A NEW COMBINED BRACKETING METHOD FOR SOLVING NONLINEAR EQUATIONS

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1 Aville online t htt://ik.org J. Mth. Comut. Si. 3 (013), No. 1, ISSN: A NEW COMBINED BRACKETING METHOD FOR SOLVING NONLINEAR EQUATIONS M.A. HAFIZ Dertment of mthemti, Fulty of Siene nd rt, Njrn Univerity, Sudi Ari Atrt: An imroved method ed on omintion of ietion, regul fli, nd roli interoltion h een develoed. A new lgorithm h een etlihed. The numer of itertion enure onvergene with reet to mny emle ulihed in the literture. Keyword: Nonliner eqution, Proli interoltion, Muller method, Bietion, Regul fli. 000 AMS Sujet Clifition: 65H05, 65B99, 65G40 1. Introdution Finding root of nonliner eqution effiiently h widered lition in numeril mthemti nd lied mthemti. If the funtion i rel, ontinuou, nd hnge ign over known intervl, the method of regul fli (or fle oition) n e ued. To eed u onvergene, ietion method utilize to divide the erh intervl in hlf with eh itertion. The eed of onvergene of the ietion method n e imroved y fitting qudrti olynomil to the funtion t the endoint nd midoint of the erh intervl. The root of thi ieted qudrti then eome one of the endoint of the new erh intervl in the net itertion. The funtion nd the oited qudrti will eome more liner over the erh intervl, thu imroving onvergene eed even more. Thi method of root finding, lled ieted diret qudrti regul fli [1]. A withing mehnim etween the ietion nd regul fli revent the low onvergene of the rooed lgorithm []. Muller [3] ued qudrti olynomil interoltion inted of liner interoltion for the root determintion, The Muller method n lo e found in the Reeived Aug 1, 01 87

2 88 M.A. HAFIZ omintion of the ietion or invere qudrti interoltion [4-5]. Reently, everl new lgorithm nd method were ulihed [6-1]. In the reent er we ue omintion of the ietion nd regul fli with the eond order olynomil interoltion tehnique ed on one end oint, midoint nd regul fli oint, thi method lled BRFC. Numeril tet on vriety of funtion how tht BRFC require fewer itertion thn other regul fli or ietion method.. Prooed zero finding method Firtly, it will e umed tht the funtion f () i ontinuou on loed intervl,. In thi intervl, the funtion h root nd the following inequlity hold 0 f f, we im to determine the root * of the nonliner eqution with let numeril evlution oile: Figure 1: Root finding y Bieted nd regul fli nd urve fitting P() f ( ) 0. (1) A new itertive vlue on the loed intervl i lulted y fitting rol to the three oint of the funtion f ( ). The firt oint i the intervl order oint

3 A NEW COMBINED BRACKETING METHOD 89 (,( )), the eond oint (, f ( )) ; (, ) i lulted y uing the ietion Fig. 1. () while the third oint (, f ( )); (, ) i lulted y uing regul fli lgorithm (Fig. 1) If we rele y After lulting nd f f., three oint (, f ( )), (, f ( )) nd (, f ( )) re finlly ville nd through thee oint, eond order olynomil n e ontruted follow: ( ) A( )( ) B ( )( ) C ( )( ) The ontnt A, B, nd C n e determined from the following ondition: (3) (4) P ( ) f ( ) A ( )( ), P ( ) f ( ) B ( )( ), P ( ) f ( ) C ( )( ). Then, A, B nd C n e determined from: (5) f ( ) f ( ) f ( ) A, B, C ( )( ) ( )( ) ( )( ) (6) To determine the net roimtion y onidering the interetion of the -i with the rol defined in eqution (4). The zero of the rol n e lulted from: 1, (7) 4 where the three rmeter, nd n e determined from: A B C, A ( ) B ( ) C ( ), f ( ). Eq. (7) give two oiilitie for 1, deending on the ign reeding the rdil term, therefore, Muller' method hooe the ign to gree with the ign of. Choen (8)

4 90 M.A. HAFIZ in thi mnner, the denomintor will e the lrget in mgnitude nd will reult in 1, eing eleted the loed zero of P() to. Thu we hve: (9) ign( ) 4 Eqution (7) n e rewritten in n itertive form y introduing i n roimte vlue of the funtion zero in the urrent itertion. A new lulted vlue i then i1. i1 i (10) ign( ) 4 Beue nd re very loer to *, ymtoti uerliner onvergene for imle root * of nonliner eqution f 0 nd the rooed method i very effetive with reet to the regul fli-ietion-roli (RBP) method []. Sine the onvergene order of the ietion nd regul-fli method i onvergene order of MRBP i lo. See [] then the 3. The new Bietion-Regul Fli Algorithm Proedure of Bietion- regul fli to find olution to f 0 given,, with 0 N m f f. Clultion reiion nd mimum numer of itertion Strt MRBP lgorithm from i =1. INPUT,,, f() nd mimum numer of itertion OUTPUT Aroimte olution * or mege of filure. N m. Ste 1. Comute the initil ietion: f f Ste. Comute the initil regul fli: Ste 3. If then et Ste 4. Clulte Funtion vlue: f ( ), f ( ), f ( ) nd f f or f 0 or f 0 or 0 Ste 5. if 0 nd to Ste 6. Set f rint *, f *

5 A NEW COMBINED BRACKETING METHOD 91 f ( ) f ( ) f ( ) A, B, C ( )( ) ( )( ) ( )( ) Ste 7. Set A B C, A ( ) B ( ) C ( ), f ( ). Ste 8. Set ign( ) 4 Ste 9. Clulte Funtion vlue: f Ste 10. if 0 f or Ste 11. Set X =ort (,,,, ) Ste 1. for j = 1 to 4 do End for Ste 13. if i < End of the MRBP lgorithm. f nd to * * rint, If f ( X j) f ( X j1) 0 then X, X j1 end if * * N m then i = i +1 nd go to te 1. ele rint, f nd to 4. Numeril emle We now omre the erformne of the reented BRFC method with different method. The numer of itertion n i lulted for the ietion (Bi), regul fli (Reg) nd Suhdolnik (Suh) lgorithm. The omrion re ed on the numer of itertion n whih re reented in Tle 1. Thi tle ontin the tet funtion ued to tet the erformne of oth method. The numeril omuttion lited in the Tle 1 were erformed on Mle Conluion Thi er reent rketing lgorithm for the itertive zero finding of nonliner eqution. The lgorithm i ed on the omintion of the ietion nd regul fli

6 9 M.A. HAFIZ Tle 1. Different rketing method y reenting the numer of itertion n N m 10, 10 Bi Reg Suh Preent BRFC No f () [, ] Numer of itertion n 1 ln [0.5, 5] in (10 ) e 1 [0.5, 8] 53 > e 1 [1, 4] [0.5, 1] in 1 [0.1, ] in 5 [0, 1] ( 1) e [0, 1.5] o [0, 1.7] ( 1) 1 [1.5, 3] e 1 [.6, 3.5] rtn 1 [1, 8] e 1 [0., 3] e in 14 3 [0, 5] [0.1,1.5] in 1 [ 1, ] in, , > , 3 48 >

7 A NEW COMBINED BRACKETING METHOD 93 with eond order olynomil interoltion tehnique. The onvergene of the lgorithm i uerliner. The rooed lgorithm n e ued good utitute for well-known rketing method. The trength of the lgorithm re reented on ome tyil emle nd omrion with other method i given. Aknowledgment * The uthor thnk Dr. A.E. Almir for hi owerful id during rering thi rtile. REFERENCES [1] G. Roert Gottlie nd Blir F. Thomon: Bieted Diret Qudrti Regul Fli. Al. Mth. Si. Vol. 4, (010), No. 15, [] A. Suhdolnik, Comined rketing method for olving nonliner eqution, Al. Mth. Lett. (01), doi: /j.ml. (01) [3] D.E. Muller, A method for olving lgeri eqution uing n utomti omuter, Mth.Tle Other Aid Comut. 10 (1956) [4] X.Wu, Imroved Muller method nd ietion method with glol nd ymtoti uerliner onvergene of oth oint nd intervl for olving nonliner eqution, Al. Mth. Comut. 166 (005) [5] F.Cotile, M.I. Gultieri, R. Lueri, A modifition of Muller method, Clolo 43 (006) [6] J.H. Chen, New modified regul fli method for nonliner eqution, Al. Mth. Comut. 184 (007) [7] P.K. Prid, D.K. Gut, A Cui onvergent itertive method for enloing imle root of nonliner eqution, Al. Mth. Comut, 187 (007) [8] P.K. Prid, D.K. Gut, An Imroved Regul Fli Method for Enloing Simle Zero of Nonliner Eqution, Al. Mth. Comut, 177 (006) [9] J.H. Chen, Z.H. Shen, On third-order onvergent regul fli method, Al. Mth. Comut. 188 (007) [10] M.S. M. Bhgt, M.A. Hfiz, Solving Nonmooth Eqution Uing Derivtive-Free Method, Bulletin of Soiety for Mthemtil Servie nd Stndrd Vol. 1 No. 3 (01), [11] M.A.Hfiz, M.S.M. Bhgt, An Effiient Two-Ste Itertive Method for Solving Sytem of Nonliner Eqution. Journl of Mthemti Reerh; 4, (4), (01) [1] M.A.Hfiz, M.S.M. Bhgt, Modified of Houeholder itertive method for olving nonliner ytem of eqution. J. Mth. Comut. Si. (01), No. 5,