Iterative Methods of Order Four for Solving Nonlinear Equations
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1 Itrativ Mods of Ordr Four for Solvig Noliar Equatios V.B. Kumar,Vatti, Shouri Domii ad Mouia,V Dpartmt of Egirig Mamatis, Formr Studt of Chmial Egirig Adhra Uivrsity Collg of Egirig A, Adhra Uivrsity Visakhapatam 000, Adhra Pradsh, Idia Abstrat I is papr, w suggst a itrativ mod of ordr four for solvig oliar quatios ad som mor ar drivd from is w mod. Th ffiiy id of is mod is. Svral ampls ar osidrd ad ompard wi istig mods. Kywords - Itrativ Mod, Noliar Equatios, Covrg Critria, Numrial Eampls, Nwto s Mod. I. INTRODUCTION Th wll kow quadrati ovrgt Nwto s mod for fidig a simpl root of o-liar quatio f 0. Whr f : D R R is a salar futio o a op itrval D, is giv by f f. = 0,,, May itrativ mods hav b dvlopd s [-] for solvig quatio. by usig svral thiqus iludig prturbatio mods ad quadratur formula. Noor [] suggstd followig algorim whih has four ordr ovrg For a giv 0, Noor s two stp algorim to omput is f y i. f y f. f f f f y ii. f y f y f f f f f. f iii. f y f f f f y f iv. f y f y y f f y f Lt ' ' b at root of quatio. i a op f y f itrval D i whih f is otiuous ad has wll dfid first drivativ ad lt b approimat to f f f y root. f f f y of. ad Rtly, Jafar ad Bhzad [] drivd fw variats of Kig s four ordr family [] i., Whr stag ad. f 0. f f y f y y.. f f y f Epadig f by Taylor s sris about, w hav Whr y is as giv i. f f f f.... Ad, som of variats suggstd by Jafar ad Bhzad [] ar..9 All abov formula ar havig four ordr ovrg ad y is as giv i. oly. I stio of is papr, w dvlop a itrativ mod for solvig. ad its ovrg ritria is disussd. Ad also, fw variats ar drivd from is w mod i sam stio. Svral umrial ampls ar osidrd ad ompard wi istig os i oludig stio. II. THE NEW ITERATIVE METHOD 0
2 Assum is small ough ad gltig highr powrs i., from. &. W hav f f f 0. Whih yilds us f f y f y f. To mak domiator largst i magitud, w tak as f f y f y f. Takig ' ' i. as approimat to root, from. ad., w ow dfi followig algorim. Algorim.: For a giv 0, omput approimat solutio by itrativ shm. f. f y f y f = 0,,,. Whr y is as giv i. This algorim is fr from sod drivativ ad rquirs two futioal valuatios ad o of its first drivativs. Th ffiiy id of is mod is. Thorm.: Lt D b a sigl zro of suffiitly diffrtiabl futio f : D R R for a op itrval D. If 0 is i viiity of, algorim. has four ordr ovrg. Proof: If ' ' b root ad b approimat to root, padig f about ' ' usig Taylor s pasio, w hav f f f f f!! iv f f v O!! f f! f! f f iv v f f O! f! f f O Whr j f j, j! f j=,,. Ad, f f O.9 Now, From. ad.0, w hav f f O.0 y O. f y f O. f f y O f f O [ O ] [ O ] O 0
3 Thus, f y 0 f O. From., w obtai f y 0 f O. f y f 0 0 {[ ]} [ ] O. f y f O. Thus, f f y f f O O O { O O O O O O O. From.,. ad. w hav rat of ovrg of mod. is four. 0
4 f y Cas.: By padig apparig i f f y f f y. f f f domiator of mod., w obtai = 0,,,.. Whr y is as giv i.. f f f y f y f y Thorm.: Lt D b a simpl zro of suffiitly diffrtiabl futio f : D R R for a op itrval f f f D. If 0 is i viiity of, Algorim. has. four ordr ovrg. Proof: As do i orm., o a asily obtai rror rlatio as Ad, abov furr yilds f f y f y f y [ O ] Whih givs us f f f f.9. Thrfor, algorim. has four ordr ovrg. Cosidrig up to first dgr, sod dgr ad ird dgr trms of prssio lyig wii brakts of formula.9, w hav followig algorims. Algorim.: For a giv 0, omput approimat solutio by itrativ shm. Algorim.: For a giv 0, omput approimat solutio by itrativ shm. f f y.0 f f = 0,,,.. Whr y is as giv i.. Thorm.: Lt D b a simpl zro of suffiitly diffrtiabl futio f : D R R for a op itrval D. If 0 is i viiity of, Algorim. has ird ordr ovrg. Proof: As do i orm., o a asily obtai rror rlatio as O Whih givs us. Thrfor, algorim. has ird ordr ovrg. Algorim.: For a giv 0, omput approimat solutio by itrativ shm. f f y f y f y f f f f. = 0,,,.. Whr y is as giv i.. Thorm.: Lt D b a simpl zro of suffiitly diffrtiabl futio f : D R R for a op itrval D. If 0 is i viiity of, Algorim. has four ordr ovrg. Proof: As do i orm., o a asily obtai rror rlatio as Whih givs us [ O ]. Thrfor, algorim. has four ordr ovrg. III. NUMERICAL EXAMPLES I is stio, svral ampls ar osidrd whih ar tak from [, ] ad all mods prstd i is papr ar big tak to tabulat omputatioal rsults blow by usig stoppig ritria f
5 TABLE. I Equatio ad its root Iitial Guss 0 Numbr of itratios tak to obtai root by applyig followig mods si os si os si sios IV. CONCLUSION Th tabulatd rsults show at mods. to.9 ar ovrgig at almost sam pa ompard to mod.. Ad, algorim. is also workig f y wll pt i as at is gativ. f REFERENCES [] A. M. Ostrowski, Solutio of Equatios ad Systms of Equatios, Aadmi Prss, Nw York, 9. [] C. Chu, Itrativ Mods Improvig Nwto s Mod by Th Dompositio Mod, Comput. Ma. Appl [] J. F. Traub, Itrativ Mods for Th Solutio Of Equatios, Prti Hall, Eglwood Cliffs NJ, 9. [] J. Kou, Sod Drivativ Fr Variats of Cauhy s Mod, Appl. Ma. Comput. 00 i prss. [] J.S. Kou, Y.T. Li ad X.H. Wag, Modifid Hally s Mod Fr from Sod Drivativ, Appl. Ma. Comput. 00, i prss. [] Jafar Biazar, Bhzad Ghabari, A Gral Four ordr Family of Mods for Solvig Noliar Equatios, MACMESE'09 Prodigs of WSEAS itratioal ofr o Mamatial ad omputatioal mods i si ad girig 009, 9-. [] M. A. Noor, W. A. Kha ad Akhtar Hussai, A Nw Modifid Hally Mod Wiout Sod Drivativs for Noliar Equatio, Appl. Ma. Comput., [] M. A. Noor, W. A. Kha, Four Ordr Itrativ Mod Fr from Sod Drivativ for Solvig Noliar Equatios, Applid Mamatial Sis, Vol., 0, o. 9, -. [9] M. A. Noor, W. A. Kha, K. I. Noor ad E. A. Said, Highr ordr Itrativ Mods Fr from Sod Drivativ for Solvig Noliar Equatios, Itr. J. Phys. Si [0] M. Aslam Noor ad V. Gupta, Modifid Housholdr Itrativ Mod Fr from Sod Drivativs for Noliar Equatios, Appl. Ma. Comput., 00, I prss. [] M. Aslam Noor, Numrial Aalysis ad Optimizatio, Ltur Nots, Mamatis Dpartmt, COMSATS Istitut of Iformatio Thology, Islamabad, Pakista, [] R. Kig, Family of Four-Ordr Mods for Noliar Equatios, SIAM J. Numr. Aal. 09, -9. [] S. Abbasbady, Improvig Nwto-Raphso Mod for Noliar Equatios by Modifid Adomia Dompositio Mod, Appl. Ma. Comput
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