Intrnational Journal of Thortical and Applid Mathmatics 08; 4(: -8 http://www.scincpublishinggroup.com/j/ijtam doi: 0.648/j.ijtam.08040. ISSN: 575-507 (Print; ISSN: 575-5080 (Onlin Comparison of Som Itrativ Mthods of Solving Nonlinar Equations Oori Charity Eblchuwu, Bn Obapo Johnson, Ali Inalgwu Michal, Auji Trhmba Fidlis Dpartmnt of Mathmatics and Statistics, Faculty of Pur and Applid Scincs, Fdral Univrsity, Wuari, Nigria Email addrss: To cit this articl: Oori Charity Eblchuwu, Bn Obapo Johnson, Ali Inalgwu Michal, Auji Trhmba Fidlis. Comparison of Som Itrativ Mthods of Solving Nonlinar Equations. Intrnational Journal of Thortical and Applid Mathmatics. Vol. 4, No., 08, pp. -8. doi: 0.648/j.ijtam.08040. Rcivd: Dcmbr, 07; Accptd: May 5, 08; Publishd: July 6, 08 Abstract: This wor focuss on nonlinar quation ( 0, it is notd that no or littl attntion is givn to nonlinar quations. Th purpos of this wor is to dtrmin th bst mthod of solving nonlinar quations. Th wor outlind four mthods of solving nonlinar quations. Unli linar quations, most nonlinar quations cannot b solvd in finit numbr of stps. Itrativ mthods ar bing usd to solv nonlinar quations. Th cost of solving nonlinar quations problms dpnd on both th cost pr itration and th numbr of itrations rquird. Drivations of ach of th mthods wr obtaind. An ampl was illustratd to show th rsults of all th four mthods and th rsults wr collctd, tabulatd and analyzd in trms of thir rrors and convrgnc rspctivly. Th rsults wr also prsntd in form of graphs. Th implication is that th highr th rat of convrgnc dtrmins how fast it will gt to th approimat root or solution of th quation. Thus, it was rcommndd that th Nwton s mthod is th bst mthod of solving th nonlinar quation f( 0 containing on variabl bcaus of its high rat of convrgnc. Kywords: Nonlinar, Itrativ Mthods, Convrgnc, Variabl. Introduction Numrical mthods ar usd to provid constructiv solutions to problms involving nonlinar quations. A nonlinar quation may hav a singl root or multipl roots. This rsarch wor will ma mphasis on solving nonlinar quation in on dimnsion and involving on unnown, : which has scalar X as solution, such that 0. Th study of this wor cannot b discussd at pa without first maing ncssary attmpts to discuss th branch of mathmatics that it ariss from which is numrical analysis. Numrical analysis, is an ara of mathmatics and computr scinc that crats, analyss and implmnt algorithm for obtaining numrical solutions to problms involving continuous variabls. Numrical algorithms ar at last as old as th Egyptian Rhind papyrus (650 BC which dscribs a root finding for solving a simpl quation li nonlinar quations in gnral, a problm that rquirs th dtrmination of valus of unnowns,... n for which (,,..., 0,,, n (Encyclopdia.com. n Nonlinar quations ar quations whos graphsolution dos not form a straight lin. In a nonlinar quation, th variabls ar ithr of dgr gratr than on or lss than on or lss than but nvr on. (Mathmatics dictionary An quation which whn plottd on a graph dos not trac a straight lin but a curv is calld nonlinar quation (www.businss dictionary.com. Nonlinar quations ar solvd by itrativ mthods. A trial solution is assumd, th trial solution is substitutd into th nonlinar quation to dtrmin th rror, or mismatch, and th mismatch is usd in som systmatic mannr to gnrat an improvd stimat of th solution. Svral mthods for finding th roots of nonlinar quations ist. Such mthods of choic for solving nonlinar quations ar Nwton s mthod, th scant mthod tc. [6] Stats (approimatly that th solution of nonlinar quation is th most difficult problm in scintific computation. Equations that can b cast in th form of a polynomial ar rfrrd to as algbraic quations. Equations
Oori Charity Eblchuwu t al.: Comparison of Som Itrativ Mthods of Solving Nonlinar Equations involving mor complicatd trms such as trigonomtric, hyprbolic, ponntial or logarithmic functions ar rfrrd to as transcndntal quations. This in gnral is rfrrd as nonlinar quations. [5] It is wll nown that whn th Jacobian of nonlinar systms is nonsingular in th nighborhood of th solution, th convrgnc of Nwton mthod is guarantd and th rat is quadratic. Violating this condition, i.. th Jacobian to b singular, th convrgnc may b unsatisfactory and may vn b lost []. is a polynomial of dgr two or thr or four, If ( act formula ar availabl. But, if f ( is a transcndntal function li a + b + sin + d log tc. Th solution is not act whn th cofficint is numrical valus, it can adopt various numrical approimat mthods to solv such algbraic and transcndntal quations. [7], profssor of mathmatics (Rtd P.S.G Collg of tchnology Coimbator. Systms of nonlinar quations ar difficult to solv in gnral. Th bst way to solv ths quations is by itrativ mthods. On of th classical mthods to solv th systm of nonlinar quations is Nwton mthod which has scond ordr rat of convrgnc. This spd is low whn w compar to third ordr mthod. [9]. Many of th compl problms in scinc and nginring contain th function of nonlinar and transcndntal natur in th quation of th form f ( 0. Numrical mthods li Nwton s mthod ar oftn usd to obtain th approimat solution of such problms bcaus it is not always possibl to obtain its act solution by usual algbraic procss [8]... Statmnt of Problm Nonlinar quations ar approimatly on of th difficult quations in scincs and lss attntion is givn to thm compard to th othr forms of quations... Aim and Objctivs of th Study Th aim of this study is to dtrmin th mthod that is th bst in solving nonlinar quations using numrical mthods Th following ar th objctivs of th study; i. Dtrmining th istnc of a solution, convrgnc and rrors of th mthods; ii. Comparing th statd mthods to dtrmin th most appropriat mthod or mthods for th givn quation.. Matrials and Mthods For th purpos of this wor, th following mthods wr compard; Nwton mthod, scant mthod, th rgular falsi mthod and th bisction mthod... Nwton Mthod Nwton mthod is on of th most popular numrical mthod is oftn rfrrd as th most powrful mthod that is usd to solv for th quation f( 0, whr f is assumd to hav a continuous drivativ f '. This mthod originatd from th Taylor s sris pansion of th function f ( about th point, such that f ( f ( + ( f '( + ( f ''( + (! Whr f and its first and scond drivativ f ' and f '' ar calculatd at. Taing th first two trms of th Taylor s sris pansion, W hav ( ( ( '( + ( W thn st ( to zro (i.. f ( 0 to find th root of th quation which yilds ( ( ( + f ' 0 ( Rarranging ( abov, w obtain th nt approimation to th root, givn ris to ( '( (4 Thus gnralizing (4 w obtain th Nwton itrativ mthod + ( '(, whr Є N (5 Eampl: f( + -0 Apply Nwton s mthod to th quation. corrct to fiv dcimal placs. F (0 -, f(, thrfor th root lis btwn 0 and. + ( ( '( f ' + ( ( + ' + + Solving th right hand sid yilds + + Convrgnc of Nwton s Mthod Hr in Nwton s mthod + + ( '( This is rally an itration mthod whr
Intrnational Journal of Thortical and Applid Mathmatics 08; 4(: -8 4 + Ø ( and Ø ( - Hnc th quation is Ø ( whr Ø ( - ( '( ( '( Th squnc,,... convrgs to th act valu if Ø ( < i.. if i.. if ( '' ( f '( ( ''( f ' < f '( < This implis that Nwton mthod convrgs if ( ''( f '( <.. Scant Mthod Th scant mthod is just a variation from th Nwton s mthod. Th Nwton mthod of solving a nonlinar quation f ( 0 is givn by th itrativ formula. ( '(. (6 + (7 On of th drawbacs of th Nwton s mthod is that on hav to valuation th drivativ of th function. To ovrcom ths drawbacs, th drivativ of th function f ( is approimatd as ( f ' ( ( Substituting quation (8 into (7 + By factoring out f ( ( ( ( + This can also b writtn as ( ( ( ( ( + f ( f ( Th abov quation is calld th scant mthod. This mthod rquirs two initial guss. (8 (9 Eampl: Solv th quation f ( scant mthod. Corrct to fiv dcimal placs. Solution, f ( 0, ( f + using Th root lis btwn 0 and and taing 0 and By scant mthod, first approimation is whn ; f ( ( 0 Whn ; Whn ; Whn 4; Whn 5; ( ( 0 ( ( ( ( ( ( ( ( ( 4 ( 4 ( 4 ( ( 5 ( 5 4 ( 5 ( 4 ( 6 ( 6 5 ( 6 ( 5 ( 7 ( 7 6 ( 7 ( 6 f ( 8 ( 8 7 f ( 8 f ( 7 f ( 9 ( 9 8 f ( f ( 4 5 4 6 5 Whn 6; 7 6 Whn 7; Whn 8; Whn 9; 8 7 9 8 0 9 9 8 ( 0 ( 0 9 ( f ( Whn 0; 0 0 9 Convrgnc of Scant Mthod Say α, + α + Whr α root of f ( 0 +, ar rrors, approimation at itration + approimation ofα at + α root of f( 0 p + + (0 If + N whr N is a constant Thn th rat of convrgnc of th mthod is + ( ( ( f ( f ( ( ( ( ( ( ( (
5 Oori Charity Eblchuwu t al.: Comparison of Som Itrativ Mthods of Solving Nonlinar Equations Say f ( α 0 and i. rror in th th itration ( α + + + α All th abov is namd ( Using ( in ( + + α + α ( ( ( ( ( (4 By man valu thorm thr ist ω in th intrval and α such that W gt f '( ω Using (, w gt f ' ( + ( ω ( ( α f α i. f ( f '( ω f ' This givs th convrgnc... Rgular Falsi Mthod (5 (6 ( ω f '( ω ( ( (7 This mthod is also bing calld th mthod of fals position. f b Considr th quation f ( 0 and lt f ( a and ( b of opposit signs. Also, lt a < b. y will mt th - ais at som point Th curv ( btwn A (a, f ( a and B (b ( chord joining th two points A (a, f ( a and B (b, ( f b. Th quation of th ( ( ( y f a f a f b a a b f b is (8 Th -coordinat of th point of intrsction of this chord for th root of - ais givs an approimat valu for th root of f ( 0, stting y 0 in th chord quation, w gt ( ( ( f a f a f b a a b ( ( ( + ( af ( a + bf ( a f a f b af a af b Thrfor, ( ( ( ( f a f b bf a af a ( ( ( ( af b bf a f b f a (9 Th valu of givs an approimat valu of th roots of 0, (a< <b. ( Now ( and f ( a ar of opposit signs or ( f ( b ar of opposit sign. If f (. ( f a < 0, thn lis btwn and a af ( f f ( a Hnc f ( f ( a In th sam way w gt, 4... Th gnral formula is + ( ( ( ( af b bf a f b f a Eampl: Solv th quation f ( rgular falsi mthod, corrct to fiv dcimal placs Solution af ( b bf ( a + f ( b f ( a Whn 0, a 0, b ( ( 0 0.5 + f ( 0.75 and (0 + using Whn A root lis btwn0.5 and such that a0.5 and btc. Convrgnc of Rgular Falsi Mthod I b th squnc of approimations obtaind from ( ( ( ( + ( And α b th act valu of th root of th quation f( 0, thn Whr approimation. Hnc ( givs, α + + + + bing th rror involvd in th ( ( ( ( α + ( α + ( α + ( α + ( α + ( α + α + α + f α + f f + + f f f f + th
Intrnational Journal of Thortical and Applid Mathmatics 08; 4(: -8 6 Thrfor f( 0.4. Bisction Mthod ( α + '( α + ''( α +... ( α + '( α + ''( α +... f f f f f f!! f ( α + f '( α + f ''( α +... f ( α + f '( α + f ''( α +...!! ''... ( f ( α + ( f ( α +! ( ( + ( f '( α + f ' ( α! f ''( α +... + f '( α + f ''( α +... Suppos w hav an quation of th form f ( 0 whos solution is in th rang (a, b is to b dtrmind. W also assum that f ( is continuous and it can b algbraic or transcndntal. If f ( a and f ( b ar of opposit signs, thn at last on root ist btwn a and b. As a first approimation, w assum that root to b 0 (mid point of th nds of th rang. Now, find th sign of f ( 0. If ( 0 is ngativ, th root lis btwn 0 and b. Any on of this is tru. This solution is found by rpatd bisction of th intrval and in ach itration picing that half which also satisfis that sign condition. Th numbr of itration rquird may b b a dtrmind from th rlation ' +... to fiv dcimal placs. Using bisction mthod ( +,corrct to fiv dcimal placs. a + b Solution Whn 0 A root lis btwn 0 and such that a 0 and b Convrgnc of Bisction Mthod Th succssiv approimation of a root α of th quation f ( 0 is said to convrg to α with ordr q I + α c α Hr q>0,, and c is som constants gratr than 0 Whn q and 0<c<, thn th convrgnc is said to b of first ordr and c is calld th rat of convrgnc.. Rsults Th gnral formula is Eampl: Solving th quation ( a + b ( +, corrct Following th ampl ( + of th nonlinar quation and illustrating th rsults by th four mthods of solving nonlinar quations f( 0 in this wor, th rsults of th mthods ar prsntd in th graphs and tabl blow. K Nwton mthod 0 + X + Tabl. Th Rsults of th four mthods. Scant Mthod o0 X + + RFMX+ Bisction mthod X + 0.00 0.7500000 6.768 0 0.500.80-0.500000000.80 -.00 0.6860500.70 0.5000000.80-0.66666 6.768 0.750000000 4.5960 -.00 0.68400 0 5 0.664000 4.590 0.6795655 5.70-0.65000000.0 - s.00 0.6800 0 5 0.6899000 7.580 0.6796658 5.80 0.687500000.6570-4.00 0.6878 7.80 6 0.680000 5.880 0.68697.6070-0.65650000 6.8770-4 5.00 0.6878 7.80 6 0.68000 0-5 0.687604.04450-0.67875000.49760-4 6.00 0.6878 7.80 6 0.6800 0 0.68958.650-0.679687500.8480-5 7.00 0.6878 7.80 6 0.6880 80 6 0.68958.7750 0.6859750 8.40-7 8.00 0.6878 7.80 6 0.6878 7.80 6 0.685740 6.79750-4 0.6864065 5.740 6 9.00 0.6878 7.80 6 0.6878 7.80 6 0.6870.97870 4 0.686787 7.0 6 0.00 0.6878 7.80 6 0.0000000 6.80-0.687690.97750-4 0.688906 7.690 6.00 0.6878 0.0000000 0.6870 5.0 5 0.687000 7.0 6
7 Oori Charity Eblchuwu t al.: Comparison of Som Itrativ Mthods of Solving Nonlinar Equations K Nwton mthod 0 + X + Scant Mthod o0 X + + RFMX+ Bisction mthod X +.00 0.6878 0.0000000 0.68755 6.9050-5 0.6850950 7.550 6.00 0.6878 0.0000000 0.68774 8.00-6 0.68980 7.740 6 4.00 0.6878 0.0000000 0.687497.50 5 0.684500 7.4970 6 5.00 0.6878 0.0000000 0.68770 7.40 6 0.687400 7.70 6 6.00 0.6878 0.0000000 0.687785.4870 5 0.684870 7.00 0.6878 0.0000000 0.687799.0 5 0.68000 8.00 0.6878 0.0000000 0.68780 9.0 6 0.6890 9.00 0.6878 0.0000000 0.68780 8.80 6 0.68880 0.00 0.6878 0.0000000 0.68700 7.70 6 0.68770 Th rrors in th Scantmthod altrnat, this mans that th rrors movs from positiv to ngativ but rmains stabl at th point of convrgnc but also divrgs at a point. Figur. Th graph of Nwton mthod. Th rror in ach th itration in th Nwton Mthod convrgs positivly and rmains positiv with grat spd and accuracy throughout th valu of. Figur. Th graph of Rgular falsi mthod. In th rgular falsi mthod th rror in ach th itration convrgs ngativly but turn positiv at th point of convrgnc and rmains positiv throughout th valu of. Figur. Th graph of Scant mthod. Figur 4. Th graph of Bisction mthod.
Intrnational Journal of Thortical and Applid Mathmatics 08; 4(: -8 8 Th rror in th itration of th bisction mthod is altrnativ, this implis that it changs from positiv to ngativ but rmains positiv at th point of convrgnc. Th tabl abov compard th rsults of all th four mthods combind and Nwton mthod was found to b th bst mthod bcaus of its fast and accurat convrgnc to th roots, th Scant mthod also shows a high and spdy convrgnc but th mthod is du to fail at any tim or prhaps divrg to anothr solution i.. whnvr +. Th rgular falsi mthod also shows a lvl of spdy convrgnc but not as compard to th Nwton and scant mthods rspctivly but crtainly abov th bisction mthod. Th Bisction mthod has th slowst convrgnc to th roots than any of th othr thr mthods but its convrgnc to th root is sur no mattr th numbr of itrations. Aftr comparing all th four mthods it will b saf to say that th rat of on itration of th Nwton mthod is quivalnt to four itrations of th bisction mthod. 4. Discussion Itrativ mthod is on in which w start from an approimation to th tru solution and if succssful, obtain bttr approimation from a computational cycl rpatd as oftn as ncssary for achiving a rquird accuracy, so that th amount of arithmtic dpnds upon th accuracy rquird. It also shows th rat of convrgnc, with rgard to convrgnc, w can summariz that a numrical mthod with a highr rat of convrgnc may rach th solution of th quation with lss itrations in comparison to anothr mthod with a slowr convrgnc. Th istnc of a solution of a nonlinar quation with on variabl is dtrmind by th intrmdiat ( 0 valu thorm. From th mthods plaind in this wor, it can b discovrd that th Nwton mthod is th bst itrativ mthod for solving th nonlinar quations with on variabl as it convrgs mor rapidly and accuratly to th root of th quation whn th initial guss is clos to th roots of th quation. It is also clar to say that for scant, at any givn tim bfor it convrgs to th root of th quation whr +, thn it mans that th mthod will fail from that point. Th rgular Falsi mthod convrgs mor slowly compar to th Nwton and scant mthod rspctivly and th bisction mthod rquirs a larg numbr of itrations bfor it convrgs to th root of th quationirrspctiv of how on starts clos to th roots of th quation. 5. Conclusion From th mthods tstd, th Nwton mthod appard to b th most robust and capabl mthod of solving th nonlinar quation f ( 0. Rsults obtaind from th four mthods abov show that th Nwton Mthod is th most fficint mthod in finding th roots of non-linar quations sing that it convrgs to th roots of th non-linar quation fastr than th othr thr mthods. That is it convrgs aftr a fw itrations unli th othr thr mthods which convrgs aftr many itrations. Rfrncs [] Aisha H.A; Fatima W.L; Waziri M.Y. (04. Intrnational journal of computr application, vol 98businss dictionary. (06. www.businss dictionary.com. copyright 00-06, wb financ, In [] Dass H.K. and Rajnish Vrma (0.Highr Enginring Mathmatics. Publishd by S. Chand and Company ltd (AN ISO 900008 Company. Ram Nw Dlhi- 0055. [] Dborah Dnt and Marcin Paaprzyci. (000.Rcnt advancs in solvrs for nonlinar algbraic Equations. School of mathmatical scincs, Univrsity of Southrn Mississippi Hattisburg. [4] Erwin Krysig (0. Advanc Enginring Mathmatics. Tnth dition. Publishd by John Wilyand sons, inc. [5] Fr dictionary. (0. Amrican Hritag dictionary of th English languag, fifth dition, copyright by Houghton Mifflin Harcourt publishing company. [6] Gibrto E. Urroz. (004.Solution of nonlinar quations. A papr documnt on solving nonlinarquation using Matlab. [7] John Ric (969, Approimation of functions: Nonlinar and Multivariat Thory, Publishr; Addison Wsly Publishing Company. [8] Kandasamy P. (0. Numrical mthods. Publishd by S. Chand and company ltd (AN ISO 900; 000 Company. Ram Nagar, nw-dlhi 0 055 [9] Masoud Allam. (00. A nw mthod for solving nonlinar quations by Taylor s Epansion. Confrnc papr, Islamic Azad Univrsity. Numrical analysis Encyclopdia Britannica onlin. <www.britannica/com/topic/numrical analysis >. [0] Sara T.M. Suliman. (009.Solving Nonlinar quations using mthods in th Hally class. Thsis for th dgr of mastr of scincs. [] Sona Tahri, Musa Mammadov (0, Solving Systms of Nonlinar Equations using a Globally Convrgnt Optimization Algorithm Transaction on Evolutionary Algorithm and Nonlinar Optimization ISSN: 9-87 Onlin Publication, Jun 0www.pcoglobal.com/gjto.htm