Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 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 Email: 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 /5 9 55 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 method @ NSP
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 ( y @ NSP
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 7 475 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
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! 4 5 6 f ( x = f ( [ e c e c e c e c e O( e ], (7 4 5 @ NSP
W A Kha et al: Higher-order iterative methods by usig Householder's 4 5 6 f ( x = f ( [ c e c e 4c e 5c e 6c e O( e ], (8 4 5 6 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 4 5 4 c e O( e ( 4 6 ad, f ( y f ( [ c e ( c c e (c 7c c 5 c e O( e ], ( 4 5 4 f ( y f ( [ c e 4( c c c e (8c 6c c c c e O( e ( 4 4 5 4 f ( y P ( x, y c e (c c c e ( c c c 4 c c e ( 4c c f f ( y 4 4 4 4c c c 6c c 6c c c c e (6c c 6c c 5 5 4 5 4 4 4 c 45c c 8c c c 4c c c c 6c c c 6 c e Oe 6 6 5 4 5 6 4 7 ( ( f ( y P ( x, y z y f ( y f ( y ( c c c c c e f 5 6 4 f ( y (c c 4c c c 6c c 6c c c c 6 c e (c c c 4 6 7 7 5 4 4 6 4c c c 88c c c 9c c 6c c c c 9c c 5 4 4 5 5 4 8 9 6cc4 4c4c 57 c c e O( e Usig (7-(4 i Algorithm 9, we have 8 5 6 4 9 x (c c4c 4cc c c cc4c e O( e Thus, we have 8 5 6 4 9 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 a @ NSP
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 6 65449684576868 985e-4 79e- NN 65449684576868 5584e-6 6 NK 65449684576868 78777e-9 56 Alg 4 65449684576868 5584e-6 6 Alg 5 65449684576868 9e-7 7 Alg 6 65449684576868 459e-7 76 Alg 7 65449684576868 757e-4 8 Alg 8 65449684576868 7655e-4 8 Alg 9 65449684576868 8e-55 9 Alg 65449684576868 4545e-5 96 JM 4 65449684576868 459e-44 4 @ NSP
W A Kha et al: Higher-order iterative methods by usig Householder's JM 4 65449684576868 - LJM 4 65449684576868 9469e-7 5 SM 4 65449684576868 796e-5 5 CM 4 65449684576868 - CM 65449684576868 7794e- 68 CM 65449684576868 8447e- 5 Example Cosider the equatio f ( x si x x, x Table (Approximate solutio of example Methods IT x f( x COC NM 5 44496485465868-48794e-4 588e-7 NN 44496485465868 45448e-9 64 NK 44496485465868 445e- 57 Alg 4 44496485465868 77e-4 64 Alg 5 44496485465868 958e-56 74 Alg 6 44496485465868 847e-48 76 Alg 7 D 44496485465868 588e-7 8 Alg 8 D 44496485465868 49796e-5 8 Alg 9 44496485465868 8466e- 9 Alg 44496485465868 8e-55 9 JM 44496485465868 588e-7 4 JM 44496485465868 49796e-5 55 @ NSP
4 W A Kha et al: Higher-order iterative methods by usig Householder's LJM 44496485465868 47644e-9 5 SM 44496485465868 56965e- 59 CM 44496485465868 755e-5 54 CM 44496485465868 9567e-6 6 CM 44496485465868 4859e-5 486 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 6 5758549867645567 959e-55 96e-8 NN 4 5758549867645567 e-6 48 NK 4 5758549867645567 456e-4 5 Alg 4 5758549867645567 e-59 9876e- 58 Alg 5 5758549867645567 -e-59 4796e-8 77 Alg 6 5758549867645567 9888e-6 684 Alg 7 5758549867645567 57867e-4 786 Alg 8 5758549867645567 -e-59 548794e-8 8 Alg 9 5758549867645567 -e-59 e-59 - Alg 5758549867645567 4968e-4 498 JM 4 5758549867645567 9768e-9 55 JM 4 5758549867645567 e-59 e-59 - LJM 4 5758549867645567 e-59 e-59 - @ NSP
W A Kha et al: Higher-order iterative methods by usig Householder's 5 SM 4 5758549867645567 56965e- 59 CM 4 5758549867645567 755e-5 54 CM 4 5758549867645567 9567e-6 6 CM 4 5758549867645567 4859e-5 486 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 5 79855664655-97e- 449e-6 99 NN 79855664655 -e-6 574e-4 56 NK 79855664655 -e-6 5589e- 466 Alg 4 79855664655 -e-6 4754e-5 568 Alg 5 79855664655 e-6 74787e-5 668 Alg 6 79855664655 e-6 46e-44 66 Alg 7 79855664655 e-6 e-6 76 Alg 8 79855664655 e-6 e-6 - Alg 9 79855664655 -e-6 e-6 9 Alg 79855664655 -e-6 e-6 - JM 79855664655 -e-6 449e-6 6 JM 79855664655 -e-6 5877e-4 458 LJM 79855664655 e-6 79e- 445 SM 79855664655 e-6 48e- 448 @ NSP
6 W A Kha et al: Higher-order iterative methods by usig Householder's CM 79855664655 -e-6 879e- 475 CM 79855664655 e-6 58e-4 59 CM 79855664655 -e-6 744e-7 479 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-56 9484e-8 NN 849e-7 579 NK 4979e-4 5 Alg 4 849e-7 569 Alg 5 998e-9 66 Alg 6 88e- 654 Alg 7 6e-7 765 Alg 8 578e-5 77 Alg 9 9755e-6 868 Alg 749e-4 86 JM 4 9484e-8 4 JM 4 579e-49 5 LJM 4 974e-4 498 SM 4 6684e-8 5 CM 4 457e-5 5 @ NSP
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 5 54446988775996 989e-5 568e-8 NN 54446988775996-8e-59 458e-4 6 NK 54446988775996 e-58 48948e- 54 Alg 4 54446988775996-8e-59 458e-4 6 Alg 5 54446988775996 e-58 4e-58 7 Alg 6 54446988775996 e-58 695e-5 74 Alg 7 54446988775996-8e-59 8e-59 - Alg 8 54446988775996-8e-59 8e-59 - Alg 9 54446988775996 e-58 e-58 - Alg 54446988775996-8e-59 8e-59 - JM 54446988775996 9484e-8 4 JM 54446988775996 e-58 6494e-6 54 LJM 54446988775996-8e-59 574e- 59 SM 54446988775996 e-58 446e- 56 CM 54446988775996-8e-59 89488e-6 5 @ NSP
8 W A Kha et al: Higher-order iterative methods by usig Householder's CM 54446988775996 e-58 54e-8 6 CM 54446988775996-8e-59 996e- 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 4 544868e-4 598 NK 5 488e-7 5 Alg 4 4 599e-8 595 Alg 5 4 e-59 59 Alg 6 4 4954e-4 69 Alg 7 4 97e-5 797 Alg 8 4 9e-4 798 Alg 9 98e-55 9 Alg 68e-4 85 JM 5 4e-8 4 JM 5 588e-54 5 LJM e-58 4869e-5 496 SM 5 79e- 499 @ NSP
W A Kha et al: Higher-order iterative methods by usig Householder's 9 CM 7 999e- 5 CM 5 - CM 5 98e-55 5 5 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 887 89 [] 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 559 568 [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 (8 477-486 [8] M Javidi, Fourth-order ad fifth-order iterative methods for oliear algebraic equatios, Math Comput Model, 5 (9 66-7 [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 55 558 [] M A Noor ad K I Noor, Iterative schemes for solvig oliear equatios, Appl Math Comput 8 (6, pp 774 779 [] 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 54 59 [4] M A Noor ad K I Noor, Predicot-corrector Halley method for oliear equatios,appl Math Comput 88 (7, pp 587 59 [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 68-7 @ NSP
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