SOME MULTI-STEP ITERATIVE ALGORITHMS FOR MINIMIZATION OF UNCONSTRAINED NON LINEAR FUNCTIONS

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1 I.J.E.M.S. VOL. : -7 ISS 9-6X SOME MULTI-STEP ITERATIVE ALGORITHMS FOR MIIMIZATIO OF UCOSTRAIED O LIEAR FUCTIOS K.Karthikea S.K.Khadar Babu M.Sudaramurth B.Rajesh Aad School o Advaced Scieces VIT Uiversit Vellore- 6. Departmet o Mathematics S.V.Uiversit Tirupath-575 ABSTRACT I this paper we propose ew iterative alorithms or miimizatio o ucostraied o liear uctios. The comparative stud is made betwee ew alorithms ad ewto s alorithms b meas o eamples. Ke words: Iterative alorithms; ewto s alorithms; Modiied homotop perturbatio method.. ITRODUCTIO Optimizatio problems with or without costraits arise i various ields such as sciece eieeri ecoomics maaemet scieces etc. where umerical iormatio is processed. I recet times ma problems i busiess situatios ad eieeri desis have bee modeled as a optimizatio problem or taki optimal decisios. I act umerical optimizatio techiques [] have made deep i to almost all braches o eieeri ad mathematics. A ucostraied miimizatio problem is the oe where a value o the vector is souht that miimizes the objective uctio. This problem ca be cosidered as particular case o the eeral costraied o-liear prorammi problem. The stud o ucostraied miimizatio techiques provide the basic uderstadi ecessar or the stud o costraied miimizatio methods ad this method ca be used to solve certai comple eieeri aalsis problem. For eample the displacemet respose liear or o-liear o a structure uder a speciied load coditio ca be oud b miimizi its potetial eer. Via Kawar et.al. itroduced ew alorithms called eteral touch techique ad orthooal itersectio techique[] or solvi the o liear equatios. Further the did the comparative stud o the ew alorithms ad ewto s alorithm. The umerical techiques or solvi oliear equatio have bee applied successull duri the last ew ears. There are ma research articles that deal with solvi oliear equatios. The homotop perturbatio method HPM was established b He i 999[] ad sstematical descriptio i [] which is a coupli o the traditioal perturbatio method ad homotop i topolo []. This method was urther developed ad improved b He ad applied to o liear oscillators with discotiuous [] oliear wave equatios[5] asmtotolo[6] boudar value problem[ [7] limit ccle ad biurcatio o oliear uctios problems[8] ad ma other subjects. Thus He s method is a uiversal oe which ca solve various kids o oliear equatios. Subsequetl ma researchers have applied the method to various liear ad o liear problems[ 9 ]. There has bee some developmet o iterative methods with hiher order coverece that required the computatio o derivatives o as low order as possible [-]. Recetl Ari Raiq ad Muhammad Raiullah[] itroduced a ew article which is some multi-step iterative methods or solvi o liear equatios based o modiied homotop perturbatio method. I this paper we itroduce two ew alorithms amel ew alorithm-i ad ew alorithm-ii or miimizatio o o liear uctios which is based o modiied homotop perturbatio method. The comparative stud is made with ew alorithms ad ewto s alorithm b meas o eamples.. EW ALGORITHMS I this sectio we itroduce two ew iterative umerical alorithms or miimizatio o oliear real valued ad twice dieretiable real uctios usi the cocept o modiied homotop perturbatio alorithms. based umerical Cosider the oliear optimizatio problem : Miimize { R : R R } where is a o-liear twice dieretiable uctio... Two step Iterative Alorithms Cosider the uctio G G. Here is the uctio to be miimized. is deied aroud the critical poit * o i ad is ive b where

2 Some Multi-Step Iterative Alorithms For Miimizatio o Ucostraied o Liear Fuctios G. I we assume that. we have G Cosider the o liear alebraic equatio = i R ---. We assume that is a simple zero o ad α is a iitial uess o suicietl close to. We have the ollowi theorem. Theorem.: Suppose is cotiuous o [a b] ad dieretiable i a b. The there eists a poit a b such that... Usi the above theorem aroud α or the equatio. we have where lies betwee ad α. We ca rewrite. i the ollowi orm ad the equatio. where c = ad = c Proo: Reer the article [] We costruct a homotop mail due to Golbabai ad Javidi [] : R [ ] R R or. which satisies c R [ ] where is ukow real parameter ad β is embeddi parameter. It is obvious that c c The embeddi parameter β icreases mootoicall rom zero to uit as the trivial problem. c is cotiuousl deormed to the oriial problem c. The modiied HPM uses the homotop parameter β as a epadi parameter to obtai [] The approimate solutio o. thereore ca be readil obtaied lim The coverece o the series. has bee proved b He i his paper []. For the applicatio o the modiied HPM to. we ca write. as ollows b epadi ito a Talor series aroud : cβ... β βθ!! ---.

3 I.J.E.M.S. VOL. : -7 ISS 9-6X Substituti.9 i to. ields....!...! c B equati terms with idetical powers o β we have....6 :.5 :. :. : c where ad Substituti. ito.5 ad letti = we obtai ---. From..5 ad. -. we have ' ' '. 8.. Substituti. -. i. we ca obtai the solutio o. as ollows This ormulatio allows us to suest the ollowi iterative method or solvi oliear equatio. Alorithm-: For a ive calculate the approimatio solutio + b the iterative scheme Alorithm-: For a ive calculate the approimatio solutio + b the iterative scheme

4 Some Multi-Step Iterative Alorithms For Miimizatio o Ucostraied o Liear Fuctios Alorithm-: For a ive calculate the approimatio solutio + b the iterative scheme Sice the equatio.7 ad.8 becomes ew Alorithm I Sice the equatio.9 ad. becomes ew Alorithm II COVERGECE AALYSIS Theorem.: Cosider the o liear equatio =. Suppose is suicietl dieretiable. The or the iterative method deied b Alorithm the coverece is atleast o order. Proo: Reer the article [] The same steps i the proo o Theorem. is ollowed to et the coverece o Alorithm which is atleast o order. Reer the article[].umerical ILLUSTRATIOS Eample.: Cosider the uctio R 5. The miimizi poit o the uctio is equal to.8667 which is obtaied i iteratios b ewto s method i iteratios b ew alorithm-i ad i iteratios b ew alorithm II or the iitial value = ad also see that o variatios i iteratio or the iitial value o = ad =. Iteratios ewto s Alorithm ew Alorithm- I ew Alorithm-II

5 I.J.E.M.S. VOL. : -7 ISS 9-6X Iteratios ewto s Alorithm ew Alorithm-I ew Alorithm-II Iteratio ewto s Alorithm ew Alorithm-I ew Alorithm-II s Eample.: Cosider the uctio R The miimizi poit o uctio is equal to which is obtaied i iteratios b ewto s Alorithm i iteratios b ew alorithm-i ad i iteratios b ew alorithm-ii or the iitial value = ad also see that o variatio o iteratios or the iitial value o = ad =. Iteratios ewto s alorithm ew Alorithm-I ew Alorithm-II Iteratios ewto s alorithm ew Alorithm-I ew Alorithm-II

6 Some Multi-Step Iterative Alorithms For Miimizatio o Ucostraied o Liear Fuctios Iteratios ewto s alorithm ew Alorithm-I ew Alorithm-II Eample.: Cosider the uctio e R. The miimizi poit o the uctio is equal to - which is obtaied i 7 iteratios b ewto s alorithm 7 iteratios b ew alorithm-i ad i 7 iteratios b ew alorithm-ii or the iitial value = ad also see that o variatios i iteratios or the iitial value o = ad =. Iteratios ewto s alorithm ew Alorithm- I ew Alorithm-II Iteratios ewto s alorithm ew Alorithm-I ew Alorithm-II

7 I.J.E.M.S. VOL. : -7 ISS 9-6X Iteratios ewto s alorithm ew Alorithm-I ew Alorithm-II COCLUSIO I this paper we propose two ew iterative alorithms amel ew alorithm I ad ew alorithm II or miimizatio o ucostraied oliear uctios ad the the comparative stud is made with ewto s Alorithm. It is clear rom the umerical results that the rate o coverece o ew alorithms is almost equal to ewto s Alorithm. B the umerical results we have the rate o coverece o ew alorithms ad ewto s alorithms are almost all same. So we have alterative alorithms or ewto s alorithm or miimizatio o ucostraied o liear uctios is established. I uture we ma eted the ew alorithms or costraied optimizatio problems 6. REFERECES [] A. Golbabai M. Javidi7 A third-order ewto tpe method or oliear equatios based o modiied homotop perturbatio method Appl. Math.Comput [] S. Abbasbad Improvi ewto_raphso method or oliear equatios b modiied Adomia decompositio method Appl. Math. Comput [] E. Babolia J. Biazar Soluti o o oliear equatios b modiied Adomia decompositio method Appl. Math. Comput [] J. Biazar A. Amirteimoori6 A improvemet to the ied poit iterative method Comput [5] M. Basto V. Semiao F.L. Calheiros6 A ew iterative method to compute oliear equatios Appl. Math. Comput [6] C. Chu 5 Iterative methods improvi ewto's method b the decompositio method Comput. Math. Appl [7] H.H.H. Homeier 5 O ewto-tpe methods with cubic coverece J. Comput. Appl. Math [8] A.Y. Özba Some ew variats o ewto's method Appl. Math. Lett [9] S. Weerakoo T.G.I. Ferado A variat o ewto's method with accelerated third-order coverece Appl. Math. Lett [] W. Rudi Priciples o Mathematical Aalsis McGraw-Hill Book Compa Ic. 95. [] E. Isaacso H.B. Keller Aalsis o umerical Methods Joh Wile & Sos Ic ew York USA 966. [] J.H. He999 Homotop perturbatio techique Comput. Methods Appl. Mech. E. 78 _ [] J.H. He A coupli method o a homotop techique ad a perturbatio techique or o-liear problems It. J. o-liear Mech

8 Some Multi-Step Iterative Alorithms For Miimizatio o Ucostraied o Liear Fuctios [] J.H. He The homotop perturbatio method or o-liear oscillators with discotiuities Appl. Math. Comput [5] J.H. He5 Applicatio o homotop perturbatio method to oliear wave equatios Chaos Solitos Fractals [6] J.H. He Asmptotolo b homotop perturbatio method Appl. Math. Comput [7] J.H. He6 Homotop perturbatio method or solvi boudar problems Phs. Lett. A. 5 _ [8] J.H. He5 Limit ccle ad biurcatio o oliear problems Chaos Solitos Fractals [] A. Raiq M. Raiullah 8 Some ew multi -step iterative methods or solvi oliear equatios usi modiied homotop perturbatio method oliear Aal. Forum []. Ari Raiq Muhammad Raiullah9 Some multistep iterative methods or solvi oliear equatios Computers ad Mathematics with Applicatios []. Moha C Joshi ad Kaa M Moudala " Optimizatio theor ad Practice " arosa Publicatio House ew Delhi. []Via Kawar Sukhjit Sih J.R.Sharma ad Mamta ew umerical techiques or solvi o-liear equatios. Idia J. pure appl. Math. 9: 9 9. [9] J.H. He A ew iterative method or solvi alebraic equatios Appl. Math. Comput