Higher-order iterative methods free from second derivative for solving nonlinear equations

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1 Iteratioal Joural of the Phsical Scieces Vol 6(8, pp , 8 April, Available olie at DOI: 5897/IJPS45 ISSN Academic Jourals Full Legth Research Paper Higher-order iterative methods free from secod derivative for solvig oliear equatios Muhammad Aslam Noor, *, Waseem Asghar Kha, Khalida Iaat Noor ad Eisa Al-said Mathematics Departmet, COMSATS Istitute of Iformatio Techolog, Park Road, Islamabad, Pakista Mathematics departmet, College of Sciece, Kig Saud Uiversit, Riadh, Saudi Arabia Accepted 4 April, I this paper, we suggest ad aale some ew higher-order iterative methods free from secod derivative for solvig oliear equatios Per iteratio, these ew methods require three evaluatios of /5 the fuctio ad two of its first-derivative with efficiec ide = 9 55 Covergece of their methods is also cosidered Several umerical eamples are give to illustrate the efficiec ad performace of these ew methods These ew iterative methods ma be viewed as a alterative to the kow methods Ke words: Noliear equatios, Newto method, covergece criteria, root fidig method, umerical eamples INTRODUCTION It is well kow that a wide class of problem which arises i several braches of pure ad applied sciece ca be studied i the geeral framework of the oliear equatios f ( =, Due to their importace, several umerical methods have bee suggested ad aaled uder certai coditios These umerical methods have bee costructed usig differet techiques such as Talor series, homotop perturbatio method ad its variat forms, quadrature formula, variatioal iteratio method, ad decompositio method (Chu, 5, 7; Ham et al, 8; Javidi, 9; Noor, 6,, a; Noor et al (7, 7a, 7b Usig the techique of updatig the solutio ad Talor series epasio, Noor et al (7b have suggested ad aaled a sith-order predictor-corrector iterative tpe Halle method for solvig the oliear equatios Ham et al (8 ad Chu (7 have also suggested a class of fifth-order ad sith-order iterative methods I the implemetatio of the method of Noor et al (7b, oe has to evaluate the secod derivative of the fuctio, which is a serious drawback of these methods To overcome these drawbacks, we modif the predictor-corrector Halle *Correspodig author moorc@ksuedusa, oormaslam@hotmailcom method b replacig the secod derivatives of the fuctio b its suitable fiite differece scheme We prove that the ew modified predictor-corrector method is of sith-order covergece We also preset the compariso of these ew methods with the methods of Noor et al (7a, Ham et al (8 ad Chu (5, 7 We discus the efficiec ide ad computatioal order of covergece of ew methods Several eamples are give to illustrate the efficiec ad performace of these ew methods These ew results ma stimulate further research i this area ITERATIVE METHODS For the sake of completeess, we recall the Newto method ad Hella method These methods are as follows: Algorithm : For a give, compute approimates solutio + b the iterative scheme: + = f ( f ( Algorithm is the well-kow Newto method, which (

2 888 It J Phs Sci has a quadratic covergece Algorithm : For a give, compute approimates solutio + f ( f ( = f ( f ( f ( + This is kow as Halle s method ad has cubic covergece (Halle, 964 Noor et al (7b have suggested the followig twostep method, usig Algorithm as predictor ad Algorithm as a corrector Algorithm : For a give, compute approimates solutio + f ( = f ( f ( f ( = f ( f ( f ( + If f ( =, the Algorithm is called the predictorcorrector Newto method ad has fourth-order covergece (Traub, 964 I order to implemet Algorithm, oe has to fid the secod derivative of this fuctio, which ma create some problems To overcome this drawback, several authors have developed ivolvig ol the first derivative This idea plas a sigificat part i developig some iterative methods free from secod derivatives To be more precise, we cosider: f ( f ( f ( = f ( + f ( Pf (, Combiig Equatios ( ad (4, we suggest the followig ew iterative method for solvig the oliear Equatio ( Algorithm 4: For a give, compute approimates solutio + f ( = f ( f ( f ( = f ( f ( P (, + f Algorithm 4 is called the ew two-step modified Halle s ( ( (4 method free from secod derivative for solvig oliear Equatio ( This method has sith-order covergece Per iteratio, this method requires two evaluatios of the fuctio ad two evaluatios of its first-derivative, so its / 4 efficiec ide equals to Followig the techique of predictor-corrector of the solutio (Che, 7; Ham et al, 8, we derive the three-step iterative method for solvig the oliear equatios Algorithm 5: For a give, compute approimates solutio + + f ( = f ( = f ( f ( f ( f ( Pf (, f ( + f ( f ( = 6 f ( f ( f ( This ew method has seveth-order covergece Per iteratio, this method requires two evaluatios of the fuctio ad two evaluatios of its first-derivative, so its /4 efficiec ide equals to Algorithm 6: For a give, compute approimates solutio + + f ( = f ( = f ( f ( f ( f ( Pf (, f ( f ( = f ( f ( Algorithm 7: For a give, compute approimates solutio + + f ( = f ( = f ( f ( f ( f ( Pf (, f ( f ( = f ( f ( f ( These ew Algorithms 6 ad 7 have eighth-order

3 Noor et al 889 covergece Per iteratio these methods requires two evaluatios of the fuctio ad two evaluatios of its firstderivative, so its efficiec ide equals to 8 55 /4 I the similar wa, we ca suggest the followig ew three-step iterative methods for solvig the oliear equatios usig the predictor-corrector techique Algorithm 8: For a give, compute approimates solutio + f ( = f ( = + f ( f ( (5 f ( f ( Pf (, f f ( + f ( f ( = f ( f ( f ( Algorithm 9: For a give, compute approimates solutio + f ( = f ( = f ( f ( f ( f ( Pf (, f ( f ( + = + f ( 4 f ( f ( + f ( f ( Algorithm : For a give, compute approimates solutio + f ( = f ( = f ( f ( f ( f ( Pf (, + f + f f f f (6 f ( f ( f ( = ( ( ( ( ( Covergece criteria Now, we cosider the covergece criteria of Algorithm 8 I a similar wa, we ca discuss the covergece of other algorithms Theorem : Let D differetiable fuctio : α be a simple ero of sufficietl f D R R for a ope iterval D Ad is iitial choice, the Algorithm 8 has ith-order covergeces Proof: If α is the root ad e be the error at th iteratio, tha e = α, usig Talor s epasio, we have: f( = f ( e + f ( e + f ( e + f ( e + f ( e!! 4! 5! ( vi f ( e + O ( e, 6! ( iv 4 ( v f ( = f ( [ e + c e + c e + c e + c e + O( e ] α, ( f ( = f ( [ + c e + c e + 4c e + 5c e + 6c e + O( e ] α, ( ( k Where: f ( α c =, k =,,, let e k = α k! f ( α From Equatios (7 ad (8, we have: f ( f ( 4 = e ce ( c c e (c 4 7c c + 4c e + ( 6c + c c c c + 4c 8c e + O ( e (9 4 5 From Equatio (9, we have: = α+ c e + (c c e + (c 7c c + 4 c e + O( e ( f ( = f ( α[ c e + ( c c e + (c 7c c + 5 c e + O( e ], ( ad, f ( = f ( α[ + c e + 4( c c c e + (8c + 6c c c c e + Oe ( ( f( P(, = c e + (cc c e + ( c + cc 4 c c e + ( 4 cc f f ( c + cc 6c c + 6cc cc e + (6 cc + 6cc c 45 c c 8c c c 4cc + cc + 6 ccc + 6 c e + Oe ( f ( P (, = f ( f ( = ( c c c c + c e f f ( + (c c + 4c c c 6c c 6c c + c c 6 c e + ( c + c c c c c + 88 c c c 9c c 6c c c c + 9c c c c4 + 4c 4c + 57c ( c e + O e ( (4

4 89 It J Phs Sci Table Approimate solutio of Eample Methods IT F( δ COC NM e-4 79e- NN e-6 6 NK e-9 56 Alg e-55 6 Alg e Alg e Alg e-44 8 Alg e-55 9 Alg e-5 96 Alg e JM e-44 4 JM LJM e-7 5 CM CM e- 68 CM e- 5 Table Approimate solutio of Eample Methods IT F( δ COC NM e-4 588e-7 NN e-9 64 NK e- 57 Alg e-7 6 Alg e Alg e Alg e- 8 Alg e- 9 Alg e-55 9 Alg e-7 9 JM e-7 4 JM e-5 55 LJM e-9 5 CM e-5 54 CM e-6 6 CM e Usig Equatios (7 to (4 i Algorithm 8, we have: = α + (c + c4c 4cc + c c + cc4c e + O( e Thus, we have: e+ = (c + c4c 4cc + c c cc4c e + O( e which shows that Algorithm 8 has ith-order covergece NUMERICAL EXAMPLES I this stud, we preset some umerical eamples to illustrate the efficiec ad the accurac of the ew developed iterative methods (Tables to 7 We compare our ew methods obtaied i Algorithm 4 to Algorithm with Newto s method (NM, method of Noor et al ((7b NN, method of Noor et al (7c, NK, methods of Chu (8 CM, CM ad CM, method of Li et al(9, LJ ad method of Javidi (9, JM ad

5 Noor et al 89 Table Approimate solutio of Eample Methods IT F( δ COC NM e-55 96e-8 NN e-6 48 NK e-4 5 Alg e-9 Alg e-59 Alg e-59 Alg e- Alg e-59 e-59 - Alg e Alg e-54 5 JM e-9 55 JM e-59 e-59 - LJM e-59 e-59 - SM e- 59 CM e-5 54 CM e-6 6 CM e Table 4 Approimate solutio of Eample 4 Methods IT F( δ COC NM e- 449e-6 99 NN e-6 574e-4 56 NK e e- 466 Alg e-9 6 Alg e-59 7 Alg e-59 8 Alg e- 8 Alg e-6 e-6 9 Alg e-6 e-6 - Alg e-9 9 JM e-6 449e-6 6 JM e e LJM e-6 79e- 445 SM e-6 48e- 448 CM e-6 879e- 475 CM e-6 58e-4 59 CM e-6 744e JM All computatios have bee doe b usig the Maple package with 5 digit floatig poit arithmetic We accept a approimate solutio rather tha the eact root, depedig o the precisio ( ε of the computer We use the followig stoppig criteria for computer programs: (i < ε, ( ii f ( < ε ad so, whe + + the stoppig criterio is satisfied, + is take as the eact root computed For umerical illustratios we 5 have used the fied stoppig criterio ε = As for the covergece criteria, it was required that the distace of two cosecutive approimatios δ Also displaed are the umber of iteratios to approimate the ero (IT, the, the value f( ad the approimate root computatioal order of covergece (COC ca be approimated usig the formula,

6 89 It J Phs Sci Table 5 Approimate solutio of Eample 5 Methods IT F( δ COC NM 7 5e e-8 NN 849e NK 4979e-4 5 Alg e-8 6 Alg 5 579e Alg 6 974e Alg e Alg e Alg 9 749e-4 86 Alg 9768e JM e-8 4 JM 4 579e-49 5 LJM 4 974e SM e-8 5 CM 4 457e-5 5 CM 4 - CM 4 56e-55 5 Table 6 Approimate solutio of Eample 6 Methods IT F( δ COC NM e-5 568e-8 NN e e-4 6 NK e e- 54 Alg e-8 6 Alg e-49 7 Alg e-4 8 Alg e-8 8 Alg e-58 e-58 - Alg e-59 8e-59 - Alg e JM e-8 4 JM e e-6 54 LJM e e- 59 SM e e- 56 CM e e-6 5 CM e-58 54e-8 6 CM e e- 499 l ( + /( COC l ( /( For the sake of compariso, all eamples are the same as i Che (5 Eample Cosider the equatio (Table f( = + 4, = Eample Cosider the equatio (Table f ( = si +, = Eample Cosider the equatio (Table f( = e +, = Eample 4 Cosider the equatio (Table 4 f ( = cos, = 7 4

7 Noor et al 89 Table 7 Approimate solutio of Eample 7 Methods IT F( δ COC NM 9 756e-5 4e-8 NN e NK 5 488e-7 5 Alg e Alg 5 579e Alg 6 974e-4 8 Alg e Alg 8 98e-55 9 Alg 9 68e-4 85 Alg 9768e JM 5 4e-8 4 JM 5 588e-54 5 LJM e e SM 5 79e- 499 CM 7 999e- 5 CM 5 - CM 5 98e-55 5 Eample 5 Cosider the equatio (Table 5 f ( = (, = 5 5 Eample 6 Cosider the equatio (Table 6 f ( =, = 6 Eample 7 Cosider the equatio (Table f ( = e, = CONCLUSIONS I this paper, we have suggested ew higher-order iterative methods free from secod derivative for solvig oliear equatio We also discussed the efficiec ide ad computatioal order of covergece of these ew methods Several eamples are give to illustrate the efficiec of Algorithms 4- Usig the idea of this paper, oe ca suggest ad aale higher-order multi-step iterative methods for solvig oliear equatios REFERENCES Chu C (5 Iterative methods improvig Newto s method b the decompositio method, Comput Math Appl, 5: Chu C (7 Some improvemets of Jarrat s methods with sith order covergeces Appl Math Comput, 9: 4 47 Halle E (964 A ew eact ad eas method for fidig the roots of equatios geerall ad without a previous reductio, Phil Ro Soc Lodo, 8: 6 47 Javidi M (9 Fourth-order ad fifth-order iterative methods for oliear algebraic equatios, Math Comput Model, 5: 66-7 Noor KI, Noor MA (7 Modified Householder iterative method for oliear equatios, Appl Math Comput, 9: Noor MA (7 New famil of iterative methods for oliear equatios, Appl Math Comput, 9: Noor MA ( Some iterative methods for solvig oliear equatios usig homotop perturbatio method, It J Comp Math, 87: 4-49 Noor MA (a O iterative methods for oliear equatios usig homotop perturbatio techique, Appl Math Iform Sci, 4: 7-5 Noor MA, Noor KI (7a Predicot-corrector Halle method for oliear equatios, Appl Math Comput, 88: Noor MA, Kha WA, Hussai A (7b A ew modified Halle method without secod derivatives for oliear equatio, Appl Math Comput, 89: 68-7 Traub JF (964 Iterative methods for solutio of equatios, Pretice- Hall, Eglewood Cliffs, NJ, USA ACKNOWLEDGEMENT This research is supported b the Visitig Professor Program of Kig Saud Uiversit, Riadh, Saudi Arabia ad Research Grat No: VPPKSU8 The authors are grateful to the referees for their costructive commets ad suggestios

Section 7.2: Direction Fields and Euler s Methods

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