Newton Homotopy Solution for Nonlinear Equations Using Maple14. Abstract

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1 Joural of Sciece ad Techology ISSN Vol. No. December 0 Newto Homotopy Solutio for Noliear Equatios Usig Maple Nor Haim Abd. Rahma, Arsmah Ibrahim, Mohd Idris Jayes Faculty of Computer ad Mathematical Scieces, Uiversiti Tekologi MARA Malaysia (UiTM), 050 Shah Alam, Selagor, Malaysia *Correspodig orhaim@ppiag.uitm.edu.my Abstract May umerical approaches have bee suggested to solve oliear problems. Some of the methods utilize successive approimatio procedure to esure every step of computig will coverge to the desired root ad oe of the most commo problems is the improper iitial values for the iterative methods. This study evaluates Palacz et.al s. (00) paper o solvig oliear equatios usig liear homotopy method i Mathematica. I this paper, the Newto-homotopy method usig start-system is implemeted i Maple, to solve several oliear problems. Comparisos of results obtaied i terms of umber of iteratios ad covergece rates show promisig applicatio of the Newto-homotopy method for oliear problems. Keywords: Homotopy, Newto-Raphso, Start-system, Polyomials, Maple. 69

2 Joural of Sciece ad Techology ISSN Vol. No. December 0. INTRODUCTION Newto-Raphso method offers fast, accurate umerical results compared to other methods ad the best method prove to coverge quadratically ad applicable to may fields of kowledge. May attempts have bee doe i fidig the solutio for the roots of equatios of the form of f ( ; () where f ( may be give eplicitly as polyomials (Nor Haim, Arsmah Mohd Idris, 00a, 00b, 00c; Li, Cheg Neta, 00; Saeed Khthr, 00; Kou, Wag Li, 00; Sheggou, Xiagke Lizhi, 009; Yu, 009; Chu Neta, 009; Chu, Bae Neta, 009; Fag He, 007,009) or, ordiary or partial differetial equatios at specific values (He, 999). However, oe of the disadvatages of the classical iterative Newto-Raphso method such as, f ( i ) i i - where, f '( i) 0 () f ' ( ) i is the usage of divisio i the formula which is ot very efficiet i terms of computatioal time (Nor Haim, Arsmah Mohd Idris, 00a,00b,00c) that could lead to diverget, whe f '( i ) 0 ad the usage of derivative is mostly difficult to solve. Aother kow method i solvig oliear problems is homotopy method, which is a global umerical method used i various sciece ad egieerig areas (Palacz et.al, 00; Hazaveh et.al, 00). Much attetio too has bee give to develop several iterative methods for solvig oliear equatios such as the Newto homotopy cotiuatio method (Saeed Khthr, 00; Rafiq Awais, 008; Abbasbady, 00). Homotopy ca guaratee to coverge by certai path if a suitable auiliary homotopy fuctio is chose (Rafiq Awais, 008). I this paper, the geeral algorithms of homotopy equatios via Newto-Raphso method are used to fid roots of polyomials. The efficiecy of these methods is illustrated ad compariso of covergece ad umber of iteratios are made.. PROJECT DESIGNS The power form is the stadard way for a polyomial i mathematical discussios ad very coveiet form for differetiatig ad itegratig. However, it may lead to loss of sigificace. So to avoid this, the shifted power is itroduced. Below is the basic theorem of a polyomial. Defiitio.: A polyomial p ( of degree is defied as ( 0 p a 0 a a a... a i () where, a i are real costats for i 0 to.. The basic idea of liear homotopy We cosider the followig o-liear algebraic equatio, f ( ad defie the cove homotopy for the fuctio H(, ) : 0,, H (, (- p( q( () 70

3 Joural of Sciece ad Techology ISSN Vol. No. December 0 where, is a embedded parameter ad 0, ; p( is the start system; q( f ( is the target system; H (,0) p( H (,) q( f (. There are basic ways to idetify a start system of a liear homotopy (Palacz et.al, 00) such as, (i) The fied-poit homotopy: 0 where 0 is a iitial approimatio of Eq.(). (ii) The Newto-homotopy: ( 0 (iii) where preferably be the highest power of of Eq.(), C is ay costat, ad the roots of p( 0 ca be easily foud. Besides the ease of Newto-homotopy, it does ot guaratee to coverge (Palacz et.al, 00; Choi Book, 99). Hereafter, our discussios will oly proceed with the type-(iii) method for its fleibility to choose the values of ad C. Below are some of the eamples of Maple algorithms used. The iteratios will follow the followig algorithms. Algorithm.: Newto-homotopy usig start-system Step: Idetify q ( f ( 0. Step: Idetify p (, such that p( - C. Step: Fid the iitial value, 0, by settig p( - C 0 such as, where p( -, H (, (- ( - 0) q( (5) p q( - q( ), H (, (- q( - q( 0) q( (6) The start-system Newto-homotopy: p( - C, H (, (- ( C ) q ( (7) Step: Simplify H(, ) (- ( -C) q( such as, Step5: Iterate H(, ) (- ( -C) q( 0, e.g. 0.0, 0., 0., 0.6, 0.8, ad.0 by usig the classical Newto Raphso (See Eq. ()), 7

4 Joural of Sciece ad Techology ISSN Vol. No. December 0. NUMERICAL EXAMPLES AND DISCUSSIONS I this sectio, some umerical eamples preseted to illustrate the efficiecy of the iterative methods such as the classical Newto-Raphso (CNR) ad the start-system homotopy (see Eq. (7)). By usig the followig test fuctios ((a), (b), (c), (d) ad (e)) ad specified the outputs to 0-digit values, we obtaied the results as show i Table. (a) f ( 8-9 (See Palacz et.al, 00); (b) f ( 6-0 ; 6 (c) f ( ; (d) (e) f ( e (See Abbasbady, 00; Saeed Khthr, 00); f5 ( cos( (See Abbasbady, 00; C.Chu et.al, 009; C.Chu Neta, 009; Kou et.al, 00; Li et.al, 00). Table shows that the efficiecy of the iterative Newto-homotopy usig start-system (NHss) gives equal or better results i terms of covergece rate as compared to the classical Newto-Raphso. It seems that the computatios coverge i less tha 5 iteratios. We also approimated usig differet powers of o the start-systems ad recorded the umber of iteratios. The followig test fuctios (f) ad (g) are used ad the approimated zeros are displayed i Table : (f) f 5 6 ( ( (See Javidi, 009); 7 (g) f ( 0 (See Saeed Khthr, 00). 7

5 Joural of Sciece ad Techology ISSN Vol. No. December 0 Table : Numerical eperimet results of the eistig method, the Classical Newto-Raphso (CNR), ad the proposed methods, the Newto-Homotopy usig start-system (NHss) usig Maple. Fuctios Used f ( q( 0 = target system ; H (, ( - ) p( q( Roots (CNR) i f ( i ) i - f ' ( ) iteratio, i i Root(s) usig Newto- Homotopy Method ; p ( = 0 = start-system H (, ) (NHss) p ( 0 Iitial guess, 0 Iteratio usig NHss, i f( 8-9 H (, ) (as i Palacz et.al, 00) -9, (),() ,- f ( 6-0 H (, ) , (),() f ( 5 - H (, ) f ( e H (, ) , (),() () * *5 f 5 ( cos( H 5 (, ) cos( *oscillatig values () cos( )

6 Joural of Sciece ad Techology ISSN Vol. No. December 0 Table : Compariso of Classical Newto-Raphso (CNR) ad Newto-homotopy usig differet start-systems (NHss). Fuctios used: f ( q( = target system Root/s usig NHss (Maple); p ( 0 = start-system Number of iteratios for NHss CNR; where 0,0.,0.,0.6,0.8.0 ; p( (f) f 5 ( ( ; (See Javidi, 009) H(, ( p( q( a) p ( 5 b) p ( 5 c) p ( 9 d) p ( 5 a), 5, 6,, 6 5 ; (5) b), 0, 8, 8, 7 8 ; () c),,,, ; () d),,,, 5 ; () (g) f 6 ( 0 ; (See Saeed Khthr, 00) a) p ( 0 b) p ( 0 a),,, 5, 5 6 ; () b),,,, ; (6) I Table, it ca be see that the fuctios that use the start-system with lower degree of (such as degree of two) coverges faster tha the higher degree.. CONCLUSIONS As demostrated i the umerical results, Newto-Homotopy usig start-system method coverges better or equal to the classical Newto-Raphso. It is also very importat to have a proper start-system to esure covergece is fast ad computig time is reduced. REFERENCES Abbasbady, S. (00). Improvig Newto-Raphso method for oliear equatios by modified Adomia decompositio method Applied Mathematics ad Computatio.5(-).pp Chu, C. Neta, B. (009). A third-order modificatio of Newto s method for multiple roots Applied Mathematics ad Computatio.pp Chu, C., Bae, H. Neta, B. (009). New families of oliear third-order solvers for fidig multiple roots Computer ad Mathematics with Applicatios 58.pp Choi, S.H. Book, N.L. (99). Ureachable roots for global homotopy cotiuatio methods AIChE J 7(7).pp Fag, L. He, G. (009). Some modificatios of Newto s method with higher-order covergece for solvig oliear equatios Joural of Computatioal ad Applied Mathematics 8.pp Feg, X. He, Y. (007). High order iterative methods without derivatives for solvig oliear equatios Applied Mathematics ad Computatio 86().pp Hazaveh K., Jeffrey D.J., Reid G.J., Watt S.M., Wittkopf A.D. (00). A eploratio of homotopy solvig i Maple WSPC Proceedig. He, J-H. (999). Homotopy perturbatio techique Computer Methods i Applied Mechaic ad Egieerig 78.pp Javidi, M. (009). Fourth-order ad fifth-order iterative method for oliear algebraic equatios Mathematical ad Computer Modellig 50.pp Kou, J., Wag, X. Li, Y. (00). Some eight-order root-fidig three-steps methods Commu Noliear Sci Numer Simulat 5.pp

7 Joural of Sciece ad Techology ISSN Vol. No. December 0 Li, S.G., Cheg, L.Z. Neta, B. (00). Some fourth-order oliear solvers with closed formulae for multiple roots Computer ad Mathematics with Applicatios 59 pp.6-5. Nor Haim Abd. Rahma, Arsmah Ibrahim Mohd Idris Jayes (00a). No-divisio Iterative Newto- Raphso i Solvig = surd[(a/r)]^(/) Proceedig of ICSTIE'0, pp.7 Nor Haim Abd Rahma, Arsmah Ibrahim, Mohd Idris Jayes (00b) Newto Homotopy Solutio for Noliear Equatios usig Maple Buku Prosidig Jilid Matematik SKASM00 sempea SKSM8. pp. 5-. Nor Haim Abd Rahma, Arsmah Ibrahim Mohd Idris Jayes (00c). No-divisio Newto-Raphso Method i Solvig Noliear Equatios Proceedigs of the d Iteratioal Coferece o Mathematical Scieces (ICMS).pp Palacz, B., Awage, J.L., Zaletyik, P. Lewis, R.H. (00). Liear homotopy solutio of oliear systems of equatios i geodesy Joural of Geodasy 8().pp doi 0.007/s Rafiq, A Awais, M. (008). Covergece o the homotopy cotiuatio method Iteratioal Joural of Appl. Math ad Mech (6).pp Saeed, R.K. Khthr, F.W. (00). Three ew iterative methods for solvig oliear equatios Australia Joural of Basic Applied Scieces (6).pp Sheggou, L., Xiagke, L. Lizhi, C. (009). "A ew fourth-order iterative method for fidig multiple roots of oliear Applied Mathematics ad Computatio 5.pp Yu, B.I. (009). A derivative free iterative method for fidig multiple roots of oliear equatios Applied Mathematics Letters.pp

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