Higher-order iterative methods by using Householder's method for solving certain nonlinear equations

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1 Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem Asghar Kha, Muhammad Aslam Noor ad Ada Rauf Departmet of Mathematics, Sukkar-Istitute of Busiess Admiistratio, Sukkur-65 Sidh, Pakista Departmet of Mathematics, COMSATS Istitute of Iformatio Techology, Islamabad, Pakista Departmet of Mathematics, Istitute of Busiess Maagemet, Karogi Creek Karachi, Pakista waseemasg@gmailcom, oormaslam@hotmailcom, adarauf@gmailcom Received: 5 Ja ; Revised: 7 Apr ; Accepted: 8 Apr Published olie: May Abstract: I this paper, we suggest ad aalyze some ew higher-order iterative methods by usig Householder s method free from secod derivative for solvig oliear equatios Here we use ew ad differet techique for implemetatio of higher-order derivative of the fuctio ad derive ew higherorder predictor-corrector iterative methods free from secod derivative The efficiecy idex equals to / Several umerical examples are give to illustrate the efficiecy ad performace of these ew methods Keywords: Noliear equatios; Newto method; Covergece criteria; Root fidig method; Numerical examples Itroductio 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( x Due to their importace; several umerical methods have bee suggested ad aalyzed uder certai coditios These umerical methods have bee costructed usig differet techiques such as Taylor series, homotopy perturbatio method [-] ad its variat forms, quadrature formula, variatioal iteratio method, ad decompositio method; see, for example [-] Usig the techique of updatig the solutio ad Taylor series expasio, Noor ad Noor [4] have suggested ad aalyzed a sixth-order predictor-corrector iterative type Halley method for solvig the oliear equatios Ham et al [7] ad Chu [4] have also suggested a class of fifth-order ad sixth-order iterative methods I the implemetatio of the method [4], oe has to evaluate the secod derivative of the fuctio, which is a serious drawback of these methods To overcome these drawbacks, we modify the predictor-corrector Halley method by replacig the secod derivatives of the fuctio by its suitable scheme We prove that the ew modified predictor-corrector method is of sixth-order covergece free from secod derivatives We also preset the compariso of the ew method with the methods of Ham et al [7] ad Chu [4] Several examples are give to illustrate the efficiecy ad robustess of the ew proposed NSP

2 8 W A Kha et al: Higher-order iterative methods by usig Householder's It has bee show that these ew iterative methods iclude a wide class of kow ad ew iterative methods as special cases Also discuss the efficiecy idex ad computatioal order of covergece of ew methods Several examples are give to illustrate the efficiecy ad performace of these ew methods We also compare these ew methods with other recet methods of the same covergece order Iterative methods We recall the Newto s method [6] ad Householder s method [,5] i Algorithm ad Algorithm, we have Algorithm For a give x by the iterative scheme: x x f ( x ( Algorithm is the well-kow Newto method, which has a quadratic covergece [6] Algorithm For a give x f ( x f ( x f ( x x x ( f ( x f ( x This is kow as Householder s method, which has cubic covergece [, 5] Noor ad Noor [], have suggested the followig two-step method, usig Algorithm method as predictor ad Algorithm as a corrector Algorithm For a give x y x f ( x f ( y f ( y f ( y x y ( f ( y f ( y If f ( y the Algorithm is called the predictor-corrector method ad has fourth-order covergece, see [6] I order to implemet this method, oe has to fid the secod derivative of this fuctio, which may create some problems To overcome this drawback, we use ew ad differet techique to reduce secod derivative of the fuctio ito the first derivative This idea plays a sigificat role i developig some ew iterative methods free from secod derivatives To be more precise, we cosider f ( y f ( x f ( y f ( y f ( x Pf ( x, y (4 y x y x Combiig ( ad (4, we suggest the followig ew iterative method for solvig the oliear equatio ( ad this is the ew motivatio of higher-order Algorithm 4 For a give x y x f ( x x f( y f ( y P ( x, y f y f ( y f ( NSP

3 W A Kha et al: Higher-order iterative methods by usig Householder's 9 Algorithm 4 is called the ew two-step modified Householder s method free from secod derivative for solvig oliear equatio ( This method has sixth-order covergece Per iteratio this method requires two evaluatios of the fuctio ad two evaluatios of its first-derivative, so its efficiecy idex /4 /m equals to 6 565, if we cosider the defiitio of efficiecy idex [8] as p, where p is the order of the method ad m is the umber of fuctioal evaluatios per iteratio required by the method Followig the techique of predictor-corrector of the solutio, see [4, 7] We derive the ew methods, we have Algorithm 5: For a give x y x f ( x x z f( y f ( y Pf ( x, y y f ( y f ( y f ( x f ( y f ( z z 6 f ( y f ( x f ( x Algorithm 6: For a give x y x f ( x f( y f ( y Pf ( x, y z y f ( y f ( y f ( x f ( y f ( z x z f ( x f ( x f ( y f ( y f ( x These ew methods have seveth-order covergece Per iteratio this method requires two evaluatios /4 of the fuctio ad two evaluatios of its first-derivative, so its efficiecy idex equals to Algorithm 7: For a give x y x f ( x f( y f ( y Pf ( x, y z y f ( y f ( y f ( x f ( z x z f ( y f ( x Algorithm 8: For a give x y x f ( x z f( y f ( y P ( x, y y f y f y f ( NSP

4 W A Kha et al: Higher-order iterative methods by usig Householder's x z f ( y f ( z f ( y f ( x f ( x These ew methods i Algorithm 7 ad Algorithm 8 have eighth-order covergece Per iteratio these methods requires two evaluatios of the fuctio ad two evaluatios of its first-derivative, so its /4 efficiecy idex equals to 8 55 I the similar way, we ca suggest the followig ew iterative methods Algorithm 9: For a give x y x f ( x x z f( y f ( y Pf ( x, y y f ( y f ( y f ( x f ( y f ( z z f ( y f ( x f ( x (5 (6 Algorithm : For a give x y x f ( x x f( y f ( y P ( x, y y f y f y f z ( ( f ( x f ( z z f ( x 4 f ( x f ( y f ( y f ( x Covergece criteria Now we cosider the covergece criteria of Algorithm 9 I a similar way, we ca discuss the covergece of other Algorithms Theorem : Let D be a simple zero of sufficietly differetiable fuctio f : D R R for a ope iterval D Ad x is iitial choice, the Algorithm 9 has ith-order covergeces Proof If is the root ad e be the error at th iteratio, tha e = x, usig Taylor s expasio, we have ( iv 4 ( v 5 f ( x f ( x e f ( x e f ( x e f ( x e f ( x e!! 4! 5! ( vi 6 7 f ( x e O( e, 6! f ( x = f ( [ e c e c e c e c e O( e ], (7 4 NSP

5 W A Kha et al: Higher-order iterative methods by usig Householder's f ( x = f ( [ c e c e 4c e 5c e 6c e O( e ], ( where ( k f ( ck k,, let e k! f x, ad ( From (7 ad (8, we have f ( x 4 = e ce ( c c e (c 4 7cc 4c e ( 6c cc f ( x 5 4 cc4 4c5 8c e O( e (9 From equatio (9, we have y c e (c c e (c 7c c c e O( e ( 4 6 ad, f ( y f ( [ c e ( c c e (c 7c c 5 c e O( e ], ( f ( y f ( [ c e 4( c c c e (8c 6c c c c e O( e ( f ( y P ( x, y c e (c c c e ( c c c 4 c c e ( 4c c f f ( y c c c 6c c 6c c c c e (6c c 6c c c 45c c 8c c c 4c c c c 6c c c 6 c e Oe ( ( f ( y P ( x, y z y f ( y f ( y ( c c c c c e f f ( y (c c 4c c c 6c c 6c c c c 6 c e (c c c c c c 88c c c 9c c 6c c c c 9c c cc4 4c4c 57 c c e O( e Usig (7-(4 i Algorithm 9, we have x (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 9 has ith-order covergece (4 4 Numerical examples I this sectio, we preset some umerical examples to illustrate the efficiecy ad the accuracy of the ew developed iterative methods i this paper (Table -Table 7 We compare our ew methods obtaied i Algorithm 4 to Algorithm with Newto s method (NM, method of Noor ad Noor ([4], NN, method of Noor et al ([6], NK, methods of Chu ([7], CM, CM ad CM, method of Siyyam ([9], SM, method of Li ad Jiao ([9], LJ ad method of Javidi ([8], JM ad JM All computatios have bee doe by usig the Maple package with 5 digit floatig poit arithmetic We accept NSP

6 W A Kha et al: Higher-order iterative methods by usig Householder's approximate solutio rather tha the exact root, depedig o the precisio ( of the computer We use the followig stoppig criteria for computer programs: i x x, ii, ad so, whe the stoppig criterio is satisfied, x is take as the exact root α computed For umerical 5 illustratios we have used the fixed stoppig criterio As for the covergece criteria, it was required that the distace of two cosecutive approximatios Also displayed are the umber of iteratios to approximate the zero (IT, the approximate root x, the value f( x ad the computatioal order of covergece (COC ca be approximated usig the formula, l ( x x /( x x COC All examples are same i [] l ( x x /( x x Example Cosider the equatio Table (Approximate solutio of example f ( x x 4x, x Methods IT x f( x COC NM e-4 79e- NN e-6 6 NK e-9 56 Alg e-6 6 Alg e-7 7 Alg e-7 76 Alg e-4 8 Alg e-4 8 Alg e-55 9 Alg e-5 96 JM e-44 NSP

7 W A Kha et al: Higher-order iterative methods by usig Householder's JM LJM e-7 5 SM e-5 5 CM CM e- 68 CM e- 5 Example Cosider the equatio f ( x si x x, x Table (Approximate solutio of example Methods IT x f( x COC NM e-4 588e-7 NN e-9 64 NK e- 57 Alg e-4 64 Alg e Alg e Alg 7 D e-7 8 Alg 8 D e-5 8 Alg e- 9 Alg e-55 9 JM e-7 4 JM e-5 NSP

8 4 W A Kha et al: Higher-order iterative methods by usig Householder's LJM e-9 5 SM e- 59 CM e-5 54 CM e-6 6 CM e Here D for diverget Example Cosider the equatio Table (Approximate solutio of example f ( x x e x x x Methods IT x f( x COC NM e-55 96e-8 NN e-6 48 NK e-4 5 Alg e e- 58 Alg e e-8 77 Alg e Alg e Alg e e-8 8 Alg e-59 e-59 - Alg e JM e-9 55 JM e-59 e-59 - LJM e-59 e-59 NSP

9 W A Kha et al: Higher-order iterative methods by usig Householder's 5 SM e- 59 CM e-5 54 CM e-6 6 CM e Example 4 Cosider the equatio f4( x cos x x, x 7 Table 4 (Approximate solutio of example 4 Methods IT x f( x COC NM e- 449e-6 99 NN e-6 574e-4 56 NK e e- 466 Alg e e Alg e e Alg e-6 46e Alg e-6 e-6 76 Alg e-6 e-6 - Alg e-6 e-6 9 Alg e-6 e-6 - JM e-6 449e-6 6 JM e e LJM e-6 79e- 445 SM e-6 48e- NSP

10 6 W A Kha et al: Higher-order iterative methods by usig Householder's CM e-6 879e- 475 CM e-6 58e-4 59 CM e-6 744e Example 5 Cosider the equatio f ( x ( x, x 5 5 Table 5 (Approximate solutio of example 5 Methods IT x f( x COC NM 7 5e e-8 NN 849e NK 4979e-4 5 Alg 4 849e Alg 5 998e-9 66 Alg 6 88e- 654 Alg 7 6e Alg 8 578e-5 77 Alg e Alg 749e-4 86 JM e-8 4 JM 4 579e-49 5 LJM 4 974e SM e-8 5 CM 4 457e-5 NSP

11 W A Kha et al: Higher-order iterative methods by usig Householder's 7 CM 4 - CM 4 56e-55 5 Example 6 Cosider the equatio f ( x x, x 6 Table 6 (Approximate solutio of example 6 Methods IT x f( x COC NM e-5 568e-8 NN e e-4 6 NK e e- 54 Alg e e-4 6 Alg e-58 4e-58 7 Alg e e-5 74 Alg e-59 8e-59 - Alg e-59 8e-59 - Alg e-58 e-58 - Alg e-59 8e-59 - JM e-8 4 JM e e-6 54 LJM e e- 59 SM e e- 56 CM e e-6 NSP

12 8 W A Kha et al: Higher-order iterative methods by usig Householder's CM e-58 54e-8 6 CM e e- 499 Example 7 Cosider the equatio f ( x e, x x 7x 7 Table 7 (Approximate solutio of example 7 Methods IT x f( x COC NM 9 756e-5 4e-8 NN e NK 5 488e-7 5 Alg e Alg 5 4 e Alg e-4 69 Alg e Alg 8 4 9e Alg 9 98e-55 9 Alg 68e-4 85 JM 5 4e-8 4 JM 5 588e-54 5 LJM e e SM 5 79e- NSP

13 W A Kha et al: Higher-order iterative methods by usig Householder's 9 CM 7 999e- 5 CM 5 - CM 5 98e Coclusios I this paper, we have suggested ew higher-order iterative methods free from secod derivative for solvig oliear equatio f( x We have discussed the efficiecy idex ad computatioal order of covergece of these ew methods Several examples are give to illustrate the efficiecy of Algorithm 4 to Algorithm Usig the idea of this paper, oe ca suggest ad aalyze higher-order multi-step iterative methods for solvig oliear equatios Results proved i this paper may stimulate further research Refereces [] S Abbasbady, Improvig Newto Raphso method for oliear equatios by modified Adomia decompositio method, Appl Math Comput 45 (, pp [] R L Burde ad JD Faires, Numerical Aalysis, PWS Publishig Compay, Bosta, [] C Chu, Iterative methods improvig Newto s method by the decompositio method, Comput Math Appl 5 (5, pp [4] C Chu, Some improvemets of Jarrat s methods with sixth order covergeces ApplMath Comput, 9 (7, 4 47 [5] A S Housholder, The Numerical Treatmet of a Sigle Noliear Equatio, McGraw-Hill, New York, 97 [6] JF Traub, Iterative Methods for Solutio of Equatios, Pretice-Hall, Eglewood Cliffs, NJ, 964 [7] Y M Ham, C Chu ad S GLee, Some higher-order modificatios of Newto s method for solvig oliear equatios, J Comput Appl Math ( [8] M Javidi, Fourth-order ad fifth-order iterative methods for oliear algebraic equatios, Math Comput Model, 5 ( [9] Y T Li ad A Q Jiao, Some variats of Newto s method with fifth-order ad fourth-order covergece for solvig oliear equatios, It J Appl Math Comput, (9-6 [] M A Noor, New family of iterative methods for oliear equatios, Appl Math Comput 9 (7, pp [] M A Noor ad K I Noor, Iterative schemes for solvig oliear equatios, Appl Math Comput 8 (6, pp [] M A Noor ad K I Noor, Three-step iterative methods for oliear equatios, Appl Math Comput 8 (6, pp 7 [] K I Noor, M A Noor ad S Momai, Modified householder iterative method for oliear equatios, Appl Math Comput 9 (7, pp [4] M A Noor ad K I Noor, Predicot-corrector Halley method for oliear equatios,appl Math Comput 88 (7, pp [5] M A Noor, Some iterative methods for solvig oliear equatios usig homotopy perturbatio method, It J Comp Math, 87 ( 4-49 [6] M A Noor, W A Kha, A Hussai, A ew modified Halley method without secod derivatives for oliear equatio, Appl Math Comput, 89 (7 NSP

14 W A Kha et al: Higher-order iterative methods by usig Householder's [7] M A Noor, Iterative methods for oliear equatios usig homotopy perturbatio techique, Appl Math Iform Sci 4 ( ( 7 5 [8] W Gautschi, Numerical Aalysis: A itroductio, Birkhauser, 997 [9] H I Siyyam, A iterative method with fifth-order covergece for oliear equatios, Appl Math Sci, (9 4-5 [] M A Noor, W A Kha, K I Noor ad E S Said, ( Higher-order iterative methods free from secod derivative for solvig oliear equatios It J Phy Sci 6(8, [] M A Noor, W A Kha, S Youus Homotopy perturbatio techique for solvig certai oliear equatios, Appl Math Sci, Vol 6,, o, [] M A Noor, W A Kha, Fourth-Order Iterative Method Free from Secod Derivative for Solvig Noliear Equatios, Appl Math Sci, Vol 6,, o 9, [] M A Noor, W A Kha, New iterative methods for solvig oliear equatio by usig homotopy perturbatio method, Appl Math Comput, 9 ( pp NSP