ROOT FINDING FOR NONLINEAR EQUATIONS

Transcription

1 ROOT FINDING FOR NONLINEAR EQUATIONS Tele Gemecu Departmet matematics, Adama Sciece ad Teclgy Uiversity, Etipia Abstract Nliear equatis /systems appear i mst sciece ad egieerig mdels Fr eample, e slvig eige value prblems, ptimizati prblems, dieretial equatis, i circuit aalysis, aalysis state equatis r a real gas, i mecaical mtis /scillatis, eater recastig, itegral equatis, image prcessig ad may ter ields egieerig desigig prcesses Nliear systems /prblems are diicult t slve maually but tey ccur aturally i luid mtis, eat traser, ave mtis, cemical reactis, etc Tis study deals it cstructi iterative metds r liear rt idig, applyig Taylr s series apprimati a liear ucti cmbied it a e crrecti term i a quadratic r cubic mdel Cmpetet iterative algritms iger rder ere ivestigated Fr test cvergece ad eiciecy, e applied basic terems ad slved sme equatis i C++ Keyrds liear equatis, Taylr s apprimati, iterative algritms r rts, errr crrecti Itrducti Nliear prblem slvig it rt idig is very cmm i sciece ad egieerig mdel applicatis Fr eample, i cemical ad electrical egieerig, evirmetal egieerig, i pysics, etc Te metd ca be direct /symblic, grapical r umerical iterative Te iterative es are derived usig iterplatis, perturbati metd, variatial tecique, ied pit metds ad s may ters, ] Tere are several eistig rt slvig metds suc as Bisecti metd, Secat metd, Regula Falsi, Net s metd ad its variats /acceleratrs Cebysev s metd, Halley s metd, Super Halley s metd,,, 7, 8, 9,,,, ] Hever, cice iitial guesses, iterval selectis, eisteces derivatives ad accelerati cvergece are sme cmm drabacs cected it algritmic cmpleities I tis article, e apply Taylr s apprimati by quadratic ad cubic mdel t derive e iterative algritms, usig a e crrecti term We als discuss etesi t iger dimesis r slvig liear system Te article is rgaized as: itrducti, basic metds based Taylr s series, etesis t D, cvergece aalysis, prcedures r cmputer cdes, test prblems, result ad discussi, cclusi ad reereces Cstructi metds based Taylr s epasi Csider te Taylr s apprimati a liear ucti abut a apprimate rt r i D / / I =,y te te Taylr s series epasi i D is epressed as +, y+ =, y+ + y +/ + y + yy + a Ad i =, y, z i D, te +, y+, z+l =, y, z+ + y +l z +/ + y +l z + l yz + yy + zz b Te liear apprimati rm, yields Net s metd :,,, ] Frm a mdiied Net s metd r multiple rt ] e gets r simple rts :,,, a

2 ISSN -58 Paper ISSN 5-5 Olie Frm te liear estimati i, e ca als derive a e metd,,, : b Assumig tat a cmbiati c & a is iger rder, e as,,, : c Tis satisies, I e ti abut te quadratic iterplati apprimati /, e btai / d Te it a e errr crrecti term = / - rm a i te rigt part d, e gets te t metds ] ] ] ] / Tere are als Halley s metd a ad Cebysev s metd b / ] a ] ] / b Ad eteded Net 5 ad Euler metd 5b, see,,,, 8,9,, 7, 9] ] ] 5,, H H Q p H 5a Suppse e pders te cubic mdel as / / / / / 7 5 / Usig a crrecti term =/ - i te rigt part, e btai Tis yields a iterati ucti 7a

3 Replacig te iger rder derivative by, e get ater algritm 7b Hever sme tese metds eed ig memry cmputer Etesis t Higer Dimesis r Nliear Systems Let us start it Net s metd t slve systems liear equatis i D Assume a liear system equatis,, 5, 7,5], g, y F, y 8, y We pe t get X =, y tat satisies I X =, y is a iitial guess ad X =,y is a eaced apprimati, te e ca apply Taylr s liear estimati as F X F X F X X X 8a Were te Jacbea matri F is J y F F X, y 9 y Te liear system 8a ca be slved by elimiati Or by Net s metd X X J F X 9a Prvided tat te iverse J F X eists Ad te iterati prcess repeats util cvergece Te cve accelerati Net s metd 9a i D t slve F is 5] X I / L X I L X ] J X F X X Wit I is idetity peratr/matri ad L F X X F J X F is called te lgaritmic degree cveity F Ad te secd partial derivative F, F is a tesr se elemets are te partial derivatives metd i D is F ji i ] Equati is super-halley s metd t slve 8 Ad Halley s i X I / L X I 5L X ] J X F X X

4 Te Cebysev s metd i D is X I / L X J X F X X Nte tat i is Net s iterati ucti i te L L i D T eted te e metds i ad, e irst rite as bel 5 ] ] Ad as 5 ] Were is Net s iterati ucti t slve i D ] CONVERGENCE ANALYSIS We sall use te llig imprtat deiiti ad terem Deiiti ] A sequece geerated by a iterative metd is said t cverge t a rt r it rder p p i tere eists c > suc tat e ce,, r sme iteger ad e r Terem Order Cvergece Assume tat as suicietly may derivatives at a rt r Te rder ay e-pit iterati ucti is a psitive iteger p, mre especially as rder p i j p r r ad r r < j < p, r ad ly i 8], see als,,,8,7] All te algritms e preseted eed a apprpriate cice ly e iitial guess i a iterval I = a, b] Ad radm cices may lead us t uecessary rs Ntice tat rm Terem abve ad cvergece ied pit iterati metd,, 8, 8], r all i a, b] Frm ic Te case > is divergece Ad eeds especial treatmet rermulatis r eed r alterative metds, ] Pr te rder cvergece p Te pr ca be de applyig terem ad deiiti r metds prs i,, 5,, 8] Pr rder cvergece algritm i 5 ] ] 5 We ca rite 5 as H Were H = 5 ] ] Ad is Net s iterati ucti Let r be a simple rt We ave r r, ad Ad r but r Dieretiatig H, e id tat r r but r S p Cversely, i p =, te e ca s tat r r but r Hece, r 5 is tird rder cverget metd T prve rder cvergece algritm i equati We ca rite as T Were T= 5 ] ]

5 Ad is Net s iterati ucti Frm ic ad r but r Dieretiatig T, e id tat r r but r S p O te ter ad, i p =, te e ca s tat r r but r Hece, is tird rder cverget metd Similarly a ad b are quadratic ad c is cubic cverget Te prs r rders te ter algritms ca be de i a similar ay We sall mae a detailed aalysis i te uture r Related ccepts are i,,, 5,, 7, 9,,, 7, 8, 9] Prcedure settig up r cmputer cdes Deie a equati t slve, 5 utput Deie derivatives, d, dd, Set errr = c c is crrecti term] Eter Iputs;, tlerace, umber Iter= 7 i errr < = tl Fr i=, I++ utput, F = ; ed D = d, DD=dd 8 else X = deie te metd] = i = i+ utput sluti /rt, ed TEST EQUATIONS, RESULTS &DISCUSSIONS We ave cse ive equatis r test eiciecy, it =,, ad rt r 78 i,,, =,,, r 7, cs cs e, 5 lg, it = -5, -, -5, r -955 i -, =,,, r 7 i,, = 5, 5,, r = i, Cmpariss ere relative t Net metd NM, Cebysev s metd CM, Halley s metd HM,,, a, b C++ implemetati as de r eac algritms ad te umber iteratis tae t cverge t a rt r t si decimal places as recrded ad ritte i te bdy te et table- uder eac metd Te stppig criteria ere usig te residual errr suc tat, r cse 8 We als ceced tis by ter stppig criteria i te literature Hit: Te triplets umbers i eac cells table- crrespd t te umber iteratis eeded r cvergece it eac te tree iitial guesses a rt r i, i,,,5 Frm te table, a algritm HM r slvig cverges at steps,, r te iitial guesses tae at,, respectively ad beig cverges at steps 5, 5, taig te same iitial guesses i 8 give Ad NM r slvig,, I te irst clum, Fuctis reers t te umber uctial evaluatis up t derivatives, ad Éiciecy e represets te cmputatial eiciecy ide idicates sless at te pit Algritms are cmpared may be raed as ast, aster r very ast relatively depedig teir Nar values i te table beig te uppermst, itermediate r te lest respectively, see ] A algritm it te least average umber iteratis Nar t cverge t a rt r uld be raed very ast cverget Te iger te Nar value, te sler is a algritm t cverge Te lesser te Nar value, te aster is a algritm t cverge Taig mre iitial guesses r mre eamples gives gd raig measure I te table, te igest value eiciecy ide is r tird rder We ca bserve tat all metds preseted i te table are better cmpetet it t 5 average umber iteratis t cverge rm bt directis at e a apprpriate iitial guess is used I is t suitably cse, te e ca epect sl cvergece ad eve divergece rm a rt 5

6 Table - Summary cmparis results, NM CH a b HM :,, 5, 5,,,,, -,, -,,-,,,, :,,,,,,,,,, 5, 5, -,,,, : -5,-,-5 7, 5,,,,,,,, 5,,,,,,,, 5, 7,9,,,, -,,, 5,,, 9,, :5,5,,,,,,,,,, 5,,5,,, 5 Nar, average iter Order p Fuctis Eiciecye Te e errr crrecti migt be ast cvergece e it cverges but des t aect te umber uctial evaluatis Cclusis I tis r, e ave applied a e crrecti term i Taylr secd ad tird rder apprimati t btai sme iterative metds r estimatig simple rts liear equatis Te crrecti tecique des t aect te umber uctial evaluatis but cvergece We ave s pssible etesis r slvig D liear systems Cmpetet metds ere ivestigated I te uture, e ill preset urter aalyses tese algritms ad ter iger rder iterative algritms it applicatis We pe tat tis result ill be very slyess ad brig abut e t perrm urter researc ACKNOWLEDGEMENTS Te autr uld lie t ta all te sta members ASTU REFERENCES ] Quarteri, A et al, Numerical matematics Tets i applied matematics; 7, Spriger-Verlag Ne Yr, Ic, USA ] Ralst, A 978, Rabiitz, P A irst curse i umerical aalysis d ed, McGra-Hill B Cmpay, Ne Yr ] Cu,C ad Neta,B9, A tird-rder mdiicati Net s metd r multiple rts, Applied Matematics ad Cmputati,p 7 79 ] Dalquist, G8, Bjrc, A, Numerical metds i scietiic cmputig Vlume I Siam - Sciety r idustrial ad applied matematics Piladelpia, USA 5] Albeau, A8, ON THE GENERALIZED HALLEY METHOD FOR SOLVING NONLINEAR EQUATIONS, ROMAI J,, 8, - ] Jai, MK,et al7, Numerical Metds r Scietiic ad Egieerig Cmputati Fit Editi Ne Age Iteratial P Limited Publisers 7] JSter, RBulirsc 99, Tets i Applied matematics, Itrducti t Numerical Aalysis d ed, spriger-verlag Ne Yr, Ic, USA 8] Gerlac, J99, Accelerated Cvergece i Net s Metd, Sciety r idustrial ad applied matematics,siam Revie, 7-7 9] Jiseg, K, et al, A uiparametric Cebysev-type metd ree rm secd derivative, Applied Matematics ad Cmputatial 79, 9- ] DPetvic, L,et al, te urt rder rt idig metds Euler type, NOVI SAD JMATH,, 57-5 ] MM Hussei 9, A te e-step iterati metds r slvig liear equatis, Wrld Applied Scieces Jural 7 special issue r applied mat, 9-95, IDOSI Publicatis ] Nr, MA, et al, variati iterati tecique r idig multiple rts liear equatis, Scietiic Researc ad Essays Vl,pp-5

7 ] Pademirli, H et al8, A rt idig algritm it it rder derivatives, Matematical ad Cmputatial Applicatis,Vl, -8 ] Segeg Li, S, ad Wag, R,, T urt rder iterative metds based Ctiued Fracti rrt idig prblems, Wrld Academy Sciece, Egieerig ad Teclgy 5] Capra, S, Applied Numerical metds it Matlab r Egieers ad Scietists rd ed, McGra Hill Educati Idia, Ne Deli ] Kalil, SA, et al7, Types Derivatives: Ccepts ad Applicatis II, Jural Matematics Researc; Vl 9, N 7] Gemecu, T7, Sme Rt Fidig Wit Etesis t Higer Dimesis, IISTE, ISSN -58, Vl7, N 8] Gemecu,T7, Sme Multiple ad Simple Real Rt Fidig Metds, IISTE, ISSN -58, Vl7, N 9] JIN Y ad KALANTARI, B cmmuicated by Peter A Clars, A cmbiatrial cstructi ig rder algritms r idig plymial rts multiplicity, America Matematical Sciety 8,