International Journal of Mathematical Archive-3(12), 2012, Available online through ISSN

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1 Iteratioal Joural o Mathematical Archive-, 0, Available olie throuh ISSN NEW ITERATIVE NUMERICAL ALGORITHMS FOR MINIMIZATION OF NONLINEAR FUNCTIONS K. Karthikea* School o Advaced Scieces, Mathematics Divisio VIT Uiversit, Vellore-604, Idia. Received o: 0--; Revised & Accepted o: -- ABSTRACT I this paper, we propose ew ew alorithms, or miimizatio o oliear uctios. The comparative stud amo the ew alorithms ad Newto s alorithm is established b meas o various eamples. Ke words: Noliear uctios; Newto s method; Halle s method; Modiied Halle s method; Order o coverece.. INTRODUCTION I recet times, ma problems i busiess situatios ad eieeri desis have bee modeled as a optimizatio problem or taki optimal decisios. Optimizatio problems with or without costraits arise i various ields such as sciece, eieeri, ecoomics, maaemet scieces, etc., where umerical iormatio is processed I act, umerical optimizatio techiques have made deep i to almost all braches o eieeri ad mathematics. Several methods [,, 6] are available or solvi ucostraied miimizatio problems. These methods ca be classiied i to two cateories as o radiet ad radiet methods. The o radiet methods require ol the objective uctio values but ot the derivatives o the uctio i idi miimum. The radiet methods require, i additio to the uctio values, the irst ad i some cases the secod derivatives o the objective uctio. Sice more iormatio about the uctio bei miimized is used throuh the use o derivatives, radiet methods are eerall more eiciet tha o radiet methods. All the ucostraied miimizatio methods are iterative i ature ad hece the start rom a iitial trial solutio ad proceed towards the miimum poit i a sequetial maer. To solve ucostraied oliear miimizatio problems arisi i the diversiied ield o eieeri ad techolo, we have several methods to et solutios. For istace, multi-step oliear cojuate radiet methods [6], ABS- MPVT alorithm [5] are used or solvi ucostraied optimizatio problems. A proimal budle method with ieact data [7] is used or miimizi ucostraied o smooth cove uctio. A ew alorithm [8] is used or solvi ucostraied optimizatio problem with the orm o sum o squares miimizatio. Ma iterative methods have bee developed or solvi oliear equatios i recet ears b usi the Talor series, decompositio techiques ad quadrature ormulae [, -5, 7, 9,, 5, 8]. Noor ad Noor [9] have suested a sith order predictor-corrector iterative tpe Halle method or solvi oliear equatios. Kou et.al. [0, ] have also suested a class o ith order iterative methods. I these methods, oe has to evaluate the secod derivative o the uctio which is a draw back o these methods. Recetl, Muhammad Aslam Noor et. al. [4] itroduced ith order modiied predictor corrector Halle method b replaci the secod derivatives o the uctio b its iite dierece scheme to overcome the above metioed drawback. I this paper, we itroduce si ew alorithms or miimizatio o o liear uctios ad comparative stud is established amo the ew alorithms with Newto s alorithm b meas o eamples.. NEW ALGORITHMS I this sectio, we itroduce si umerical alorithms or miimizi oliear real valued ad thrice dieretiable real uctios. Cosider the oliear optimizatio problem: Miimize {, R, : R R } dieretiable uctio. Cosider the uctio G where G is deied aroud the critical poit * o i 0 where is a oliear thrice. Here is the uctio to be miimized. ad is ive b Correspodi author: K. Karthikea* School o Advaced Scieces, Mathematics Divisio VIT Uiversit, Vellore-604, Idia. Iteratioal Joural o Mathematical Archive-, Dec

2 K. Karthikea*/ New Iterative Numerical Alorithms or Miimizatio o Noliear Fuctios/IJMA-, Dec.-0. I we assume that 0 G., we have G 0 i 0. Cosider the equatio 0. whose oe or more roots are to be oud. represets the raph o the uctio is kow or the desired root o the equatio 0. iitial estimate 0 ad assume that a Here we cosider iterative techiques to id the simple root o a o liear equatio 0 where : D R R or a ope iterval D is a scalar uctio. Let α be a simple real zero o a real uctio ad let 0 be a iitial approimatio to α. B usi Talor s series, we have !. From the above equatio. we have the ollowi ew methods. New method I For a ive 0, we et b the ollowi iterative schemes. Sice the equatio. becomes New Alorithm I.4 The order o coverece o the ew alorithm I is three which is based o Halle s method [7, ]. New method II We itroduce New method II which is based o Noor ad Noor [9] two step method. For a ive 0, we et b the ollowi iterative schemes.5 Sice the equatio.5 becomes New Alorithm II.6 I the equatio.5, to make it ree rom secod derivative o the uctio we cosider.7 0, IJMA. All Rihts Reserved 489 0

3 K. Karthikea*/ New Iterative Numerical Alorithms or Miimizatio o Noliear Fuctios/IJMA-, Dec.-0. 0, IJMA. All Rihts Reserved 4894 Combii.5 ad.7 we have the ollowi ew method which is two step modiied Halle s method or the uctio. The order o coverece is ith order which is clear rom the ollowi theorem.. New method III For a ive 0, we et b the ollowi iterative schemes.8 Sice the equatio.8 becomes New Alorithm III.9 We itroduce the ollowi ew method-iv which is based o Noor et. al. [] a two step Halle method o ith order o coveret. New method IV For a ive 0, we et b the ollowi iterative schemes.0 Sice the equatio.0 becomes New Alorithm IV. We itroduce the ollowi ew method-v ad VI which are based o the ith order coverece methods o Kou et. al. [0, ] New method V For a ive 0, we et b the ollowi iterative schemes. Sice the equatio. becomes

4 K. Karthikea*/ New Iterative Numerical Alorithms or Miimizatio o Noliear Fuctios/IJMA-, Dec.-0. New Alorithm V. New method VI For a ive 0, we et b the ollowi iterative schemes.4 Sice the equatio.4 becomes New Alorithm VI.5. CONVERGENCE ANALYSIS Here we cosider the coverece criteria o the New method III ad hece we have the coverece aalsis o alorithm-iii. α I be a simple zero o suicietl dieretiable uctio I R R Theorem.: Let I. I 0 is suicietl close to α, the alorithm.8 has ith order o coverece. : or a ope iterval Proo: The proo o this theorem ollows as i coverece theorem [4] ad hece the order o coverece o the alorithm NUMERICAL ILLUSTRATIONS Eample 4.: Cosider the uctio 5. The miimized value o the uctio is The ollowi table depicts the umber o iteratios eeded to covere to the miimized value or all the ew alorithms with three iitial values 0, 0 ad 0. Table I: shows a compariso betwee the New iterative Alorithms ad Newto s Alorithms For iitial value For iitial value For iitial value Newto s Alorithm 5 5 New Alorithm-I New Alorithm-II 4 New Alorithm-III 5 New Alorithm-IV 6 New Alorithm-V 7 New Alorithm-VI Eample 4.: Cosider the uctio e. The miimized value o the uctio is -. The ollowi table depicts the umber o iteratios eeded to covere to the miimized value or all the ew alorithms with three iitial values 0, 0 ad 0. 0, IJMA. All Rihts Reserved 4895

5 K. Karthikea*/ New Iterative Numerical Alorithms or Miimizatio o Noliear Fuctios/IJMA-, Dec.-0. Table II: shows a compariso betwee the New iterative Alorithms ad Newto s Alorithms For iitial value For iitial value For iitial value Newto s Alorithm New Alorithm-I New Alorithm-II 4 4 New Alorithm-III 5 New Alorithm-IV New Alorithm-V New Alorithm-VI Eample 4.: Cosider the uctio 4 5. The miimized value o the uctio is The ollowi table depicts the umber o iteratios eeded to covere to the miimized value or all the ew alorithms with three iitial values 0, 0 ad 0. Table III: shows a compariso betwee the New iterative Alorithms ad Newto s Alorithms For iitial value For iitial value For iitial value Newto s Alorithm New Alorithm-I 5 5 New Alorithm-II 4 New Alorithm-III New Alorithm-IV New Alorithm-V New Alorithm-VI Eample 4.4: Cosider the uctio 0. The miimized value o the uctio is The ollowi table depicts the umber o iteratios eeded to covere to the miimized value or all the ew alorithms with three iitial values 0, 0 ad 0. Table IV: shows a compariso betwee the New iterative Alorithms ad Newto s Alorithms For iitial value For iitial value For iitial value Newto s Alorithm New Alorithm-I 4 5 New Alorithm-II 4 New Alorithm-III 5 New Alorithm-IV 6 New Alorithm-V 7 New Alorithm-VI Eample 4.5: Cosider the uctio e. The miimized value o the uctio is The ollowi table depicts the umber o iteratios eeded to covere to the miimized value or all the ew alorithms with three iitial values 0, 0 0, ad 0. Table V: shows a compariso betwee the New iterative Alorithms ad Newto s Alorithms For iitial value For iitial value For iitial value Newto s Alorithm 4 New Alorithm-I New Alorithm-II 4 New Alorithm-III 5 New Alorithm-IV 6 New Alorithm-V 7 New Alorithm-VI 0, IJMA. All Rihts Reserved 4896

6 K. Karthikea*/ New Iterative Numerical Alorithms or Miimizatio o Noliear Fuctios/IJMA-, Dec CONCLUSION I this paper, we have itroduced si umerical alorithms amel, New Alorithm I, New Alorithm II, New Alorithm III, New Alorithm IV, New Alorithm V, New Alorithm VI or miimizatio o o liear uctios. From the above illustratios it is clear that the rate o coverece o these ew alorithms is aster tha Newto s Alorithm. I real lie problems, the variables ca ot be chose arbitraril rather the have to satis certai speciied coditios called costraits. Such problems are kow as costraied optimizatio problems. I ear uture, we have a pla to eted the proposed ew alorithms to costraied optimizatio problems. REFERENCES []. S. Amat, S. Busquier ad J.M. Gutierrez, Geometric costructio o iterative uctios to solve oliear equatios, J. Comput. Appl. Math., 5700, []. Adrei, N, A scaled oliear cojuate radiet alorithm or ucostraied Optimizatio, Optimizatio, , []. M. Aslam Noor ad K. Iaat Noor, Fith-order iterative methods or solvi oliear equatios, Appl. Math. Comput. 006 doi:0.06/j.amc [4]. J.A. Ezquerro ad M.A. Heradez, A uiparametric Halle-tpe iteratio with ree secod derivative, It. J. pure Appl. Math., 6 00, 0 4. [5]. J.A. Ezquerro ad M.A. Heradez, O Halle-tpe iteratios with ree secod derivative, J. Comput. Appl. Math., 70004, [6]. J.A.Ford, Y. Narushima ad H.Yabe, Multi-step oliear cojuate radiet methods or costraied miimizatio, Computatioal Optimizatio ad applicatio, 40, 9-6. [7]. E.Halle, A ew eact ad eas method or idi the roots o equatios eerall ad without a previous reductio, Phil. Ro. Soc. Lodo [8].Y.Hu, H. Su, ad J. Chu, Coerece proceedis IEEE iteratioal coerece o sstems, ma ad cberetics, 7004, 608 6, [9]. K. Iaat Noor, M. Aslam Noor, Predictor corrector Halle method or oliear Equatios, Appl. Math. Comput., i press, doi:0.06/j.amc..0. [0]. Jishe Kou ad Yitia Li, Improvemets o Chebshev Halle methods with ith order coverece, Appl. Math. Comput. 006 doi:0.06/j.amc []. Jishe Kou, Yitia Li ad Xiuhua Wa, A amil o ith-order iteratios composed o Newto ad third-order methods, Appl. Math. Comput.006 doi:0. 06/ j.amc []. Moha C Joshi ad Kaa M Moudala, 004, Optimizatio theor ad Practice, Narosa Publicatio House, New Delhi. []. A. Melma, Geometr ad coverece o Halle s method, SIAM Rev., 9 4, 997, [4]. Muhammad Aslam Noor, Waseem Ashar Kha ad Akhtar Hussai, A ew modiied Halle method without secod derivatives or oliear equatio, Appl. Math. Comput 89, 007, [5].Pa, L.P., Spedicato, E., Xia, Z.Q. ad Wa, W, A method or solvi the sstem o liear equatios ad liear iequalities, Mathematical ad Computer Modelli,.465-6, 007, [6]. G.V. Reklaitis, A. Ravidra ad K.M. Rasdell, 98, Eieeri optimizatio methods ad applicatios, Joh Wile ad sos, New York, [7]. J. She, Z.Q Xia ad L.P. Pa, A proimal budle method with ieact data or cove o dieretiable miimizatio, Noliear Aalsis, Theor, ad Applicatios, , [8]. J.F. Traub, 964, Iterative or Solutio o Equatios, Pretice-Hall, Elewood, Clis, NJ. Source o support: Nil, Colict o iterest: Noe Declared 0, IJMA. All Rihts Reserved 4897