Itertiol Jourl of Mthemtics d Sttistics Ivetio (IJMSI) E-ISSN: 31 767 P-ISSN: 31-759 Volume Issue 8 August. 01 PP-01-06 Some New Itertive Methods Bsed o Composite Trpezoidl Rule for Solvig Nolier Equtios Ogbereyivwe Oghovese 1, Emuefe O. Joh 1 ( ( Deprtmet of Mthemtics d Sttistics, Delt Stte Polytechic, Ozoro, Nigeri) (Deprtmet of Geerl Studies, Petroleum Triig Istitute, Effuru, Delt Stte, Nigeri) ABSTRACT: I this pper, ew two steps fmily of itertive methods of order two d three costructed bsed o composite trpezoidl rule d fudmetl theorem of clculus, for solvig olier equtios. Severl umericl exmples re give to illustrte the efficiecy d performce of the itertive methods; the methods re lso compred with well kow existig itertive method. KEYWORDS: Nolier equtios, computtiol order of covergece, Newto s method I. INTRODUCTION Solvig olier equtios is oe of the most predomit problems i umericl lysis. A clssicl d very populr method for solvig olier equtios is the Newto s method. Some historicl poits o this method c be foud i [1]. Recetly, some methods hve bee proposed d lyzed for solvig olier equtios [-13]. Some of these methods hve bee suggested either by usig qudrture formuls, homotopy, decompositio or Tylor s series [-13]. Motivted by these techiques pplied by vrious uthors [-13] d refereces therei, i costructig umerous itertive methods for solvig olier equtios, we suggest two steps fmily of itertive method bsed o composite trpezoidl rule d fudmetl theorem of clculus for solvig olier equtios. We lso cosidered the covergece lysis of these methods. Severl exmples of fuctios, some of which re sme s i [-13] were used to illustrte the performce of the methods d compriso with other existig methods. We use the followig defiitios: II. PRELIMINARIES Defiitio 1. (See Deis d Schble [] ) Let α R, x R, = 0,1,, The, the sequece x is sid to coverge to α if lim x α = 0 (1) If, i dditio, there exists costt c 0, iteger x 0 0, d p 0 such tht for ll x 0, x +1 α c x α p () the x is sid to coverge to α with q-order t lest p. If p =, the covergece is sid to be of order. Defiitio (See Gru-Schez et l. [1]) The computtiol locl order of covergece, ρ, (CLOC) of sequece x 0 is defied by ρ = log e log e, (3) 1 where x 1 d x re two cosecutive itertios er the roots α d e = x 1 α. Nottio 1: (See [6]) The ottio e = x α is the error i the th itertio. The equtio e +1 = ce p + O e p+1, () is clled the error equtio. By substitutig e = x α for ll i y itertive method d simplifyig, we obti the error equtio for tht method. The vlue of p obtied is clled the order of this method. 1 P g e
III. DEVELOPMENT OF THE METHODS Cosider olier equtio f x = 0 (5) By the Fudmetl Theorem of Clculus, if f(x) is cotiuous t every poit of [, b] d F is y tiderivtives of f(x) o [, b], the b f(x) dx = F b F (6) Differetitig both side of (6) with respect to x, we hve; f x = f b f (7) where f(b) d f() re derivtives of F(b) d F() respectively. Recll the Composite Trpezoidl rule give by; 1 b b f(x) dx = f + f i + f(b) (8) i=1 If = i (8) we hve; b b + b f(x) dx = f + f + f(b) (9) Differetitig (9) with respect to x, we hve; f x = b + b f + f + f b (10) Equtig (7) d (10) we hve; f b f = b + b f + f + f b (11) From(5), we hve; x = f() + b f x f x f f x f (1) Usig(1), oe c suggest the followig itertive method for solvig the olier equtio(5). Algorithm 1: Give iitil pproximtio x 0 (close to α the root of (5)), we fid the pproximte solutio x +1 by the implicit itertive method: x +1 = x f(x x + x +1 ) f x x f x f +1 f x x f, = 0,1,, (13) x The implicit itertive method i (13) is predictor-corrector scheme, with Newto s method s the predictor, d Algorithm 1 s the corrector. The first cosequece of (13) is the suggested two-step itertive method for solvig (5) stted s follows: Algorithm : Give iitil pproximtio x 0 (close to α the root of (5)), we fid the pproximte solutio x +1 by the itertive schemes: y = x f(x ) x +1 = x f(x x + y ) f x y x f y f x y x f From(1), we hve tht; y x = f(x ) Usig (15) i (1b) we suggest other ew itertive scheme s follows: (1), = 0,1,, (1b) Algorithm 3: Give iitil pproximtio x 0 (close to α the root of (5), we fid the pproximte solutio x +1 by the itertive schemes: (15) P g e
x +1 = x f(x ) + f(x ) f y f x f(x ) + y, = 0,1,, (16) From (5) d (11) we c hve the fixed poit formultio give by f() x = f + f + x (17 ) + f x The formultio (17 ) eble us to suggest the followig itertive method for solvig olier equtios. Algorithm : Give iitil pproximtio x 0 (close to α the root of (5), we fid the pproximte solutio x +1 by the itertive schemes: y = x f(x ) f (x ) f(x ) x +1 = x + + y, = 0,1,, (18) + f (y ) I the ext sectio, we preset the covergece lysis of Algorithm d. Similr procedures c be pplied to lyze the covergece of Algorithm 3. IV. CONVERGENCE ANALYSIS OF THE METHODS Theorem 1: Let α I be simple zero of sufficietly differetible fuctio f: I R R for ope itervl I. If x 0 is sufficietly close to α, the the itertive method defied by (1) is of order two d it stisfies the followig error equtio: where e +1 = α 3c e + c 3 + 6c 3 c e 3 + O e 3 (19 ) c = f (α) f (α) (0) Proof Let α be simple zero of f, d e = x α. Usig Tylor expsio roud x = α d tkig ito ccout f α = 0, we get f x = f α e + c e + c 3 e 3 + c e +, (1) = f α 1 + c e + 3c 3 e + c e 3 + 5c 5 e + () where c k = fk (α), k =,3,, k!f (α) (3) Usig 1 d, we hve; f(x ) f x = e c e + (c c 3 )e 3 + (7c c 3 c 3 3c )e + () But Hece, y = x f x 5 = α + c e + (c c 3 )e 3 (7c c 3 c 3 3c )e + (6) y x = e + c e + (c c 3 )e 3 (7c c 3 c 3 3c )e + (7) From 6, we hve; f y = f α 1 + c e + c c 3 c 3 e 3 + ( 11c c 3 + 6c c + 8c )e + (8) Combiig () d (8), we hve; f y = c e + 3c 3 + 6c e + 16c 3 c + 16c c 3 e 3 + (9) From (7) d (9) we hve; y x f y = e + 3c e + (5c 3 10c )e 3 + 30c c 3 + 30c 3 + 7c e + (30) 3 P g e
From the reltio; x + y we hve; f x + y + y = x f x = α + 1 e + 1 c e c c 3 e 3 1 7c c 3 c 3 3c e + (31 ) = f x f x = f(α) 1 e + 3 c e + 1 c + 9 8 c 3 e 3 + 5 c 3 17 8 c c 3 + 5 16 c e + 3c + 57 8 c 3c 9 c 3 13 c c + 65 3 c 5 e 5 + (3) = f(α) 1 + c e + c + 3 c 3 e + c 3 + 7 c c 3 + 1 c e 3 Usig () d (33) we hve; + 9 c c + c 37 c c 3 + 3c 3 + 5 16 c 5 e + (33) + y = 1 c e + 3c 3 c 3 3c 3 e + (3) Ad (7) with (3) gives; x y x f + y = e + c e + 3 c + c 3 c e 3 + (35) Usig 9, (30) d (35) i x +1 = x f x y x f y y x f x + y = α 3c e + c 3 + 6c 3 c e 3 + O e (36) Thus, we observe tht the Algorithm is secod order coverget. Theorem : Let α I be simple zero of sufficietly differetible fuctio f: I R R for ope itervl I. If x 0 is sufficietly close to α, the the itertive method defied by (18) is of order three d it stisfies the followig error equtio: where c 3 = f (α) 3!f (α) Proof Usig, (33) d (8) we hve; e +1 = α + 1 8 c 3 + c e 3 + O e (37 ) + + y + f y = f α + c e + 9 c 3 + c e + 5c + 11c c 3 8c 3 e 3 + ( 11c c 3 + 6c c + 8c )e + (39) d from (1) we hve; (38) = f α e + c e + c 3 e 3 + c e + (0) Combiig (39) d (0) i (18) gives; P g e
e +1 = x f x + + y + f y = α + 1 8 c 3 + c e 3 + O e (1) This mes the method defied by (18) is of third-order. Tht completes the proof. V. NUMERICAL EXAMPLES I this sectio, we preset some exmples to illustrte the efficiecy of our developed methods which re give by the Algorithm 1. We compre the performce of Algorithm (AL) d Algorithm (AL) with tht of Newto Method (NM). All computtios re crried out with double rithmetic precisio. Displyed i Tble 1 re the umber of itertios (NT) required to chieve the desired pproximte root x d respective Computtiol Locl Order of Covergece (CLOC), ρ. The followig stoppig criteri were used. i. x +1 x < ε ii. f x +1 < ε (37) where ε = 10 15. We used the followig fuctios, some of which re sme s i [-,6-1,1] f 1 x = x 1 3 1 f x = cos x x f 3 x = x 3 10 f (x) = x e x 3x + f 5 x = x + e x 1 f 6 (x) = x 3 + x 10 f 7 x = lx + x 5 f 8 x = e x six + l (x + 1) (38) Tble 1: Compriso betwee methods depedig o the umber of itertios (IT) d Computtiol Locl Order of Covergece. f(x) x 0 Number of itertios (NT) Computtiol Locl Order of Covergece (CLOC) NM AL AL NM AL AL f 1 3.5 7 8 5 1.99999 1.9189.99158 f 1.7 5 3.191 1.89891 3.5551 f 3 1.5 6 3.05039 1.97063 3.18850 f 5 6.1719.10516 3.355 f 5 9 3 5.03511 1.0301 3.17161 f 6 5 6 3.06888 1.97365 3.18 f 7 7 5 3.38378.16311.17738 f 8 0.5 6 1 5 1.9071 1.86901.00000 The computtiol results preseted i Tble 1 shows tht the suggested methods re comprble with Newto Method. This mes tht; the ew methods (Algorithm i prticulr) c be cosidered s sigifict improvemet of Newto Method, hece; they c serve s ltertive to other secod d third order coverget respectively, methods of solvig olier equtios. VI. CONCLUSION We derived two step fmily of itertive methods bsed o composite trpezoidl rule d fudmetl theorem of clculus, for solvig olier equtios. Covergece proof is preseted i detil for lgorithm d d they re of order two d three respectively. Alysis of efficiecy showed tht these methods c be used s ltertive to other existig order two d three itertive methods for zero of olier equtios. Filly, we hoped tht this study mkes cotributio to solve olier equtios. 5 P g e
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