Iteratioal Joural o Matheatical Archive-4(9), 03, -5 Available olie through www.ija.io ISSN 9 5046 THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED NEWTON RAPHSON METHOD FOR SOLVING NONLINEAR EQUATIONS M. S. Kuar, Mylapalli * ad V. B. Kuar, Vatti Departet o Egieerig Matheatics, Gita Istitute o Techology, Gita Uiversity, Visakhapata 530 045, Adhra Pradesh, Idia ad Departet o Egieerig Matheatics, College o Egieerig, Adhra Uiversity, Visakhapata 530 003, Adhra Pradesh, Idia (Received o: 5-07-3; Revised & Accepted o: -08-3) ABSTRACT I this paper, we preset the rate o covergece o Geeralized Extrapolated Newto Raphso (GEN-R) Method to id the ultiple roots o o-liear equatios. Nuerical exaples are give to illustrate the perorace o the preseted ethods. Key Words: N R Method, Iterative ethod. AMS Classiicatio: 65 H 0. SECTION : INTRODUCTION Oe o the ost iportat probles i Nuerical Aalysis is solvig o-liear equatios. To solve these equatios, we ca use iterative ethods such as Newto-Raphso (NR) Method ad its variats. I this paper, we cosider iterative ethods to id the ultiple root o a o-liear equatio (x) = 0 (.) where ay be algebraic, trascedetal or cobied o both. Let η be a root o the equatio (.) with ultiplicity ad let >. The, (η) = (η) = = ( - ) (η) = 0 ad (η) 0 (.) ad also the equatio (.) ca be expressed as (x) = (x-η) k(x) = 0 (.3) where k(x) is bouded ad k(η) 0. As oted by Jai et. al [ whe the equatio (.) has a ultiple root, ost o the ethods exists or solvig a siple root o (x)=0 have oly liear rate o covergece. I η be a root o (.) with ultiplicity, the the classical Geeralized Newto - Raphso ethod (GN-R) is deied as ( x ) x + = x (=0,, ) (.4) ( x ) This is a powerul ad well-kow iterative ethod kow to covergece quadratically. Correspodig author: M. S. Kuar, Mylapalli * Departet o Egieerig Matheatics, Gita Istitute o Techology, Gita Uiversity, Visakhapata 530 045, Adhra Pradesh, Idia Iteratioal Joural o Matheatical Archive- 4(9), Sept. 03
M. S. Kuar*, Mylapalli ad V. B. Kuar, Vatti / THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED / IJMA- 4(9), Sept.-03. The Geeralized Extrapolated Newto Raphso Method (GEN-R) cosidered by Vatti VBK et al. [3 or the ultiple root o equatio (.) is give as ( ) x x + = x α (.5) ( x ) As it is well-kow that ay iterative ethod o the or x + = φ (x ) coverges i φ (x ) < or all x i I. Hece the ethod (.5) coverges uder the coditio µ = -α + α ω < or all x i I (.6) where ω = ( x ) ( x) [ ( ) x (.7) The uctio (x), (.3) i the iediate eighborhood o x =η, ca be writte as (x) = k. (x - η) where k ~ k (η) is eectively costat. The (x) = k. (x - η) - (x)= k. (-) (x - η) - Thus, we have [ = k. ( x η). ( ) kx ( η) [ k ( x η ) [ = (.8) We eed to id a real value o α or each iteratio, which iiizes µ o (.6). Sice ω o (.7) is positive ad real or all x, i the iediate viciity o η with the act (.8), we have i geeral ( x ) ( x ) ω [ ( x ) (.9) The process o iiizig µ o (.6) keepig i view o (.8) with respect to α usig the procedure give i Youg [5 ad i Kuar ad Koeru [4, is give the optial choice or α as α ( opt) = + [ = + ω (.0) With the optial value o α, µ o (.0) takes the or ( ω ) µ = + ( ω ) (.) which is always less tha uity as log as ω < ad hece the covergece o the GEN-R ethod (.5) is assured. 03, IJMA. All Rights Reserved
M. S. Kuar*, Mylapalli ad V. B. Kuar, Vatti / THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED / IJMA- 4(9), Sept.-03. I this paper, we develop the rate o covergece o Geeralized Extrapolated Newto - Raphso Method or ultiple roots o the equatio (.) i sectio. Ad i sectio 3, we cosider soe exaples or idig a ultiple root o equatio (.) usig the ethods discussed i this paper. It is assued throughout this paper that (x), (x), (x) are cotiuous i I : a < x < b such that (a) (b) < 0 ad (x) ad (x) have the sae sig or all successive approxiatios o x startig with x 0. SECTION : CONVERGENCE ANALYSIS OF GEN-R METHOD Let η be the ultiple root o the equatio (x) =0 ad ε +, ε be errors whe x +, x are the (+) th approxiates. The, we have ad th X + = η+ ε + X = η+ ε (.) Substitutig these values o x + ad x i (.), we have where ε + + [ [ ( k)[ k = ε + 3 4 5 6 7 ε ε ε V ε V ε V ε + ε V + + + + + + +... 6 4 0 70 5040 3 4 5 6 ε ε V ε V ε V ε + ε V + + + + + +... 6 4 0 70 3 4 5 ( k) ε ε V ε V ε ε V ε 6 7 + + + + + + V ε + V... 6 4 0 70 5040 3 4 5 6 7 ε ε ε V ε V ε V ε + ε V + + + + + + +... 6 4 0 70 5040 ε + ε V + ε 3 4 5 6 V ε V ε V ε + + + + V +... 6 4 0 70 3 3 ( ) 5 5 4 ( ) V + ε + ε + ε + + ε + +... 3 4 3 ( k) ( ) ε ( ) ( ( ) V + + ε + ) + ε + +... 3 ( ) ( ) ε + ε + 3 ε V + + +... 3 03, IJMA. All Rights Reserved 3
M. S. Kuar*, Mylapalli ad V. B. Kuar, Vatti / THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED / IJMA- 4(9), Sept.-03. Sipliyig urther ad eglectig the higher powers o ε, we obtaied (i) Whe = 3 + 4 6 (ii) Whe = 3 V + 4 4 (iii) Whe =3 3 V V + 59 60 I geeral, oe ca have ε + ~ ε 3. k where, ( + ) ( + ) k = k k where k ad k are costats. Thereore, ε + ε 3 Hece the GEN-R ethod has a cubic rate o covergece. SECTION- 3: NUMERICAL EXAMPLES We cosider soe exaples give i Sastry[ ad Jai [ or idig the ultiple roots o a equatio usig the ethods discussed i this paper ad the successive approxiatios o the roots are tabulated below util the uctioal value becoes egligible. Table 3. Fidig double root o (x) = x 3 x x + with x =4 Method (.4) Method (.5) x + x +.6970769.89965.078870497.0056096 3.0046886.000000005 4.000000566 -- Table 3. Fidig the triple root o (x) = x 4 x 3 3x + 5x lies i (0, ) with x = 0. Method (.4) Method (.5) x + x +. 0.9375.00408633 0.999993656 3.0000093 --- 03, IJMA. All Rights Reserved 4
M. S. Kuar*, Mylapalli ad V. B. Kuar, Vatti / THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED / IJMA- 4(9), Sept.-03. CONCLUSIONS It ca be see ro the above tabulated results that the ethod (.5), GEN-R takes less iteratios tha that o the (.4), GN-R ethod ad the preseted ethod (.5) coverges ore rapidly tha the classical Geeralized Newto - Raphso ethod (GN-R). Thereore, the ethod (.5) has better covergece eiciecy. REFERENCES [ Jai. M.K, Iyegar. S.R. Kad Jai. R.K., (004), Nuerical Methods o Scietiic ad Egieerig Coputatio, New age Iteratioal publishers, New Delhi, Idia. [ Sastry. S. S, (005), Itroductory Methods o Nuerical Aalysis, Pretice Hall o Idia, New Delhi, Idia. [3 Vatti. V. B. K ad Mylapalli. M. S. K., Geeralized Extrapolated Newto Raphso Method South East Asia Joural o Matheatics ad atheatical scieces, 00, Vol.8 o.3, pp. 43-48. [4 Vatti. V. B. K. ad Koeru. S. R., Extrapolated accelerated Gauss-Seidel Methods, It. Joural o Math. Cop., 987, Vol.. [5 Youg. D. M., (97), Iterative Solutio o large liear systes, Acadeic Press, New York ad Lodo. Source o support: Nil, Colict o iterest: Noe Declared 03, IJMA. All Rights Reserved 5