Riza A Ozyapıcı and E Misirli ldquoMultiplicative finitedifference methodsrdquo Quarterly of Applied Mathematics vol 67no 4 pp 745ndash754 2009

[8] A Ozyapici and E Misirli ldquoExponential approximations onmultiplicative calculusrdquo Proceedings of the Jangjeon Mathemati-cal Society vol 12 no 2 pp 227ndash236 2009

[9] A Ozyapici and B Bilgehan ldquoFinite product representation viamultiplicative calculus and its applications to exponential signalprocessingrdquo Numerical Algorithms vol 71 no 2 pp 475ndash4892016

[10] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[11] J Englehardt J Swartout and C Loewenstine ldquoA new theoret-ical discrete growth distribution with verification for microbialcounts in waterrdquo Risk Analysis vol 29 no 6 pp 841ndash856 2009

[12] H Ozyapıcı I Dalcı and A Ozyapıcı ldquoIntegrating accountingand multiplicative calculus an effective estimation of learningcurverdquo Computational and Mathematical Organization Theory2016

[13] B Bilgehan ldquoEfficient approximation for linear and non-linearsignal representationrdquo IET Signal Processing vol 9 no 3 pp260ndash266 2015

[14] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007

[15] WKasprzak B Lysik andM RybaczukDimensions InvariantsModels and Fractals Ukrainian Society on Fracture MechanicsSPOLOM Wroclaw-Lviv Poland 2004

[16] M Grossman Bigeometric Calculus A System with a Scale-FreeDerivative Archimedes Foundation RockportMass USA1983

[17] M Riza and B Eminagaı ldquoBigeometric calculusmdasha modellingtoolrdquo httpsarxivorgabs14022877v1

[18] M A Noor W A Khan and A Hussain ldquoA newmodified Hal-ley method without second derivatives for nonlinear equationrdquoAppliedMathematics and Computation vol 189 no 2 pp 1268ndash1273 2007

[19] A Ramli M L Abdullah and M Mamat ldquoBroydenrsquos methodfor solving fuzzy nonlinear equationsrdquo Advances in FuzzySystems vol 2010 Article ID 763270 6 pages 2010

[20] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquo Mathematical and ComputationalApplications vol 12 no 2 pp 117ndash124 2007

[21] E Misirli and Y Gurefe Multiplicative calculus and its applica-tions [MS thesis] Turkish Council of Higher Education 2009Thesis No 252639

[22] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[23] M E Davis and R J Davis Fundamentals of Chemical ReactionEngineering MGraw-Hill Companies 2003

Submit your manuscripts athttpwwwhindawicom

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

Page 2: Research Article Effective Root-Finding Methods for ...downloads.hindawi.com/journals/jmath/2016/8174610.pdfResearch Article Effective Root-Finding Methods for Nonlinear Equations

2 Journal of Mathematics

lowast differentiable on 119860 Then for any 119909 119909 + ℎ isin 119860 there exists anumber 120579 isin (0 1) such that

119891 (119909 + ℎ) =119899

prod119898=0

(119891lowast(119898) (119909))ℎ119898

119898

sdot (119891lowast(119899+1) (119909 + 120579ℎ))ℎ119899+1

(119899+1)

(3)

Definition 3 Given 119909 isin R+ = (0infin) the multiplicativeabsolute value of 119909 is denoted by |119909|lowastsuch that

|119909|lowast =

119909 if 119909 ge 11

119909if 119909 lt 1

(4)

(2) Volterra (Bigeometric) Multiplicative Calculus

Definition 4 Let 119891 be a positive function over the openinterval (119886 119887) If the limit

119891120587 (119909) =119889120587119891 (119909)

119889119909= limℎrarr0

(119891 ((1 + ℎ) 119909)

119891 (119909))1ℎ

(5)

exists then 119891 is said to be Volterra type differentiable at 119909 isin(119886 119887)

In [7] the relation between these two multiplicativecalculi can be given as

119891120587 (119909) = (119891lowast

(119909))119909

(6)

119891120587120587 (119909) = (119891lowastlowast

(119909))1199092

(119891lowast)119909

(7)

119891120587(3) (119909) = (119891lowast(3)

(119909))1199093

(119891lowastlowast (119909))31199092

(119891lowast)119909

(8)

Furthermore necessary concepts onVolterra calculus caneasily be derived by using the above relations (6) and (8)together with multiplicative concepts

Taylor expansion for one variable cannot be obtainedeasily in Volterra calculus Few factors of the Volterra typeTaylor expansion are deduced respectively in [14] and [7] as

119891 (119909 + ℎ)

= 119891 (119909) 119891120587

(119909)ℎ119909minusℎ

2

21199092

+sdotsdotsdot 119891120587120587 (119909)ℎ2

21199092

minussdotsdotsdot sdot sdot sdot (9)

Recently the closed form of Volterra Taylor theorem (asBigeometric Taylor theorem) was presented in the paper [17]

(3) Zero-Finding Methods andTheir Applications The processof determining roots of nonlinear functions is involvedin many applications in various fields such as image andaudio processing mathematics fuzzy systems and fluidmechanics In fuzzy systems it is important to be able to solvenonlinear systems of equations In fluid mechanics root-finding methods arise in finding depth of water In the caseof image processing it is applied to detect the significant localchanges (zero crossings) of the intensity levels under thework

of edge detection It is known that accurate prompt resultsyield best selection of localization which produce smoothimage In the case of digital audio processing zero crossingpoints represent the sample at zero amplitude At any otherpoint the amplitude of the wave is rising towards its peak orsinking towards zero It is very important to determine theexact zero crossing in a shortest time interval especially inthe case of looping where the joins between the audio mustbe as smooth as possible

As mentioned above the zeros of functions (especiallynonlinear functions) are very significant in real applicationsas well as mathematical applications such as critical pointsof nonlinear functions Therefore using numerical methodsare essential in relation to these problems Newton-RaphsonChebyshevrsquos Halley Broyden and perturbed methods aresome of the important methods for approximating the zerosof functions The papers [18ndash20] and many papers thereinpresent recent developments and applications of the ordinaryroot-findingmethodsThemain problem of these methods isthat the initial quest of each method should be sufficientlyclose to the root Moreover the rate of convergence of thesemethods is different in many applications

It is also possible to create alternative methods basedon multiplicative calculi which provide larger intervals forinitial assumptions with more improved rate of convergenceFirstly the multiplicative root-finding methods have beendiscussed in master thesis [21] For many applications theuse of multiplicative based methods will reduce the amountof computational complexity and the time consumption

2 Newton-Raphson Methods viaMultiplicative Calculi

In this section multiplicative root-finding methods based onNewton-Raphsonmethod are alternatively introducedThesemethods will provide better performance formany problems

21 Multiplicative Newton-RaphsonMethod (MN) The start-ing point of multiplicative Newton formulae will be thecorresponding multiplicative Taylor theorems In order toobtain Volterra taylor theorem the relationships betweenmultiplicative andVolterra derivatives stated inRiza et al [7]were applied

Theorem 5 (multiplicative Newton-Raphson) Assume that119891 isin 1198622[119886 119887] and there exist a number 119898 isin [119886 119887] such that119891(119898) = 1 If119891lowast(119898) = 1 and 119892(119909) = 119909minus ln119891(119909) ln119891lowast(119909) thenthere exist a 120575 gt 0 such that the sequence 119898

119896infin119896=1

defined byiteration

119898119896= 119892 (119898

119896minus1) = 119898

119896minus1minus

ln119891 (119898119896minus1)

ln119891lowast (119898119896minus1)

forall119896 = 1 2 3

(10)

will converge to119898 for any initial value1198980isin [119898 minus 120575119898 + 120575]

Proof Since 119891(119898) = 1 we have 119892(119898) = 119898 To find the point119909 such that 119891(119909) = 1 is accomplished by finding a fixed

Journal of Mathematics 3

point of function 119892(119909) The multiplicative Taylor theorem(Theorem 2) of degree of 1 with its reminder term can begiven as

119891 (119909) asymp 119891 (1198980) 119891lowast (119898

0)(119909minus119898

0)

119891lowastlowast (119888)(119909minus119898

0)2

2 (11)

where 119888 isin [1198980 119909] Substituting 119909 = 119898 into (11) gives

119891 (1198980) 119891lowast (119898

0)(119898minus119898

0)

119891lowastlowast (119888)(119898minus119898

0)2

2 cong 1 (12)

whenever 119891(119898) = 1 If 1198980is close enough to 119898 we can

consider first two terms of (12) as

119891 (1198980) 119891lowast (119898

0)(119898minus119898

0)

asymp 1 (13)

Thus

119898 = 1198980minus (

ln119891 (1198980)

ln119891lowast (1198980)) (14)

Letting119898119896rarr 119898 and119898

119896minus1rarr 1198980yields (10)

22 Volterra (Bigeometric) Newton-Raphson Methods (AVM)The following theorem states the iteration of root-findingalgorithms with the convergent criteria in the framework ofVolterra calculus

Theorem 6 (Volterra Newton-Raphson) Assume that 119891 isin

1198622[119892] and there exist a number V isin [119886 119887] such that119891(V) = 1 If119891120587(V) = 1 and 119892(119909) = 119909 minus 119909 ln119891(119909) ln119891120587(119909) then there exista 120575 gt 0 such that the sequence V

119896infin119896=1

defined by iteration

V119896= 119892 (V

119896minus1) = V119896minus1minus V119896minus1

ln119891 (V119896minus1)

ln119891120587 (V119896minus1)

for 119896 = 1 2

(15)

will converge to V for any initial value V0isin [V minus 120575 V + 120575]

Proof Insight can be easily given of the proof of multiplica-tive Newton-Raphson method

It is very important to note that methods (10) and (15)are identical because first two terms of multiplicative andVolterra Taylor theorems in the derivation ofNewtonmethodyield same iterations

Alternatively the following theorem can be considered asa modification of Volterra Newton-Raphson method

Theorem 7 (alternative Volterra Newton-Raphson) Assumethat 119891 isin 1198622[119892] and there exist a number V isin [119886 119887] such that119891(V) = 1 If 119891120587(V) = 1 119892(119909) = 119909 minus 119909 ln119891(119909) ln119891120587(119909)and ℎ(119909) = (1199092)(32 minus ln119891(119909) ln119891120587(119909) + (12)(1 minusln119891(119909) ln119891120587(119909))2) then there exists 120575 gt 0 such that thesequence V

119896infin119896=1

defined by iteration

V119896= ℎ (V

119896minus1) =

V119896minus1

2(3

2minus

ln119891 (V119896minus1)

ln119891120587 (V119896minus1)

+1

2(1 minus

ln119891 (V119896minus1)

ln119891120587 (V119896minus1))

2

) for 119896 = 1 2

(16)

will converge to V for any initial value V0isin [V minus 120575 V + 120575]

Proof Insight can be easily given of the proof of multiplica-tive Newton-Raphson method

3 Perturbed Root-Finding Methods viaMultiplicative Calculi

In this section multiplicative perturbed root-finding meth-ods are derived based on corresponding Taylor theoremsThe effort to derive these methods will provide betterapproximation with less computational time and complexityThe number of perturbed terms in corresponding ordinaryperturbed methods (see [20]) should be increased to havethe same approximations as multiplicative perturbed meth-ods for many nonlinear equations The ordinary perturbedmethod (OP) with two terms can be given as (see [20 22])

119909119899+1= 119909119899minus [

119891 (119909119899)

1198911015840 (119909119899)+1

2

11989110158401015840 (119909119899) (119891 (119909

119899))2

(1198911015840 (119909119899))3

] (17)

The multiplicative perturbed methods with two terms arealternatively derived Numerical results tabulated in nextsection demonstrate efficiency of the proposed method forthe solutions of nonlinear equations

31 Multiplicative Perturbed Method We will start withthe multiplicative Taylor theorem to derive correspondingmultiplicative perturbedmethodThe first three factors in themultiplicative Taylor theorem can be given as

119891 (119909 + ℎ) asymp 119891 (119909) 119891lowast

(119909)ℎ 119891lowastlowast (119909)

ℎ2

2 (18)

Assume that there exists at least a value 119909 such that thefunction 119891(119909) = 1 Representing the root value withperturbation terms as 119909 = 119909

0+ 1205761199091+ 12059821199092and substituting

them into (18) give

119891 (1199090+ 1205761199091+ 12059821199092) asymp 119891 (119909

0) [119891lowast (119909

0)]1205761199091

sdot (119891lowast (1199090))1205982

1199092 (119891lowastlowast (119909

0))1205762

1199092

12

(119891lowastlowast (1199090))1205763

11990911199092

sdot (119891lowastlowast (1199090))1205764

1199092

22

(19)

Omitting the last two terms in (19) and setting 1 give

119891 (1199090) 119891lowast (119909

0)1205761199091 = 1

(119891lowast (1199090))1205982

1199092 (119891lowastlowast (119909

0))1205762

1199092

12

= 1

(20)

The solutions of the equations in (20) are respectively

1205761199091= minus

ln119891 (1199090)

ln119891lowast (1199090)

12059821199092=minus (12) (ln119891 (119909

0) ln119891lowast (119909

0))2 ln119891lowastlowast (119909

0)

ln119891lowast (1199090)

(21)

4 Journal of Mathematics

Let the initial estimation of 119909 where 119891(119909) = 1 to be denotedas 1199090 According to the equations in (21) the second iterative

value 1199091can be computed as

1199091= 1199090minus

ln119891 (1199090)

ln119891lowast (1199090)minus1

2

ln119891lowastlowast (1199090) (ln119891 (119909

0))2

(ln119891lowast (1199090))3

(22)

Let 1199091= 119909119899+1

and 1199090= 119909119899in (22) Then

119909119899+1= 119909119899minus

ln119891 (119909119899)

ln119891lowast (119909119899)minus1

2

ln119891lowastlowast (119909119899) (ln119891 (119909

119899))2

(ln119891lowast (119909119899))3

(23)

Equation (23) expresses multiplicative perturbed iteration(MP)

32 Volterra (Bigeometric) Perturbed Method (VP) We willderive Volterra perturbed method in the framework ofVolterra calculus The starting point is (9) in the form

119891 (119909 + ℎ) asymp 119891 (119909) 119891120587

(119909)ℎ119909 119891120587120587 (119909)

ℎ2

21199092

(24)

Assume that there exists at least a value 119909 such that thefunction119891(119909) = 1 Let the root value with perturbation termsbe represented as 119909 = 119909

0+ 1205761199091+ 12059821199092and substituting it in

(24) gives

119891 (1199090+ 1205761199091+ 12059821199092) asymp 119891 (119909

0) 119891120587 (119909

0)12057611990911199090

sdot 119891120587 (1199090)1205982

11990921199090 119891120587120587 (119909

0)1205982

1199092

12(1199090)2

sdot 119891120587120587 (1199090)1205763

11990911199092(1199090)2

119891120587120587 (1199090)1205764

1199092

22(1199090)2

(25)

Omitting the last two terms in (25) and setting it to 1 we have

1205761199091= 1199090[minus

ln119891 (1199090)

ln119891120587 (1199090)]

12059821199092= minus1199090[ln119891120587120587 (119909

0) (ln119891 (119909

0))2

2 (ln119891120587 (1199090))3

]

(26)

If the initial estimation of 119891(119909) = 1 is 1199090 the second iterative

value 1199091can be computed by using (26) as

1199091

= 1199090[1 minus

ln119891 (1199090)

ln119891120587 (1199090)minus1

2

ln119891120587120587 (1199090) (ln119891 (119909

0))2

(ln119891120587 (1199090))3

] (27)

Let 1199091= 119909119899+1

and 1199090= 119909119899in (27) Then we obtain

119909119899+1

= 119909119899[1 minus

ln119891 (119909119899)

ln119891120587 (119909119899)minus1

2

ln119891120587120587 (119909119899) (ln119891 (119909

119899))2

(ln119891120587 (119909119899))3

] (28)

Equation (28) expresses Volterra perturbed iteration (VP)

4 Convergence Criterion of Proposed Methods

The selection of initial value is also very important for theconvergence of the multiplicative root-finding algorithms Inthis section the criterion of the convergence of the proposedmethods will be given which will also lead to a selection ofthe starting point for any given problem Before giving theconvergence of the multiplicative Newton-Raphson methodsdue to the initial value the fixed point theorems shouldbe derived in the multiplicative sense Additionally multi-plicative Rolle and Mean Value theorems discussed in [2]should be considered Moreover Volterra Rolle and VolterraMeanValue theorems which were firstly given in [16] shouldexplicitly be derived

Theorem 8 Suppose that 119892lowast is defined over (119886 119887) and let apositive constant 119896 lt 119890 (27182818 )with |119892lowast(119909)119892(119909)|lowast le 119896 lt119890 for all 119909 isin (119886 119887)Then 119892 has unique fixed point 119865 in [119886 119887]

Proof Let 1198651and 1198652be two fixed pointsThen by multiplica-

tive Mean Value theorem there exist 119888 isin (119886 119887) such that

(119890119892(119888))lowast

= (119890119892(1198652)

119890119892(1198651))

1(1198652minus1198651)

= 119890ln(11989011986521198901198651)1(1198652minus1198651)

= 119890 (29)

which contradicts the statementTherefore119892(119909) has a uniquefixed point 119865 in [119886 119887]

By the assumption in multiplicative Newton-Raphsonmethod 119891(119898) = 1 and thus 119892lowast(119898) = 1 The above theoremproduces a sufficient condition for initial value 119898

0to yield a

sequence 119898119896 for 119896 = 0 1 2 for the root of 119891(119898) = 1 so

that1198980isin (119898 minus 120575119898 + 120575) and 120575 can be selected such that

(119890119892(119909))lowast

lt 119890 (30)

for all 119909 isin (119898 minus 120575119898 + 120575)Hence the condition in (30) is convergence criteria of the

initial value of the proposed multiplicative methodAnalogously it is possible to derive similar condition for

the corresponding Volterra method

Theorem 9 (Volterra Rollersquos theorem) Assume that 119891 isin119862[119886 119887] and 119891 is positive on [119886 119887] Let 119891120587(119909) exist for all 119909 isin(119886 119887) If 119891(119886) = 119891(119887) = 1 then exist119888 isin (119886 119887) such that

119891120587 (119888) = 1 (31)

Proof It can easily be given by proof of multiplicative Rollersquostheorem

Theorem 10 (Volterra Mean Value theorem) Assume that119891 isin 119862[119886 119887] and 119891 is positive on [119886 119887] Let 119891120587(119909) exist for all119909 isin (119886 119887)Then exist119888 isin (119886 119887) such that

119891120587 (119888) = [119891 (119887)

119891 (119886)]1(ln 119887minusln 119886)

(32)

Proof Let us consider the function

119865 (119905) = 119891 (119886) [119891 (119887)

119891 (119886)](ln 119905minusln 119886)(ln 119887minusln 119886)

(33)

Journal of Mathematics 5

Then 119865(119886) = 119891(119886) and 119865(119887) = 119891(119887) Let 119866(119905) = 119865(119905)119891(119905)so that 119866(119886) = 119866(119887) = 1 By Volterra Rollersquos theorem exist119888 isin(119886 119887) such that

119866120587 (119888) =119865120587 (119905)

119891120587 (119905)= 1 (34)

Hence 119865120587(119905) = 119891120587(119905) = 119890119905[(ln119891(119887)minusln119891(119886))119905(ln 119887minusln 119886)] whichgives formula (32)

Theorem 11 Suppose that 119892120587 is defined over (119886 119887) and let apositive constant 119870 le 119890 with |[119892(119909)120587]|lowast le 119870 le 119890119909 for all 119909 isin(119886 119887) where 119890 = 27182818 Then 119892 has unique fixed point119865 in [119886 119887]

Proof It can be easily shown by using Mean Value theoremin Volterra calculus

Sufficient condition for initial value V0to yield a conver-

gence sequence V119896 for 119896 = 0 1 2 for the root of 119891(V) = 1

is that V0isin (119898 minus 120575119898 + 120575) and 120575 can be selected such that

[119892 (119909)120587] lt 119890119909 (35)

for all 119909 isin (119898 minus 120575119898 + 120575)Hence condition in (35) is convergence criteria of the

initial value of the proposed Volterra method

5 Some Numerical Results ofProposed Methods

In this section some examples will be considered to reveal theapplicability of the introduced methods Numerical resultsare reported which indicate that the proposed methods mayallow a considerable saving in both the number of step sizesand reduction of computational cost Besides the examplesconsisting of different type of functions reveal the advantagesof proposed methods compared to the ordinary methods

51 Comparisons of Multiplicative and Ordinary Methods Itis important to note that finding a zero of the function ℎ(119909)at 119909 = 119903 is accomplished by finding the multiplicative root119903 such that 119891(119903) = ℎ(119903) + 1 = 1 The numerical results ofsome nonlinear equations using proposed and ordinary root-finding methods are listed in Tables 1 and 2

Displayed in Table 1 is the number of function evaluationsrequired such that |1 minus 119891(119909

119899)119891(119909

119890)| lt 10minus15 The functions

in Table 1 with their roots 119909119890 respectively are

1198911(119909) = 10119890

minus1199092

minus 1 119909119890= 16796306104284499

1198912(119909) = sin119909 minus 1

2119909 119909

119890= 18954942670339809

1198913(119909) = tan119909 minus tanh119909

119909119890= 39266023120479185

Table 1 OP represents ordinary perturbation iterations MP rep-resents multiplicative perturbation iterations and VP representsVolterra perturbation iterations 119909

0represent initial value NC states

that the given method does not converge for the given zero offunction 119909

119890

119891 (119909119899) 119909

0OP MP VP

1198911

25 NC 4 4

055 NC 6 6

1198912

07 NC 5 NC2 3 3 4

1198913

45 5 4 3

35 3 3 3

1198914

25 8 4 6

075 NC 3 4

1198915

265 NC 6 6

225 4 4 4

1198916

175 5 3 4

25 NC 4 6

Table 2 NM represents ordinary Newton-Raphson iterations MNrepresents multiplicative Newton-Raphson iterations and AVNrepresents alternative Volterra Newton-Raphson iterations 119909

0rep-

resent initial value NC states that the given method does notconverge for the given zero of function 119909

119890

119891(119909119899) 119909

0NM MN AVN

1198911

25 NC 5 5

075 NC 8 11

1198912

16 5 5 5

2 4 4 4

1198913

45 7 5 5

35 6 4 5

1198914

25 12 4 5

075 11 4 5

1198915

3 7 6 6

225 4 3 4

1198916

175 7 6 6

25 NC 6 5

1198914(119909) = 119909119890

1199092

minus sin2119909 + 3 cos119909 minus 4

119909119890= 10651360157761873

1198915(119909) = 119890

minus119909 + cos119909 119909119890= 17461395304080125

1198916(119909) = 119909 minus ln (1199092 minus 3) minus 2

119909119890= 17461395304080125

(36)

According to the obtained results multiplicative root-finding algorithms can be used effectively and efficientlyin real applications mentioned in section one Moreoverthese methods yield better approximations for the nonlinearequations especially when the equations involve exponentiallogarithmic and hyperbolic function On the other hand the

6 Journal of Mathematics

Table 3The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (38) 119909119890= 05576473009191445

119891(119909119899) 119909

0OP MP VP NM MN AVN

175 NC 4 4 NC 6 6

125 NC 4 7 14 5 7

Table 4The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (41) 119909119890= 21954

119891(119909119899) 119909

0OP MP VP NM MN AVN

19 NC 6 NC 7 5 5

16 3 3 4 3 3 4

ordinary methods can give more accurate results especiallyfor polynomial equations Thus the method should beselected according to the functions appeared in the equations

52 Realistic Applications In this subsection two exampleswill be considered to demonstrate possible impacts of theintroduced methods to science and engineering

Example 12 The process in which chemicals interact to formnew chemicals with different compositions is called chemicalreactionsThis process is the results of chemical properties ofthe element or compound causing changes in compositionThese chemical changes are chemistsrsquo main purpose It isimportant to mention a remark from [23] where ldquothe twoquestions that must be answered for a chemically reactingsystem are (1) what changes are expected to occur and (2)how fast will they occurrdquo This is an indication that math-ematical representations are very important for chemicalinvestigations Consequently a chemical reaction is mainlyrepresented by a chemical equation which represents thechange from reactants to products This process generallyinvolves nonlinear functions so it is compulsory to useand apply numerical approaches Exemplarily suppose thata chemical reaction is made and the concentration of aparticular ion at the time 119905 is given by a nonlinear function

119891 (119909) = 5119890minus3119909 + 119890minus5119909 (37)

If we are interested in when this concentration will be one-half of its value at initial time 0 we need to solve this problemnumerically If 119891(0) = 2 as an initial assumption this will beequivalent to finding a root of nonlinear equation

5119890minus3119909 + 119890minus5119909 minus 1 = 0 (38)

Usually chemists tend to use ordinary numerical methodsto estimate the root of (38) However the multiplicativebase methods whose effectiveness was proved in problemsthat involve exponential functions should not be disregardedin many applications The superiority of the multiplicativenumerical methods can be easily observed for this problemby Table 3

Example 13 The Rayleigh function which corresponds toRayleigh distribution

119891 (119909) =119909

120590119890minus1199092

21205902

for arbitrary 119886 119887 119888 isin 119877 (39)

plays an important role in magnetic resonance imaging(MRI) and probability theory It is a striking example in orderto show the efficiency of the multiplicative and Volterra root-finding methods Assume that the density function of thecontinuous random variable119883 on the interval [0infin) is givenby

119891 (119909) = 119909119890minus1199092

2 where 120590 = 1 (40)

It may be interesting to consider the root of the equation

120572 minus 119891 (119909) = 120572 minus 119909119890minus1199092

2 = 0 0 le 120572 le 1 (41)

which estimates the point119909 according to the given probability120572 of an event in119883 We attempt to use root-finding algorithmsfor (41) for 120572 = 019720178928946303 which will be given byTable 4

6 Conclusion

In this study the multiplicative and Volterra based root-finding methods are presented These methods were testedfor some nontrivial problems and compared with the originalroot-finding method The results show that in certain prob-lems the multiplicative andor Volterra methods give moreaccurate results compared to the original root-finding meth-ods Especially the examples showed that the nature of theunderlying calculus plays an important role in approximatingthe zeros of the function

The selection of the initial value is very important for theconvergence of the iteration Two theorems in Section 4 indi-cate the conditions for convergence related to multiplicativeand Volterra methods respectively Section 4 also highlightsoptimal selection of the initial value Evidently in certain sit-uations the multiplicative andor Volterra methods convergefaster compared to the ordinary methods Particularly it canbe easily observed that multiplicative and Volterra Newton-Raphson methods are more accurate than ordinary Newton-Raphson method in many applications Therefore these

Journal of Mathematics 7

methods based on multiplicative calculi have proven theirimportance in the process of numerical approximations ofnonlinear equations Also the numerical results obtained inthe paper encourage the usage of multiplicative and Volterramethods for solving nonlinear equations

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Grossman and R Katz Non-Newtonian Calculus Lee PressPigeon Cove Mass USA 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] L Florack and H V Assen ldquoMultiplicative calculus in biomedi-cal image analysisrdquo Journal ofMathematical Imaging andVisionvol 42 no 1 pp 64ndash75 2012

[4] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[5] D A Filip and C Piatecki ldquoA non-Newtonian examinationof the theory of exogenous economic growthrdquo CNCSISmdashUEFISCSU (project number PNII IDEI 23662008) and Lab-oratoire dEconomie dOrleans (LEO) 2010

[6] E Misirli and Y Gurefe ldquoMultiplicative adams bashforthmdashmoulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011

[7] M Riza A Ozyapıcı and E Misirli ldquoMultiplicative finitedifference methodsrdquo Quarterly of Applied Mathematics vol 67no 4 pp 745ndash754 2009

[8] A Ozyapici and E Misirli ldquoExponential approximations onmultiplicative calculusrdquo Proceedings of the Jangjeon Mathemati-cal Society vol 12 no 2 pp 227ndash236 2009

[9] A Ozyapici and B Bilgehan ldquoFinite product representation viamultiplicative calculus and its applications to exponential signalprocessingrdquo Numerical Algorithms vol 71 no 2 pp 475ndash4892016

[10] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[11] J Englehardt J Swartout and C Loewenstine ldquoA new theoret-ical discrete growth distribution with verification for microbialcounts in waterrdquo Risk Analysis vol 29 no 6 pp 841ndash856 2009

[12] H Ozyapıcı I Dalcı and A Ozyapıcı ldquoIntegrating accountingand multiplicative calculus an effective estimation of learningcurverdquo Computational and Mathematical Organization Theory2016

[13] B Bilgehan ldquoEfficient approximation for linear and non-linearsignal representationrdquo IET Signal Processing vol 9 no 3 pp260ndash266 2015

[14] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007

[15] WKasprzak B Lysik andM RybaczukDimensions InvariantsModels and Fractals Ukrainian Society on Fracture MechanicsSPOLOM Wroclaw-Lviv Poland 2004

[16] M Grossman Bigeometric Calculus A System with a Scale-FreeDerivative Archimedes Foundation RockportMass USA1983

[17] M Riza and B Eminagaı ldquoBigeometric calculusmdasha modellingtoolrdquo httpsarxivorgabs14022877v1

[18] M A Noor W A Khan and A Hussain ldquoA newmodified Hal-ley method without second derivatives for nonlinear equationrdquoAppliedMathematics and Computation vol 189 no 2 pp 1268ndash1273 2007

[19] A Ramli M L Abdullah and M Mamat ldquoBroydenrsquos methodfor solving fuzzy nonlinear equationsrdquo Advances in FuzzySystems vol 2010 Article ID 763270 6 pages 2010

[20] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquo Mathematical and ComputationalApplications vol 12 no 2 pp 117ndash124 2007

[21] E Misirli and Y Gurefe Multiplicative calculus and its applica-tions [MS thesis] Turkish Council of Higher Education 2009Thesis No 252639

[22] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[23] M E Davis and R J Davis Fundamentals of Chemical ReactionEngineering MGraw-Hill Companies 2003

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Effective Root-Finding Methods for ...downloads.hindawi.com/journals/jmath/2016/8174610.pdfResearch Article Effective Root-Finding Methods for Nonlinear Equations

Journal of Mathematics 3

point of function 119892(119909) The multiplicative Taylor theorem(Theorem 2) of degree of 1 with its reminder term can begiven as

119891 (119909) asymp 119891 (1198980) 119891lowast (119898

0)(119909minus119898

0)

119891lowastlowast (119888)(119909minus119898

0)2

2 (11)

where 119888 isin [1198980 119909] Substituting 119909 = 119898 into (11) gives

119891 (1198980) 119891lowast (119898

0)(119898minus119898

0)

119891lowastlowast (119888)(119898minus119898

0)2

2 cong 1 (12)

whenever 119891(119898) = 1 If 1198980is close enough to 119898 we can

consider first two terms of (12) as

119891 (1198980) 119891lowast (119898

0)(119898minus119898

0)

asymp 1 (13)

Thus

119898 = 1198980minus (

ln119891 (1198980)

ln119891lowast (1198980)) (14)

Letting119898119896rarr 119898 and119898

119896minus1rarr 1198980yields (10)

22 Volterra (Bigeometric) Newton-Raphson Methods (AVM)The following theorem states the iteration of root-findingalgorithms with the convergent criteria in the framework ofVolterra calculus

Theorem 6 (Volterra Newton-Raphson) Assume that 119891 isin

1198622[119892] and there exist a number V isin [119886 119887] such that119891(V) = 1 If119891120587(V) = 1 and 119892(119909) = 119909 minus 119909 ln119891(119909) ln119891120587(119909) then there exista 120575 gt 0 such that the sequence V

119896infin119896=1

defined by iteration

V119896= 119892 (V

119896minus1) = V119896minus1minus V119896minus1

ln119891 (V119896minus1)

ln119891120587 (V119896minus1)

for 119896 = 1 2

(15)

will converge to V for any initial value V0isin [V minus 120575 V + 120575]

Proof Insight can be easily given of the proof of multiplica-tive Newton-Raphson method

It is very important to note that methods (10) and (15)are identical because first two terms of multiplicative andVolterra Taylor theorems in the derivation ofNewtonmethodyield same iterations

Alternatively the following theorem can be considered asa modification of Volterra Newton-Raphson method

Theorem 7 (alternative Volterra Newton-Raphson) Assumethat 119891 isin 1198622[119892] and there exist a number V isin [119886 119887] such that119891(V) = 1 If 119891120587(V) = 1 119892(119909) = 119909 minus 119909 ln119891(119909) ln119891120587(119909)and ℎ(119909) = (1199092)(32 minus ln119891(119909) ln119891120587(119909) + (12)(1 minusln119891(119909) ln119891120587(119909))2) then there exists 120575 gt 0 such that thesequence V

119896infin119896=1

defined by iteration

V119896= ℎ (V

119896minus1) =

V119896minus1

2(3

2minus

ln119891 (V119896minus1)

ln119891120587 (V119896minus1)

+1

2(1 minus

ln119891 (V119896minus1)

ln119891120587 (V119896minus1))

2

) for 119896 = 1 2

(16)

will converge to V for any initial value V0isin [V minus 120575 V + 120575]

Proof Insight can be easily given of the proof of multiplica-tive Newton-Raphson method

3 Perturbed Root-Finding Methods viaMultiplicative Calculi

In this section multiplicative perturbed root-finding meth-ods are derived based on corresponding Taylor theoremsThe effort to derive these methods will provide betterapproximation with less computational time and complexityThe number of perturbed terms in corresponding ordinaryperturbed methods (see [20]) should be increased to havethe same approximations as multiplicative perturbed meth-ods for many nonlinear equations The ordinary perturbedmethod (OP) with two terms can be given as (see [20 22])

119909119899+1= 119909119899minus [

119891 (119909119899)

1198911015840 (119909119899)+1

2

11989110158401015840 (119909119899) (119891 (119909

119899))2

(1198911015840 (119909119899))3

] (17)

The multiplicative perturbed methods with two terms arealternatively derived Numerical results tabulated in nextsection demonstrate efficiency of the proposed method forthe solutions of nonlinear equations

31 Multiplicative Perturbed Method We will start withthe multiplicative Taylor theorem to derive correspondingmultiplicative perturbedmethodThe first three factors in themultiplicative Taylor theorem can be given as

119891 (119909 + ℎ) asymp 119891 (119909) 119891lowast

(119909)ℎ 119891lowastlowast (119909)

ℎ2

2 (18)

Assume that there exists at least a value 119909 such that thefunction 119891(119909) = 1 Representing the root value withperturbation terms as 119909 = 119909

0+ 1205761199091+ 12059821199092and substituting

them into (18) give

119891 (1199090+ 1205761199091+ 12059821199092) asymp 119891 (119909

0) [119891lowast (119909

0)]1205761199091

sdot (119891lowast (1199090))1205982

1199092 (119891lowastlowast (119909

0))1205762

1199092

12

(119891lowastlowast (1199090))1205763

11990911199092

sdot (119891lowastlowast (1199090))1205764

1199092

22

(19)

Omitting the last two terms in (19) and setting 1 give

119891 (1199090) 119891lowast (119909

0)1205761199091 = 1

(119891lowast (1199090))1205982

1199092 (119891lowastlowast (119909

0))1205762

1199092

12

= 1

(20)

The solutions of the equations in (20) are respectively

1205761199091= minus

ln119891 (1199090)

ln119891lowast (1199090)

12059821199092=minus (12) (ln119891 (119909

0) ln119891lowast (119909

0))2 ln119891lowastlowast (119909

0)

ln119891lowast (1199090)

(21)

4 Journal of Mathematics

Let the initial estimation of 119909 where 119891(119909) = 1 to be denotedas 1199090 According to the equations in (21) the second iterative

value 1199091can be computed as

1199091= 1199090minus

ln119891 (1199090)

ln119891lowast (1199090)minus1

2

ln119891lowastlowast (1199090) (ln119891 (119909

0))2

(ln119891lowast (1199090))3

(22)

Let 1199091= 119909119899+1

and 1199090= 119909119899in (22) Then

119909119899+1= 119909119899minus

ln119891 (119909119899)

ln119891lowast (119909119899)minus1

2

ln119891lowastlowast (119909119899) (ln119891 (119909

119899))2

(ln119891lowast (119909119899))3

(23)

Equation (23) expresses multiplicative perturbed iteration(MP)

32 Volterra (Bigeometric) Perturbed Method (VP) We willderive Volterra perturbed method in the framework ofVolterra calculus The starting point is (9) in the form

119891 (119909 + ℎ) asymp 119891 (119909) 119891120587

(119909)ℎ119909 119891120587120587 (119909)

ℎ2

21199092

(24)

Assume that there exists at least a value 119909 such that thefunction119891(119909) = 1 Let the root value with perturbation termsbe represented as 119909 = 119909

0+ 1205761199091+ 12059821199092and substituting it in

(24) gives

119891 (1199090+ 1205761199091+ 12059821199092) asymp 119891 (119909

0) 119891120587 (119909

0)12057611990911199090

sdot 119891120587 (1199090)1205982

11990921199090 119891120587120587 (119909

0)1205982

1199092

12(1199090)2

sdot 119891120587120587 (1199090)1205763

11990911199092(1199090)2

119891120587120587 (1199090)1205764

1199092

22(1199090)2

(25)

Omitting the last two terms in (25) and setting it to 1 we have

1205761199091= 1199090[minus

ln119891 (1199090)

ln119891120587 (1199090)]

12059821199092= minus1199090[ln119891120587120587 (119909

0) (ln119891 (119909

0))2

2 (ln119891120587 (1199090))3

]

(26)

If the initial estimation of 119891(119909) = 1 is 1199090 the second iterative

value 1199091can be computed by using (26) as

1199091

= 1199090[1 minus

ln119891 (1199090)

ln119891120587 (1199090)minus1

2

ln119891120587120587 (1199090) (ln119891 (119909

0))2

(ln119891120587 (1199090))3

] (27)

Let 1199091= 119909119899+1

and 1199090= 119909119899in (27) Then we obtain

119909119899+1

= 119909119899[1 minus

ln119891 (119909119899)

ln119891120587 (119909119899)minus1

2

ln119891120587120587 (119909119899) (ln119891 (119909

119899))2

(ln119891120587 (119909119899))3

] (28)

Equation (28) expresses Volterra perturbed iteration (VP)

4 Convergence Criterion of Proposed Methods

The selection of initial value is also very important for theconvergence of the multiplicative root-finding algorithms Inthis section the criterion of the convergence of the proposedmethods will be given which will also lead to a selection ofthe starting point for any given problem Before giving theconvergence of the multiplicative Newton-Raphson methodsdue to the initial value the fixed point theorems shouldbe derived in the multiplicative sense Additionally multi-plicative Rolle and Mean Value theorems discussed in [2]should be considered Moreover Volterra Rolle and VolterraMeanValue theorems which were firstly given in [16] shouldexplicitly be derived

Theorem 8 Suppose that 119892lowast is defined over (119886 119887) and let apositive constant 119896 lt 119890 (27182818 )with |119892lowast(119909)119892(119909)|lowast le 119896 lt119890 for all 119909 isin (119886 119887)Then 119892 has unique fixed point 119865 in [119886 119887]

Proof Let 1198651and 1198652be two fixed pointsThen by multiplica-

tive Mean Value theorem there exist 119888 isin (119886 119887) such that

(119890119892(119888))lowast

= (119890119892(1198652)

119890119892(1198651))

1(1198652minus1198651)

= 119890ln(11989011986521198901198651)1(1198652minus1198651)

= 119890 (29)

which contradicts the statementTherefore119892(119909) has a uniquefixed point 119865 in [119886 119887]

By the assumption in multiplicative Newton-Raphsonmethod 119891(119898) = 1 and thus 119892lowast(119898) = 1 The above theoremproduces a sufficient condition for initial value 119898

0to yield a

sequence 119898119896 for 119896 = 0 1 2 for the root of 119891(119898) = 1 so

that1198980isin (119898 minus 120575119898 + 120575) and 120575 can be selected such that

(119890119892(119909))lowast

lt 119890 (30)

for all 119909 isin (119898 minus 120575119898 + 120575)Hence the condition in (30) is convergence criteria of the

initial value of the proposed multiplicative methodAnalogously it is possible to derive similar condition for

the corresponding Volterra method

Theorem 9 (Volterra Rollersquos theorem) Assume that 119891 isin119862[119886 119887] and 119891 is positive on [119886 119887] Let 119891120587(119909) exist for all 119909 isin(119886 119887) If 119891(119886) = 119891(119887) = 1 then exist119888 isin (119886 119887) such that

119891120587 (119888) = 1 (31)

Proof It can easily be given by proof of multiplicative Rollersquostheorem

Theorem 10 (Volterra Mean Value theorem) Assume that119891 isin 119862[119886 119887] and 119891 is positive on [119886 119887] Let 119891120587(119909) exist for all119909 isin (119886 119887)Then exist119888 isin (119886 119887) such that

119891120587 (119888) = [119891 (119887)

119891 (119886)]1(ln 119887minusln 119886)

(32)

Proof Let us consider the function

119865 (119905) = 119891 (119886) [119891 (119887)

119891 (119886)](ln 119905minusln 119886)(ln 119887minusln 119886)

(33)

Journal of Mathematics 5

Then 119865(119886) = 119891(119886) and 119865(119887) = 119891(119887) Let 119866(119905) = 119865(119905)119891(119905)so that 119866(119886) = 119866(119887) = 1 By Volterra Rollersquos theorem exist119888 isin(119886 119887) such that

119866120587 (119888) =119865120587 (119905)

119891120587 (119905)= 1 (34)

Hence 119865120587(119905) = 119891120587(119905) = 119890119905[(ln119891(119887)minusln119891(119886))119905(ln 119887minusln 119886)] whichgives formula (32)

Theorem 11 Suppose that 119892120587 is defined over (119886 119887) and let apositive constant 119870 le 119890 with |[119892(119909)120587]|lowast le 119870 le 119890119909 for all 119909 isin(119886 119887) where 119890 = 27182818 Then 119892 has unique fixed point119865 in [119886 119887]

Proof It can be easily shown by using Mean Value theoremin Volterra calculus

Sufficient condition for initial value V0to yield a conver-

gence sequence V119896 for 119896 = 0 1 2 for the root of 119891(V) = 1

is that V0isin (119898 minus 120575119898 + 120575) and 120575 can be selected such that

[119892 (119909)120587] lt 119890119909 (35)

for all 119909 isin (119898 minus 120575119898 + 120575)Hence condition in (35) is convergence criteria of the

initial value of the proposed Volterra method

5 Some Numerical Results ofProposed Methods

In this section some examples will be considered to reveal theapplicability of the introduced methods Numerical resultsare reported which indicate that the proposed methods mayallow a considerable saving in both the number of step sizesand reduction of computational cost Besides the examplesconsisting of different type of functions reveal the advantagesof proposed methods compared to the ordinary methods

51 Comparisons of Multiplicative and Ordinary Methods Itis important to note that finding a zero of the function ℎ(119909)at 119909 = 119903 is accomplished by finding the multiplicative root119903 such that 119891(119903) = ℎ(119903) + 1 = 1 The numerical results ofsome nonlinear equations using proposed and ordinary root-finding methods are listed in Tables 1 and 2

Displayed in Table 1 is the number of function evaluationsrequired such that |1 minus 119891(119909

119899)119891(119909

119890)| lt 10minus15 The functions

in Table 1 with their roots 119909119890 respectively are

1198911(119909) = 10119890

minus1199092

minus 1 119909119890= 16796306104284499

1198912(119909) = sin119909 minus 1

2119909 119909

119890= 18954942670339809

1198913(119909) = tan119909 minus tanh119909

119909119890= 39266023120479185

Table 1 OP represents ordinary perturbation iterations MP rep-resents multiplicative perturbation iterations and VP representsVolterra perturbation iterations 119909

0represent initial value NC states

that the given method does not converge for the given zero offunction 119909

119890

119891 (119909119899) 119909

0OP MP VP

1198911

25 NC 4 4

055 NC 6 6

1198912

07 NC 5 NC2 3 3 4

1198913

45 5 4 3

35 3 3 3

1198914

25 8 4 6

075 NC 3 4

1198915

265 NC 6 6

225 4 4 4

1198916

175 5 3 4

25 NC 4 6

Table 2 NM represents ordinary Newton-Raphson iterations MNrepresents multiplicative Newton-Raphson iterations and AVNrepresents alternative Volterra Newton-Raphson iterations 119909

0rep-

resent initial value NC states that the given method does notconverge for the given zero of function 119909

119890

119891(119909119899) 119909

0NM MN AVN

1198911

25 NC 5 5

075 NC 8 11

1198912

16 5 5 5

2 4 4 4

1198913

45 7 5 5

35 6 4 5

1198914

25 12 4 5

075 11 4 5

1198915

3 7 6 6

225 4 3 4

1198916

175 7 6 6

25 NC 6 5

1198914(119909) = 119909119890

1199092

minus sin2119909 + 3 cos119909 minus 4

119909119890= 10651360157761873

1198915(119909) = 119890

minus119909 + cos119909 119909119890= 17461395304080125

1198916(119909) = 119909 minus ln (1199092 minus 3) minus 2

119909119890= 17461395304080125

(36)

According to the obtained results multiplicative root-finding algorithms can be used effectively and efficientlyin real applications mentioned in section one Moreoverthese methods yield better approximations for the nonlinearequations especially when the equations involve exponentiallogarithmic and hyperbolic function On the other hand the

6 Journal of Mathematics

Table 3The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (38) 119909119890= 05576473009191445

119891(119909119899) 119909

0OP MP VP NM MN AVN

175 NC 4 4 NC 6 6

125 NC 4 7 14 5 7

Table 4The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (41) 119909119890= 21954

119891(119909119899) 119909

0OP MP VP NM MN AVN

19 NC 6 NC 7 5 5

16 3 3 4 3 3 4

ordinary methods can give more accurate results especiallyfor polynomial equations Thus the method should beselected according to the functions appeared in the equations

52 Realistic Applications In this subsection two exampleswill be considered to demonstrate possible impacts of theintroduced methods to science and engineering

Example 12 The process in which chemicals interact to formnew chemicals with different compositions is called chemicalreactionsThis process is the results of chemical properties ofthe element or compound causing changes in compositionThese chemical changes are chemistsrsquo main purpose It isimportant to mention a remark from [23] where ldquothe twoquestions that must be answered for a chemically reactingsystem are (1) what changes are expected to occur and (2)how fast will they occurrdquo This is an indication that math-ematical representations are very important for chemicalinvestigations Consequently a chemical reaction is mainlyrepresented by a chemical equation which represents thechange from reactants to products This process generallyinvolves nonlinear functions so it is compulsory to useand apply numerical approaches Exemplarily suppose thata chemical reaction is made and the concentration of aparticular ion at the time 119905 is given by a nonlinear function

119891 (119909) = 5119890minus3119909 + 119890minus5119909 (37)

If we are interested in when this concentration will be one-half of its value at initial time 0 we need to solve this problemnumerically If 119891(0) = 2 as an initial assumption this will beequivalent to finding a root of nonlinear equation

5119890minus3119909 + 119890minus5119909 minus 1 = 0 (38)

Usually chemists tend to use ordinary numerical methodsto estimate the root of (38) However the multiplicativebase methods whose effectiveness was proved in problemsthat involve exponential functions should not be disregardedin many applications The superiority of the multiplicativenumerical methods can be easily observed for this problemby Table 3

Example 13 The Rayleigh function which corresponds toRayleigh distribution

119891 (119909) =119909

120590119890minus1199092

21205902

for arbitrary 119886 119887 119888 isin 119877 (39)

plays an important role in magnetic resonance imaging(MRI) and probability theory It is a striking example in orderto show the efficiency of the multiplicative and Volterra root-finding methods Assume that the density function of thecontinuous random variable119883 on the interval [0infin) is givenby

119891 (119909) = 119909119890minus1199092

2 where 120590 = 1 (40)

It may be interesting to consider the root of the equation

120572 minus 119891 (119909) = 120572 minus 119909119890minus1199092

2 = 0 0 le 120572 le 1 (41)

which estimates the point119909 according to the given probability120572 of an event in119883 We attempt to use root-finding algorithmsfor (41) for 120572 = 019720178928946303 which will be given byTable 4

6 Conclusion

In this study the multiplicative and Volterra based root-finding methods are presented These methods were testedfor some nontrivial problems and compared with the originalroot-finding method The results show that in certain prob-lems the multiplicative andor Volterra methods give moreaccurate results compared to the original root-finding meth-ods Especially the examples showed that the nature of theunderlying calculus plays an important role in approximatingthe zeros of the function

The selection of the initial value is very important for theconvergence of the iteration Two theorems in Section 4 indi-cate the conditions for convergence related to multiplicativeand Volterra methods respectively Section 4 also highlightsoptimal selection of the initial value Evidently in certain sit-uations the multiplicative andor Volterra methods convergefaster compared to the ordinary methods Particularly it canbe easily observed that multiplicative and Volterra Newton-Raphson methods are more accurate than ordinary Newton-Raphson method in many applications Therefore these

Journal of Mathematics 7

methods based on multiplicative calculi have proven theirimportance in the process of numerical approximations ofnonlinear equations Also the numerical results obtained inthe paper encourage the usage of multiplicative and Volterramethods for solving nonlinear equations

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Grossman and R Katz Non-Newtonian Calculus Lee PressPigeon Cove Mass USA 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] L Florack and H V Assen ldquoMultiplicative calculus in biomedi-cal image analysisrdquo Journal ofMathematical Imaging andVisionvol 42 no 1 pp 64ndash75 2012

[4] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[5] D A Filip and C Piatecki ldquoA non-Newtonian examinationof the theory of exogenous economic growthrdquo CNCSISmdashUEFISCSU (project number PNII IDEI 23662008) and Lab-oratoire dEconomie dOrleans (LEO) 2010

[6] E Misirli and Y Gurefe ldquoMultiplicative adams bashforthmdashmoulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011

[7] M Riza A Ozyapıcı and E Misirli ldquoMultiplicative finitedifference methodsrdquo Quarterly of Applied Mathematics vol 67no 4 pp 745ndash754 2009

[8] A Ozyapici and E Misirli ldquoExponential approximations onmultiplicative calculusrdquo Proceedings of the Jangjeon Mathemati-cal Society vol 12 no 2 pp 227ndash236 2009

[9] A Ozyapici and B Bilgehan ldquoFinite product representation viamultiplicative calculus and its applications to exponential signalprocessingrdquo Numerical Algorithms vol 71 no 2 pp 475ndash4892016

[10] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[11] J Englehardt J Swartout and C Loewenstine ldquoA new theoret-ical discrete growth distribution with verification for microbialcounts in waterrdquo Risk Analysis vol 29 no 6 pp 841ndash856 2009

[12] H Ozyapıcı I Dalcı and A Ozyapıcı ldquoIntegrating accountingand multiplicative calculus an effective estimation of learningcurverdquo Computational and Mathematical Organization Theory2016

[13] B Bilgehan ldquoEfficient approximation for linear and non-linearsignal representationrdquo IET Signal Processing vol 9 no 3 pp260ndash266 2015

[14] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007

[15] WKasprzak B Lysik andM RybaczukDimensions InvariantsModels and Fractals Ukrainian Society on Fracture MechanicsSPOLOM Wroclaw-Lviv Poland 2004

[16] M Grossman Bigeometric Calculus A System with a Scale-FreeDerivative Archimedes Foundation RockportMass USA1983

[17] M Riza and B Eminagaı ldquoBigeometric calculusmdasha modellingtoolrdquo httpsarxivorgabs14022877v1

[18] M A Noor W A Khan and A Hussain ldquoA newmodified Hal-ley method without second derivatives for nonlinear equationrdquoAppliedMathematics and Computation vol 189 no 2 pp 1268ndash1273 2007

[19] A Ramli M L Abdullah and M Mamat ldquoBroydenrsquos methodfor solving fuzzy nonlinear equationsrdquo Advances in FuzzySystems vol 2010 Article ID 763270 6 pages 2010

[20] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquo Mathematical and ComputationalApplications vol 12 no 2 pp 117ndash124 2007

[21] E Misirli and Y Gurefe Multiplicative calculus and its applica-tions [MS thesis] Turkish Council of Higher Education 2009Thesis No 252639

[22] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[23] M E Davis and R J Davis Fundamentals of Chemical ReactionEngineering MGraw-Hill Companies 2003

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 4: Research Article Effective Root-Finding Methods for ...downloads.hindawi.com/journals/jmath/2016/8174610.pdfResearch Article Effective Root-Finding Methods for Nonlinear Equations

4 Journal of Mathematics

Let the initial estimation of 119909 where 119891(119909) = 1 to be denotedas 1199090 According to the equations in (21) the second iterative

value 1199091can be computed as

1199091= 1199090minus

ln119891 (1199090)

ln119891lowast (1199090)minus1

2

ln119891lowastlowast (1199090) (ln119891 (119909

0))2

(ln119891lowast (1199090))3

(22)

Let 1199091= 119909119899+1

and 1199090= 119909119899in (22) Then

119909119899+1= 119909119899minus

ln119891 (119909119899)

ln119891lowast (119909119899)minus1

2

ln119891lowastlowast (119909119899) (ln119891 (119909

119899))2

(ln119891lowast (119909119899))3

(23)

Equation (23) expresses multiplicative perturbed iteration(MP)

32 Volterra (Bigeometric) Perturbed Method (VP) We willderive Volterra perturbed method in the framework ofVolterra calculus The starting point is (9) in the form

119891 (119909 + ℎ) asymp 119891 (119909) 119891120587

(119909)ℎ119909 119891120587120587 (119909)

ℎ2

21199092

(24)

Assume that there exists at least a value 119909 such that thefunction119891(119909) = 1 Let the root value with perturbation termsbe represented as 119909 = 119909

0+ 1205761199091+ 12059821199092and substituting it in

(24) gives

119891 (1199090+ 1205761199091+ 12059821199092) asymp 119891 (119909

0) 119891120587 (119909

0)12057611990911199090

sdot 119891120587 (1199090)1205982

11990921199090 119891120587120587 (119909

0)1205982

1199092

12(1199090)2

sdot 119891120587120587 (1199090)1205763

11990911199092(1199090)2

119891120587120587 (1199090)1205764

1199092

22(1199090)2

(25)

Omitting the last two terms in (25) and setting it to 1 we have

1205761199091= 1199090[minus

ln119891 (1199090)

ln119891120587 (1199090)]

12059821199092= minus1199090[ln119891120587120587 (119909

0) (ln119891 (119909

0))2

2 (ln119891120587 (1199090))3

]

(26)

If the initial estimation of 119891(119909) = 1 is 1199090 the second iterative

value 1199091can be computed by using (26) as

1199091

= 1199090[1 minus

ln119891 (1199090)

ln119891120587 (1199090)minus1

2

ln119891120587120587 (1199090) (ln119891 (119909

0))2

(ln119891120587 (1199090))3

] (27)

Let 1199091= 119909119899+1

and 1199090= 119909119899in (27) Then we obtain

119909119899+1

= 119909119899[1 minus

ln119891 (119909119899)

ln119891120587 (119909119899)minus1

2

ln119891120587120587 (119909119899) (ln119891 (119909

119899))2

(ln119891120587 (119909119899))3

] (28)

Equation (28) expresses Volterra perturbed iteration (VP)

4 Convergence Criterion of Proposed Methods

The selection of initial value is also very important for theconvergence of the multiplicative root-finding algorithms Inthis section the criterion of the convergence of the proposedmethods will be given which will also lead to a selection ofthe starting point for any given problem Before giving theconvergence of the multiplicative Newton-Raphson methodsdue to the initial value the fixed point theorems shouldbe derived in the multiplicative sense Additionally multi-plicative Rolle and Mean Value theorems discussed in [2]should be considered Moreover Volterra Rolle and VolterraMeanValue theorems which were firstly given in [16] shouldexplicitly be derived

Theorem 8 Suppose that 119892lowast is defined over (119886 119887) and let apositive constant 119896 lt 119890 (27182818 )with |119892lowast(119909)119892(119909)|lowast le 119896 lt119890 for all 119909 isin (119886 119887)Then 119892 has unique fixed point 119865 in [119886 119887]

Proof Let 1198651and 1198652be two fixed pointsThen by multiplica-

tive Mean Value theorem there exist 119888 isin (119886 119887) such that

(119890119892(119888))lowast

= (119890119892(1198652)

119890119892(1198651))

1(1198652minus1198651)

= 119890ln(11989011986521198901198651)1(1198652minus1198651)

= 119890 (29)

which contradicts the statementTherefore119892(119909) has a uniquefixed point 119865 in [119886 119887]

By the assumption in multiplicative Newton-Raphsonmethod 119891(119898) = 1 and thus 119892lowast(119898) = 1 The above theoremproduces a sufficient condition for initial value 119898

0to yield a

sequence 119898119896 for 119896 = 0 1 2 for the root of 119891(119898) = 1 so

that1198980isin (119898 minus 120575119898 + 120575) and 120575 can be selected such that

(119890119892(119909))lowast

lt 119890 (30)

for all 119909 isin (119898 minus 120575119898 + 120575)Hence the condition in (30) is convergence criteria of the

initial value of the proposed multiplicative methodAnalogously it is possible to derive similar condition for

the corresponding Volterra method

Theorem 9 (Volterra Rollersquos theorem) Assume that 119891 isin119862[119886 119887] and 119891 is positive on [119886 119887] Let 119891120587(119909) exist for all 119909 isin(119886 119887) If 119891(119886) = 119891(119887) = 1 then exist119888 isin (119886 119887) such that

119891120587 (119888) = 1 (31)

Proof It can easily be given by proof of multiplicative Rollersquostheorem

Theorem 10 (Volterra Mean Value theorem) Assume that119891 isin 119862[119886 119887] and 119891 is positive on [119886 119887] Let 119891120587(119909) exist for all119909 isin (119886 119887)Then exist119888 isin (119886 119887) such that

119891120587 (119888) = [119891 (119887)

119891 (119886)]1(ln 119887minusln 119886)

(32)

Proof Let us consider the function

119865 (119905) = 119891 (119886) [119891 (119887)

119891 (119886)](ln 119905minusln 119886)(ln 119887minusln 119886)

(33)

Journal of Mathematics 5

Then 119865(119886) = 119891(119886) and 119865(119887) = 119891(119887) Let 119866(119905) = 119865(119905)119891(119905)so that 119866(119886) = 119866(119887) = 1 By Volterra Rollersquos theorem exist119888 isin(119886 119887) such that

119866120587 (119888) =119865120587 (119905)

119891120587 (119905)= 1 (34)

Hence 119865120587(119905) = 119891120587(119905) = 119890119905[(ln119891(119887)minusln119891(119886))119905(ln 119887minusln 119886)] whichgives formula (32)

Theorem 11 Suppose that 119892120587 is defined over (119886 119887) and let apositive constant 119870 le 119890 with |[119892(119909)120587]|lowast le 119870 le 119890119909 for all 119909 isin(119886 119887) where 119890 = 27182818 Then 119892 has unique fixed point119865 in [119886 119887]

Proof It can be easily shown by using Mean Value theoremin Volterra calculus

Sufficient condition for initial value V0to yield a conver-

gence sequence V119896 for 119896 = 0 1 2 for the root of 119891(V) = 1

is that V0isin (119898 minus 120575119898 + 120575) and 120575 can be selected such that

[119892 (119909)120587] lt 119890119909 (35)

for all 119909 isin (119898 minus 120575119898 + 120575)Hence condition in (35) is convergence criteria of the

initial value of the proposed Volterra method

5 Some Numerical Results ofProposed Methods

In this section some examples will be considered to reveal theapplicability of the introduced methods Numerical resultsare reported which indicate that the proposed methods mayallow a considerable saving in both the number of step sizesand reduction of computational cost Besides the examplesconsisting of different type of functions reveal the advantagesof proposed methods compared to the ordinary methods

51 Comparisons of Multiplicative and Ordinary Methods Itis important to note that finding a zero of the function ℎ(119909)at 119909 = 119903 is accomplished by finding the multiplicative root119903 such that 119891(119903) = ℎ(119903) + 1 = 1 The numerical results ofsome nonlinear equations using proposed and ordinary root-finding methods are listed in Tables 1 and 2

Displayed in Table 1 is the number of function evaluationsrequired such that |1 minus 119891(119909

119899)119891(119909

119890)| lt 10minus15 The functions

in Table 1 with their roots 119909119890 respectively are

1198911(119909) = 10119890

minus1199092

minus 1 119909119890= 16796306104284499

1198912(119909) = sin119909 minus 1

2119909 119909

119890= 18954942670339809

1198913(119909) = tan119909 minus tanh119909

119909119890= 39266023120479185

Table 1 OP represents ordinary perturbation iterations MP rep-resents multiplicative perturbation iterations and VP representsVolterra perturbation iterations 119909

0represent initial value NC states

that the given method does not converge for the given zero offunction 119909

119890

119891 (119909119899) 119909

0OP MP VP

1198911

25 NC 4 4

055 NC 6 6

1198912

07 NC 5 NC2 3 3 4

1198913

45 5 4 3

35 3 3 3

1198914

25 8 4 6

075 NC 3 4

1198915

265 NC 6 6

225 4 4 4

1198916

175 5 3 4

25 NC 4 6

Table 2 NM represents ordinary Newton-Raphson iterations MNrepresents multiplicative Newton-Raphson iterations and AVNrepresents alternative Volterra Newton-Raphson iterations 119909

0rep-

resent initial value NC states that the given method does notconverge for the given zero of function 119909

119890

119891(119909119899) 119909

0NM MN AVN

1198911

25 NC 5 5

075 NC 8 11

1198912

16 5 5 5

2 4 4 4

1198913

45 7 5 5

35 6 4 5

1198914

25 12 4 5

075 11 4 5

1198915

3 7 6 6

225 4 3 4

1198916

175 7 6 6

25 NC 6 5

1198914(119909) = 119909119890

1199092

minus sin2119909 + 3 cos119909 minus 4

119909119890= 10651360157761873

1198915(119909) = 119890

minus119909 + cos119909 119909119890= 17461395304080125

1198916(119909) = 119909 minus ln (1199092 minus 3) minus 2

119909119890= 17461395304080125

(36)

According to the obtained results multiplicative root-finding algorithms can be used effectively and efficientlyin real applications mentioned in section one Moreoverthese methods yield better approximations for the nonlinearequations especially when the equations involve exponentiallogarithmic and hyperbolic function On the other hand the

6 Journal of Mathematics

Table 3The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (38) 119909119890= 05576473009191445

119891(119909119899) 119909

0OP MP VP NM MN AVN

175 NC 4 4 NC 6 6

125 NC 4 7 14 5 7

Table 4The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (41) 119909119890= 21954

119891(119909119899) 119909

0OP MP VP NM MN AVN

19 NC 6 NC 7 5 5

16 3 3 4 3 3 4

ordinary methods can give more accurate results especiallyfor polynomial equations Thus the method should beselected according to the functions appeared in the equations

52 Realistic Applications In this subsection two exampleswill be considered to demonstrate possible impacts of theintroduced methods to science and engineering

Example 12 The process in which chemicals interact to formnew chemicals with different compositions is called chemicalreactionsThis process is the results of chemical properties ofthe element or compound causing changes in compositionThese chemical changes are chemistsrsquo main purpose It isimportant to mention a remark from [23] where ldquothe twoquestions that must be answered for a chemically reactingsystem are (1) what changes are expected to occur and (2)how fast will they occurrdquo This is an indication that math-ematical representations are very important for chemicalinvestigations Consequently a chemical reaction is mainlyrepresented by a chemical equation which represents thechange from reactants to products This process generallyinvolves nonlinear functions so it is compulsory to useand apply numerical approaches Exemplarily suppose thata chemical reaction is made and the concentration of aparticular ion at the time 119905 is given by a nonlinear function

119891 (119909) = 5119890minus3119909 + 119890minus5119909 (37)

If we are interested in when this concentration will be one-half of its value at initial time 0 we need to solve this problemnumerically If 119891(0) = 2 as an initial assumption this will beequivalent to finding a root of nonlinear equation

5119890minus3119909 + 119890minus5119909 minus 1 = 0 (38)

Usually chemists tend to use ordinary numerical methodsto estimate the root of (38) However the multiplicativebase methods whose effectiveness was proved in problemsthat involve exponential functions should not be disregardedin many applications The superiority of the multiplicativenumerical methods can be easily observed for this problemby Table 3

Example 13 The Rayleigh function which corresponds toRayleigh distribution

119891 (119909) =119909

120590119890minus1199092

21205902

for arbitrary 119886 119887 119888 isin 119877 (39)

plays an important role in magnetic resonance imaging(MRI) and probability theory It is a striking example in orderto show the efficiency of the multiplicative and Volterra root-finding methods Assume that the density function of thecontinuous random variable119883 on the interval [0infin) is givenby

119891 (119909) = 119909119890minus1199092

2 where 120590 = 1 (40)

It may be interesting to consider the root of the equation

120572 minus 119891 (119909) = 120572 minus 119909119890minus1199092

2 = 0 0 le 120572 le 1 (41)

which estimates the point119909 according to the given probability120572 of an event in119883 We attempt to use root-finding algorithmsfor (41) for 120572 = 019720178928946303 which will be given byTable 4

6 Conclusion

In this study the multiplicative and Volterra based root-finding methods are presented These methods were testedfor some nontrivial problems and compared with the originalroot-finding method The results show that in certain prob-lems the multiplicative andor Volterra methods give moreaccurate results compared to the original root-finding meth-ods Especially the examples showed that the nature of theunderlying calculus plays an important role in approximatingthe zeros of the function

The selection of the initial value is very important for theconvergence of the iteration Two theorems in Section 4 indi-cate the conditions for convergence related to multiplicativeand Volterra methods respectively Section 4 also highlightsoptimal selection of the initial value Evidently in certain sit-uations the multiplicative andor Volterra methods convergefaster compared to the ordinary methods Particularly it canbe easily observed that multiplicative and Volterra Newton-Raphson methods are more accurate than ordinary Newton-Raphson method in many applications Therefore these

Journal of Mathematics 7

methods based on multiplicative calculi have proven theirimportance in the process of numerical approximations ofnonlinear equations Also the numerical results obtained inthe paper encourage the usage of multiplicative and Volterramethods for solving nonlinear equations

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Grossman and R Katz Non-Newtonian Calculus Lee PressPigeon Cove Mass USA 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] L Florack and H V Assen ldquoMultiplicative calculus in biomedi-cal image analysisrdquo Journal ofMathematical Imaging andVisionvol 42 no 1 pp 64ndash75 2012

[4] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[5] D A Filip and C Piatecki ldquoA non-Newtonian examinationof the theory of exogenous economic growthrdquo CNCSISmdashUEFISCSU (project number PNII IDEI 23662008) and Lab-oratoire dEconomie dOrleans (LEO) 2010

[6] E Misirli and Y Gurefe ldquoMultiplicative adams bashforthmdashmoulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011

[7] M Riza A Ozyapıcı and E Misirli ldquoMultiplicative finitedifference methodsrdquo Quarterly of Applied Mathematics vol 67no 4 pp 745ndash754 2009

[8] A Ozyapici and E Misirli ldquoExponential approximations onmultiplicative calculusrdquo Proceedings of the Jangjeon Mathemati-cal Society vol 12 no 2 pp 227ndash236 2009

[9] A Ozyapici and B Bilgehan ldquoFinite product representation viamultiplicative calculus and its applications to exponential signalprocessingrdquo Numerical Algorithms vol 71 no 2 pp 475ndash4892016

[10] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[11] J Englehardt J Swartout and C Loewenstine ldquoA new theoret-ical discrete growth distribution with verification for microbialcounts in waterrdquo Risk Analysis vol 29 no 6 pp 841ndash856 2009

[12] H Ozyapıcı I Dalcı and A Ozyapıcı ldquoIntegrating accountingand multiplicative calculus an effective estimation of learningcurverdquo Computational and Mathematical Organization Theory2016

[13] B Bilgehan ldquoEfficient approximation for linear and non-linearsignal representationrdquo IET Signal Processing vol 9 no 3 pp260ndash266 2015

[14] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007

[15] WKasprzak B Lysik andM RybaczukDimensions InvariantsModels and Fractals Ukrainian Society on Fracture MechanicsSPOLOM Wroclaw-Lviv Poland 2004

[16] M Grossman Bigeometric Calculus A System with a Scale-FreeDerivative Archimedes Foundation RockportMass USA1983

[17] M Riza and B Eminagaı ldquoBigeometric calculusmdasha modellingtoolrdquo httpsarxivorgabs14022877v1

[18] M A Noor W A Khan and A Hussain ldquoA newmodified Hal-ley method without second derivatives for nonlinear equationrdquoAppliedMathematics and Computation vol 189 no 2 pp 1268ndash1273 2007

[19] A Ramli M L Abdullah and M Mamat ldquoBroydenrsquos methodfor solving fuzzy nonlinear equationsrdquo Advances in FuzzySystems vol 2010 Article ID 763270 6 pages 2010

[20] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquo Mathematical and ComputationalApplications vol 12 no 2 pp 117ndash124 2007

[21] E Misirli and Y Gurefe Multiplicative calculus and its applica-tions [MS thesis] Turkish Council of Higher Education 2009Thesis No 252639

[22] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[23] M E Davis and R J Davis Fundamentals of Chemical ReactionEngineering MGraw-Hill Companies 2003

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

Page 5: Research Article Effective Root-Finding Methods for ...downloads.hindawi.com/journals/jmath/2016/8174610.pdfResearch Article Effective Root-Finding Methods for Nonlinear Equations

Journal of Mathematics 5

Then 119865(119886) = 119891(119886) and 119865(119887) = 119891(119887) Let 119866(119905) = 119865(119905)119891(119905)so that 119866(119886) = 119866(119887) = 1 By Volterra Rollersquos theorem exist119888 isin(119886 119887) such that

119866120587 (119888) =119865120587 (119905)

119891120587 (119905)= 1 (34)

Hence 119865120587(119905) = 119891120587(119905) = 119890119905[(ln119891(119887)minusln119891(119886))119905(ln 119887minusln 119886)] whichgives formula (32)

Theorem 11 Suppose that 119892120587 is defined over (119886 119887) and let apositive constant 119870 le 119890 with |[119892(119909)120587]|lowast le 119870 le 119890119909 for all 119909 isin(119886 119887) where 119890 = 27182818 Then 119892 has unique fixed point119865 in [119886 119887]

Proof It can be easily shown by using Mean Value theoremin Volterra calculus

Sufficient condition for initial value V0to yield a conver-

gence sequence V119896 for 119896 = 0 1 2 for the root of 119891(V) = 1

is that V0isin (119898 minus 120575119898 + 120575) and 120575 can be selected such that

[119892 (119909)120587] lt 119890119909 (35)

for all 119909 isin (119898 minus 120575119898 + 120575)Hence condition in (35) is convergence criteria of the

initial value of the proposed Volterra method

5 Some Numerical Results ofProposed Methods

In this section some examples will be considered to reveal theapplicability of the introduced methods Numerical resultsare reported which indicate that the proposed methods mayallow a considerable saving in both the number of step sizesand reduction of computational cost Besides the examplesconsisting of different type of functions reveal the advantagesof proposed methods compared to the ordinary methods

51 Comparisons of Multiplicative and Ordinary Methods Itis important to note that finding a zero of the function ℎ(119909)at 119909 = 119903 is accomplished by finding the multiplicative root119903 such that 119891(119903) = ℎ(119903) + 1 = 1 The numerical results ofsome nonlinear equations using proposed and ordinary root-finding methods are listed in Tables 1 and 2

Displayed in Table 1 is the number of function evaluationsrequired such that |1 minus 119891(119909

119899)119891(119909

119890)| lt 10minus15 The functions

in Table 1 with their roots 119909119890 respectively are

1198911(119909) = 10119890

minus1199092

minus 1 119909119890= 16796306104284499

1198912(119909) = sin119909 minus 1

2119909 119909

119890= 18954942670339809

1198913(119909) = tan119909 minus tanh119909

119909119890= 39266023120479185

Table 1 OP represents ordinary perturbation iterations MP rep-resents multiplicative perturbation iterations and VP representsVolterra perturbation iterations 119909

0represent initial value NC states

that the given method does not converge for the given zero offunction 119909

119890

119891 (119909119899) 119909

0OP MP VP

1198911

25 NC 4 4

055 NC 6 6

1198912

07 NC 5 NC2 3 3 4

1198913

45 5 4 3

35 3 3 3

1198914

25 8 4 6

075 NC 3 4

1198915

265 NC 6 6

225 4 4 4

1198916

175 5 3 4

25 NC 4 6

Table 2 NM represents ordinary Newton-Raphson iterations MNrepresents multiplicative Newton-Raphson iterations and AVNrepresents alternative Volterra Newton-Raphson iterations 119909

0rep-

resent initial value NC states that the given method does notconverge for the given zero of function 119909

119890

119891(119909119899) 119909

0NM MN AVN

1198911

25 NC 5 5

075 NC 8 11

1198912

16 5 5 5

2 4 4 4

1198913

45 7 5 5

35 6 4 5

1198914

25 12 4 5

075 11 4 5

1198915

3 7 6 6

225 4 3 4

1198916

175 7 6 6

25 NC 6 5

1198914(119909) = 119909119890

1199092

minus sin2119909 + 3 cos119909 minus 4

119909119890= 10651360157761873

1198915(119909) = 119890

minus119909 + cos119909 119909119890= 17461395304080125

1198916(119909) = 119909 minus ln (1199092 minus 3) minus 2

119909119890= 17461395304080125

(36)

According to the obtained results multiplicative root-finding algorithms can be used effectively and efficientlyin real applications mentioned in section one Moreoverthese methods yield better approximations for the nonlinearequations especially when the equations involve exponentiallogarithmic and hyperbolic function On the other hand the

6 Journal of Mathematics

Table 3The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (38) 119909119890= 05576473009191445

119891(119909119899) 119909

0OP MP VP NM MN AVN

175 NC 4 4 NC 6 6

125 NC 4 7 14 5 7

Table 4The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (41) 119909119890= 21954

119891(119909119899) 119909

0OP MP VP NM MN AVN

19 NC 6 NC 7 5 5

16 3 3 4 3 3 4

ordinary methods can give more accurate results especiallyfor polynomial equations Thus the method should beselected according to the functions appeared in the equations

52 Realistic Applications In this subsection two exampleswill be considered to demonstrate possible impacts of theintroduced methods to science and engineering

Example 12 The process in which chemicals interact to formnew chemicals with different compositions is called chemicalreactionsThis process is the results of chemical properties ofthe element or compound causing changes in compositionThese chemical changes are chemistsrsquo main purpose It isimportant to mention a remark from [23] where ldquothe twoquestions that must be answered for a chemically reactingsystem are (1) what changes are expected to occur and (2)how fast will they occurrdquo This is an indication that math-ematical representations are very important for chemicalinvestigations Consequently a chemical reaction is mainlyrepresented by a chemical equation which represents thechange from reactants to products This process generallyinvolves nonlinear functions so it is compulsory to useand apply numerical approaches Exemplarily suppose thata chemical reaction is made and the concentration of aparticular ion at the time 119905 is given by a nonlinear function

119891 (119909) = 5119890minus3119909 + 119890minus5119909 (37)

If we are interested in when this concentration will be one-half of its value at initial time 0 we need to solve this problemnumerically If 119891(0) = 2 as an initial assumption this will beequivalent to finding a root of nonlinear equation

5119890minus3119909 + 119890minus5119909 minus 1 = 0 (38)

Usually chemists tend to use ordinary numerical methodsto estimate the root of (38) However the multiplicativebase methods whose effectiveness was proved in problemsthat involve exponential functions should not be disregardedin many applications The superiority of the multiplicativenumerical methods can be easily observed for this problemby Table 3

Example 13 The Rayleigh function which corresponds toRayleigh distribution

119891 (119909) =119909

120590119890minus1199092

21205902

for arbitrary 119886 119887 119888 isin 119877 (39)

plays an important role in magnetic resonance imaging(MRI) and probability theory It is a striking example in orderto show the efficiency of the multiplicative and Volterra root-finding methods Assume that the density function of thecontinuous random variable119883 on the interval [0infin) is givenby

119891 (119909) = 119909119890minus1199092

2 where 120590 = 1 (40)

It may be interesting to consider the root of the equation

120572 minus 119891 (119909) = 120572 minus 119909119890minus1199092

2 = 0 0 le 120572 le 1 (41)

which estimates the point119909 according to the given probability120572 of an event in119883 We attempt to use root-finding algorithmsfor (41) for 120572 = 019720178928946303 which will be given byTable 4

6 Conclusion

In this study the multiplicative and Volterra based root-finding methods are presented These methods were testedfor some nontrivial problems and compared with the originalroot-finding method The results show that in certain prob-lems the multiplicative andor Volterra methods give moreaccurate results compared to the original root-finding meth-ods Especially the examples showed that the nature of theunderlying calculus plays an important role in approximatingthe zeros of the function

The selection of the initial value is very important for theconvergence of the iteration Two theorems in Section 4 indi-cate the conditions for convergence related to multiplicativeand Volterra methods respectively Section 4 also highlightsoptimal selection of the initial value Evidently in certain sit-uations the multiplicative andor Volterra methods convergefaster compared to the ordinary methods Particularly it canbe easily observed that multiplicative and Volterra Newton-Raphson methods are more accurate than ordinary Newton-Raphson method in many applications Therefore these

Journal of Mathematics 7

methods based on multiplicative calculi have proven theirimportance in the process of numerical approximations ofnonlinear equations Also the numerical results obtained inthe paper encourage the usage of multiplicative and Volterramethods for solving nonlinear equations

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Grossman and R Katz Non-Newtonian Calculus Lee PressPigeon Cove Mass USA 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] L Florack and H V Assen ldquoMultiplicative calculus in biomedi-cal image analysisrdquo Journal ofMathematical Imaging andVisionvol 42 no 1 pp 64ndash75 2012

[4] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[5] D A Filip and C Piatecki ldquoA non-Newtonian examinationof the theory of exogenous economic growthrdquo CNCSISmdashUEFISCSU (project number PNII IDEI 23662008) and Lab-oratoire dEconomie dOrleans (LEO) 2010

[6] E Misirli and Y Gurefe ldquoMultiplicative adams bashforthmdashmoulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011

[7] M Riza A Ozyapıcı and E Misirli ldquoMultiplicative finitedifference methodsrdquo Quarterly of Applied Mathematics vol 67no 4 pp 745ndash754 2009

[8] A Ozyapici and E Misirli ldquoExponential approximations onmultiplicative calculusrdquo Proceedings of the Jangjeon Mathemati-cal Society vol 12 no 2 pp 227ndash236 2009

[9] A Ozyapici and B Bilgehan ldquoFinite product representation viamultiplicative calculus and its applications to exponential signalprocessingrdquo Numerical Algorithms vol 71 no 2 pp 475ndash4892016

[10] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[11] J Englehardt J Swartout and C Loewenstine ldquoA new theoret-ical discrete growth distribution with verification for microbialcounts in waterrdquo Risk Analysis vol 29 no 6 pp 841ndash856 2009

[12] H Ozyapıcı I Dalcı and A Ozyapıcı ldquoIntegrating accountingand multiplicative calculus an effective estimation of learningcurverdquo Computational and Mathematical Organization Theory2016

[13] B Bilgehan ldquoEfficient approximation for linear and non-linearsignal representationrdquo IET Signal Processing vol 9 no 3 pp260ndash266 2015

[14] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007

[15] WKasprzak B Lysik andM RybaczukDimensions InvariantsModels and Fractals Ukrainian Society on Fracture MechanicsSPOLOM Wroclaw-Lviv Poland 2004

[16] M Grossman Bigeometric Calculus A System with a Scale-FreeDerivative Archimedes Foundation RockportMass USA1983

[17] M Riza and B Eminagaı ldquoBigeometric calculusmdasha modellingtoolrdquo httpsarxivorgabs14022877v1

[18] M A Noor W A Khan and A Hussain ldquoA newmodified Hal-ley method without second derivatives for nonlinear equationrdquoAppliedMathematics and Computation vol 189 no 2 pp 1268ndash1273 2007

[19] A Ramli M L Abdullah and M Mamat ldquoBroydenrsquos methodfor solving fuzzy nonlinear equationsrdquo Advances in FuzzySystems vol 2010 Article ID 763270 6 pages 2010

[20] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquo Mathematical and ComputationalApplications vol 12 no 2 pp 117ndash124 2007

[21] E Misirli and Y Gurefe Multiplicative calculus and its applica-tions [MS thesis] Turkish Council of Higher Education 2009Thesis No 252639

[22] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[23] M E Davis and R J Davis Fundamentals of Chemical ReactionEngineering MGraw-Hill Companies 2003

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

Page 6: Research Article Effective Root-Finding Methods for ...downloads.hindawi.com/journals/jmath/2016/8174610.pdfResearch Article Effective Root-Finding Methods for Nonlinear Equations

6 Journal of Mathematics

Table 3The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (38) 119909119890= 05576473009191445

119891(119909119899) 119909

0OP MP VP NM MN AVN

175 NC 4 4 NC 6 6

125 NC 4 7 14 5 7

Table 4The number of function evaluations required such that |1minus119891(119909119899)119891(119909

119890)| lt 10minus15 NC states that the givenmethod does not converge

for the given root of (41) 119909119890= 21954

119891(119909119899) 119909

0OP MP VP NM MN AVN

19 NC 6 NC 7 5 5

16 3 3 4 3 3 4

ordinary methods can give more accurate results especiallyfor polynomial equations Thus the method should beselected according to the functions appeared in the equations

52 Realistic Applications In this subsection two exampleswill be considered to demonstrate possible impacts of theintroduced methods to science and engineering

Example 12 The process in which chemicals interact to formnew chemicals with different compositions is called chemicalreactionsThis process is the results of chemical properties ofthe element or compound causing changes in compositionThese chemical changes are chemistsrsquo main purpose It isimportant to mention a remark from [23] where ldquothe twoquestions that must be answered for a chemically reactingsystem are (1) what changes are expected to occur and (2)how fast will they occurrdquo This is an indication that math-ematical representations are very important for chemicalinvestigations Consequently a chemical reaction is mainlyrepresented by a chemical equation which represents thechange from reactants to products This process generallyinvolves nonlinear functions so it is compulsory to useand apply numerical approaches Exemplarily suppose thata chemical reaction is made and the concentration of aparticular ion at the time 119905 is given by a nonlinear function

119891 (119909) = 5119890minus3119909 + 119890minus5119909 (37)

If we are interested in when this concentration will be one-half of its value at initial time 0 we need to solve this problemnumerically If 119891(0) = 2 as an initial assumption this will beequivalent to finding a root of nonlinear equation

5119890minus3119909 + 119890minus5119909 minus 1 = 0 (38)

Usually chemists tend to use ordinary numerical methodsto estimate the root of (38) However the multiplicativebase methods whose effectiveness was proved in problemsthat involve exponential functions should not be disregardedin many applications The superiority of the multiplicativenumerical methods can be easily observed for this problemby Table 3

Example 13 The Rayleigh function which corresponds toRayleigh distribution

119891 (119909) =119909

120590119890minus1199092

21205902

for arbitrary 119886 119887 119888 isin 119877 (39)

plays an important role in magnetic resonance imaging(MRI) and probability theory It is a striking example in orderto show the efficiency of the multiplicative and Volterra root-finding methods Assume that the density function of thecontinuous random variable119883 on the interval [0infin) is givenby

119891 (119909) = 119909119890minus1199092

2 where 120590 = 1 (40)

It may be interesting to consider the root of the equation

120572 minus 119891 (119909) = 120572 minus 119909119890minus1199092

2 = 0 0 le 120572 le 1 (41)

which estimates the point119909 according to the given probability120572 of an event in119883 We attempt to use root-finding algorithmsfor (41) for 120572 = 019720178928946303 which will be given byTable 4

6 Conclusion

In this study the multiplicative and Volterra based root-finding methods are presented These methods were testedfor some nontrivial problems and compared with the originalroot-finding method The results show that in certain prob-lems the multiplicative andor Volterra methods give moreaccurate results compared to the original root-finding meth-ods Especially the examples showed that the nature of theunderlying calculus plays an important role in approximatingthe zeros of the function

The selection of the initial value is very important for theconvergence of the iteration Two theorems in Section 4 indi-cate the conditions for convergence related to multiplicativeand Volterra methods respectively Section 4 also highlightsoptimal selection of the initial value Evidently in certain sit-uations the multiplicative andor Volterra methods convergefaster compared to the ordinary methods Particularly it canbe easily observed that multiplicative and Volterra Newton-Raphson methods are more accurate than ordinary Newton-Raphson method in many applications Therefore these

Journal of Mathematics 7

methods based on multiplicative calculi have proven theirimportance in the process of numerical approximations ofnonlinear equations Also the numerical results obtained inthe paper encourage the usage of multiplicative and Volterramethods for solving nonlinear equations

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Grossman and R Katz Non-Newtonian Calculus Lee PressPigeon Cove Mass USA 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] L Florack and H V Assen ldquoMultiplicative calculus in biomedi-cal image analysisrdquo Journal ofMathematical Imaging andVisionvol 42 no 1 pp 64ndash75 2012

[4] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[5] D A Filip and C Piatecki ldquoA non-Newtonian examinationof the theory of exogenous economic growthrdquo CNCSISmdashUEFISCSU (project number PNII IDEI 23662008) and Lab-oratoire dEconomie dOrleans (LEO) 2010

[6] E Misirli and Y Gurefe ldquoMultiplicative adams bashforthmdashmoulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011

[7] M Riza A Ozyapıcı and E Misirli ldquoMultiplicative finitedifference methodsrdquo Quarterly of Applied Mathematics vol 67no 4 pp 745ndash754 2009

[8] A Ozyapici and E Misirli ldquoExponential approximations onmultiplicative calculusrdquo Proceedings of the Jangjeon Mathemati-cal Society vol 12 no 2 pp 227ndash236 2009

[9] A Ozyapici and B Bilgehan ldquoFinite product representation viamultiplicative calculus and its applications to exponential signalprocessingrdquo Numerical Algorithms vol 71 no 2 pp 475ndash4892016

[10] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[11] J Englehardt J Swartout and C Loewenstine ldquoA new theoret-ical discrete growth distribution with verification for microbialcounts in waterrdquo Risk Analysis vol 29 no 6 pp 841ndash856 2009

[12] H Ozyapıcı I Dalcı and A Ozyapıcı ldquoIntegrating accountingand multiplicative calculus an effective estimation of learningcurverdquo Computational and Mathematical Organization Theory2016

[13] B Bilgehan ldquoEfficient approximation for linear and non-linearsignal representationrdquo IET Signal Processing vol 9 no 3 pp260ndash266 2015

[14] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007

[15] WKasprzak B Lysik andM RybaczukDimensions InvariantsModels and Fractals Ukrainian Society on Fracture MechanicsSPOLOM Wroclaw-Lviv Poland 2004

[16] M Grossman Bigeometric Calculus A System with a Scale-FreeDerivative Archimedes Foundation RockportMass USA1983

[17] M Riza and B Eminagaı ldquoBigeometric calculusmdasha modellingtoolrdquo httpsarxivorgabs14022877v1

[18] M A Noor W A Khan and A Hussain ldquoA newmodified Hal-ley method without second derivatives for nonlinear equationrdquoAppliedMathematics and Computation vol 189 no 2 pp 1268ndash1273 2007

[19] A Ramli M L Abdullah and M Mamat ldquoBroydenrsquos methodfor solving fuzzy nonlinear equationsrdquo Advances in FuzzySystems vol 2010 Article ID 763270 6 pages 2010

[20] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquo Mathematical and ComputationalApplications vol 12 no 2 pp 117ndash124 2007

[21] E Misirli and Y Gurefe Multiplicative calculus and its applica-tions [MS thesis] Turkish Council of Higher Education 2009Thesis No 252639

[22] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[23] M E Davis and R J Davis Fundamentals of Chemical ReactionEngineering MGraw-Hill Companies 2003

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

Page 7: Research Article Effective Root-Finding Methods for ...downloads.hindawi.com/journals/jmath/2016/8174610.pdfResearch Article Effective Root-Finding Methods for Nonlinear Equations

Journal of Mathematics 7

methods based on multiplicative calculi have proven theirimportance in the process of numerical approximations ofnonlinear equations Also the numerical results obtained inthe paper encourage the usage of multiplicative and Volterramethods for solving nonlinear equations

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Grossman and R Katz Non-Newtonian Calculus Lee PressPigeon Cove Mass USA 1972

[2] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008

[3] L Florack and H V Assen ldquoMultiplicative calculus in biomedi-cal image analysisrdquo Journal ofMathematical Imaging andVisionvol 42 no 1 pp 64ndash75 2012

[4] A Uzer ldquoMultiplicative type complex calculus as an alternativeto the classical calculusrdquo Computers amp Mathematics with Appli-cations vol 60 no 10 pp 2725ndash2737 2010

[5] D A Filip and C Piatecki ldquoA non-Newtonian examinationof the theory of exogenous economic growthrdquo CNCSISmdashUEFISCSU (project number PNII IDEI 23662008) and Lab-oratoire dEconomie dOrleans (LEO) 2010

[6] E Misirli and Y Gurefe ldquoMultiplicative adams bashforthmdashmoulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011

[7] M Riza A Ozyapıcı and E Misirli ldquoMultiplicative finitedifference methodsrdquo Quarterly of Applied Mathematics vol 67no 4 pp 745ndash754 2009

[8] A Ozyapici and E Misirli ldquoExponential approximations onmultiplicative calculusrdquo Proceedings of the Jangjeon Mathemati-cal Society vol 12 no 2 pp 227ndash236 2009

[9] A Ozyapici and B Bilgehan ldquoFinite product representation viamultiplicative calculus and its applications to exponential signalprocessingrdquo Numerical Algorithms vol 71 no 2 pp 475ndash4892016

[10] A E Bashirov E Mısırlı Y Tandogdu and A Ozyapıcı ldquoOnmodeling with multiplicative differential equationsrdquo AppliedMathematics vol 26 no 4 pp 425ndash438 2011

[11] J Englehardt J Swartout and C Loewenstine ldquoA new theoret-ical discrete growth distribution with verification for microbialcounts in waterrdquo Risk Analysis vol 29 no 6 pp 841ndash856 2009

[12] H Ozyapıcı I Dalcı and A Ozyapıcı ldquoIntegrating accountingand multiplicative calculus an effective estimation of learningcurverdquo Computational and Mathematical Organization Theory2016

[13] B Bilgehan ldquoEfficient approximation for linear and non-linearsignal representationrdquo IET Signal Processing vol 9 no 3 pp260ndash266 2015

[14] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007

[15] WKasprzak B Lysik andM RybaczukDimensions InvariantsModels and Fractals Ukrainian Society on Fracture MechanicsSPOLOM Wroclaw-Lviv Poland 2004

[16] M Grossman Bigeometric Calculus A System with a Scale-FreeDerivative Archimedes Foundation RockportMass USA1983

[17] M Riza and B Eminagaı ldquoBigeometric calculusmdasha modellingtoolrdquo httpsarxivorgabs14022877v1

[18] M A Noor W A Khan and A Hussain ldquoA newmodified Hal-ley method without second derivatives for nonlinear equationrdquoAppliedMathematics and Computation vol 189 no 2 pp 1268ndash1273 2007

[19] A Ramli M L Abdullah and M Mamat ldquoBroydenrsquos methodfor solving fuzzy nonlinear equationsrdquo Advances in FuzzySystems vol 2010 Article ID 763270 6 pages 2010

[20] M Pakdemirli H Boyacı and H A Yurtsever ldquoPerturbativederivation and comparisons of root-finding algorithms withfourth order derivativesrdquo Mathematical and ComputationalApplications vol 12 no 2 pp 117ndash124 2007

[21] E Misirli and Y Gurefe Multiplicative calculus and its applica-tions [MS thesis] Turkish Council of Higher Education 2009Thesis No 252639

[22] M Pakdemirli and H Boyacı ldquoGeneration of root findingalgorithms via perturbation theory and some formulasrdquoAppliedMathematics and Computation vol 184 no 2 pp 783ndash7882007

[23] M E Davis and R J Davis Fundamentals of Chemical ReactionEngineering MGraw-Hill Companies 2003

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

Page 8: Research Article Effective Root-Finding Methods for ...downloads.hindawi.com/journals/jmath/2016/8174610.pdfResearch Article Effective Root-Finding Methods for Nonlinear Equations

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