Bulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics,

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1 Bulletin of the Trnsilvni University of Brşov Vol (59), No Series III: Mthemtics, Informtics, Physics, ON THE CONVERGENCE OF SOME MODIFIED NEWTON METHODS THROUGH COMPUTER ALGEBRA Ernest SCHEIBER Abstrct For some modified Newton methods to solve non-liner eqution the convergence is estblished nd the convergence order is computed using Computer Algebr Softwre. 22 ACM Subject Clssifiction: G..5, K..3. Key words: roots of nonliner equtions, modified Newton method, computer uses in eduction. Introduction We present the possibility to estblish the convergence nd compute the convergence order of method to solve non-liner eqution using Computer Algebr Softwre (CAS). The pplied procedure is bsed on well known convergence result, i.e. [?]. Severl modified Newton methods re known. Some of them re derived from different qudrture formuls [], [2], [7], [5]. We shll pply the convergence result to these methods, but the computtions re mde using CAS. The sme rgumenttion for convergence ws used in [4] nd [3], too. It implies tht the convergence occurs only when the initil pproximtion is properly chosen nd tht the convergence order is 3. In [9] we used the sme pproch for some methods to simultneously compute ll the roots of polynomil. We give unitry simplified presenttion of the convergence results for severl modified Newton methods with the usge of Mthemtic CAS [2]. The note is orgnized s follows. In Section 2 we recll the convergence result tht will be used. In Section 3 the convergence conditions re verified for some modified Newton methods using Mthemtic. e-mil: scheiber@unitbv.ro

2 28 Ernest Scheiber 2 A convergence frmework Let Ω C n be n open convex subset, T : Ω C n, T (z) = (T (z),..., T n (z)) T n m times differentible opertor such tht T (m) (z) is continuous nd the sequence (z (k) ) k N defined by z (k+) = T (z (k) ), z (k) = (z (k), z(k) 2,..., z(k) n ) T () z (k+) i = T i (z (k) ), i {, 2,..., n}, k N. In C n we shll use the mx norm z = mx{ z, z 2,..., z n }. We remind result enbling to estblish the convergence of such methods nd lower bound of their convergence order [?]. The min ingredient of the convergence theorem is the following well known result, but for completeness we shll give the proof of the result tht we shll use. Theorem. [] Let X, Y be normed spces, D n open convex subset of X nd T : D Y n m times Frèchet differentible opertor. Then, for ny x, y D T (y) T (x) j= Using this result, we hve Theorem 2. Let α Ω. If. T (α) = α, j! T (j) (x) (y x)... (y x) }{{} j times 2. T (α) = T (α) =... = T () (α) = y x m sup T (m) (ζ). (2) ζ [x,y] then there exists r > such tht for ny z () C n, z () α < r, the sequence z (k+) = T (z (k) ), k N, () converges to α. Proof. Let r > be such tht V = {z C n : z α r } Ω nd C = mx z V T (m) (z). There exists < r r such tht C r m < r ( ) C r <. We denote V = {z C n : z α r}. If z V, then (2) nd the present hypothesis implies T (z) α = T (z) T (α) z α m sup ζ [α,z] j= j! T (j) (α) (z α)... (z α) }{{} j times T (m) (ζ) C r m < r,

3 Modified Newton methods through computer lgebr 29 thus T (z) V. For z = z (k) from the bove reltions we obtin z (k+) α = T (z (k) ) α C z(k) α m. (3) Using recursively the inequlity (3), we find = z (k) α C z(k ) α m C ( ) +m C z (k 2) α m2... ( ) m k C r m k = ( (C ( ) m C z(k 2) α m = ( ) +m+...+m k C z () α mk ) r ) m k r, for k. x Let lim k x k = x. If lim k+ x k x k x = ρ, with < ρ <, then r is the r convergence order of the sequence (x k ) k N. From the inequlity (3) it results tht the convergence order of the sequence (z (k) ) k N is t lest m. 3 Modified Newton methods Let there be differentible function F : Ω R R nd the non-liner eqution F (x) =, (4) such tht F () =, F (). The itertion formul of modified Newton methods is Denoting x n+ = x n F (x n) F (x n ) G(x n), n N. (5) T (x) = x F (x) G(x), in order to prove the third order convergence of the method (5) the following reltions must be verified T () = T () =, T () = T (3) (). (6) The convergence occurs when the initil pproximtion x is properly chosen. The Hlley s method [3] is defined by x n+ = x n 2F (x n )F (x n ) 2F 2 (x n ) F (x n )F (x n ), G(x) = 2F 2 (x) 2F 2 (x) F (x)f (x).

4 2 Ernest Scheiber If T (x) = x 2F (x)f (x 2F 2 (x) F (x)f (x) then the computtion in Mthemtic 2 is performed by the code below: F [] = T [x ]:=x (2F [x]f [x])/(2f [x] 2 F [x]f [x]) T [x]/.x D[T [x], x]/.x D[T [x], {x, 2}]/.x Simplify[D[T [x], {x, 3}]/.x ] 3F [] 2 2F []F (3) [] 2F [] 2 The sme scheme will be pplied to the following methods. In [] Weerkoon nd Fernndo hd introduced the third order convergence method x n+ = x n For 2F (x n ) ( ), G(x) = F (x n ) + F x n F (xn) F (x n) T 2(x) = x 2F (x) ( ) + F x F (x) we found F [] = T2[x ]:=x 2F [x]/(f [x] + F [x F [x]/f [x]]) T2[x]/.x D[T2[x], x]/.x D[T2[x], {x, 2}]/.x Simplify[D[N[x], {x, 3}]/.x ] 3T [] 2 +T []T (3) [] 2T [] 2 Frontini nd Sormni [2] considered the method 2 ( ). (7) + F x F (x) F (x n ) x n+ = x n ( ), G(x) = F x n F (xn) 2F (x n) ( F x F (x) 2 ). (8) 2 The settings nd given commnds re printed with bold chrcters.

5 Modified Newton methods through computer lgebr 2 So, for the Mthemtic code is F [] = T3[x ]:=x F [x]/f [x F [x]/(2f [x])] T3[x]/.x D[T3[x], x]/.x D[T3[x], {x, 2}]/.x D[T3[x], {x, 3}]/.x 3F [] 2 2F [] 2 F (3) [] 4F [] F (x) T 3(x) = x ( ). F x F (x) 2 In [4], [] the following method is defined x n+ = x n F (x n) 2 F (x n ) + ( ), (9) F x n F (xn) G(x) = 2 F (x n) + ( ). F x F (x) The corresponding Mthemtic code for this method is F [] = T4[x ]:=x F [x]/2(/f [x] + /F [x F [x]/f [x]]) T4[x]/.x D[T4[x], x]/.x D[T4[x], {x, 2}]/.x D[T4[x], {x, 3}]/.x ( ) 3F [] 2 F (3) [] 2F [] 2 F [] 3 2 F [] F [] 2 F (3) [] F [] 3 F [] 2 In [5] the following method is introduced where M N. Now x n+ = x n G(x) = 2MF (x n ) ( ), () 2M k= F x n F (xn) k.5 F (x n) 2M 2M ( 2M k= F x F (x) ). k.5 2M

6 22 Ernest Scheiber The required computtions is given by the code F [] = T5[x ] = x 2MF [x]/sum[f [x F [x]/f [x](k /2)/(2M)], {k,, 2M}] 2MF [x] x [ ] 2M k= F T5[x]/.x D[T5[x], x] + 2MF [x] 2M k= x ( 2 +k )F [x] 2MF [x] D[T5[x], x]/.x D[T5[x], {x, 2}]/.x D[T5[x], {x, 3}]/.x 2M ( 2 +k 2M + ( 2 +k)f [x]f [ [x] )F 2MF [x] 2 x ( ] +k)f [x] 2 2MF [x] ( [ 2M k= F x ( 2 ]) +k)f [x] 2MF 2 [x] ( 3F [] 2 4MF [] 2 + F (3) [] 2MF [] + 3F [] ( [ 2MF [x] 2M k= F x ( 2 ] +k)f [x] 2MF [x] )) F [] 2 4MF [] 3 24M 2 F [] 2 F []F (3) []+6M 2 F []F (3) [] 96M 3 F [] 3 Even the Lguerre method to compute root of polynomil [8], [6] my be presented in the sme mnner. Let be P (x) = m i= (x x i) polynomil hving only simple roots. The Lguerre method is defined by x n+ = x n P (x n) P (x n ) m + m m, n N, x C. P (x n)p (x n) P 2 (x n) The squre root of complex number is chosen to hve non-negtive rel prt. In this cse G(x) = m + m m. P (x)p (x) P 2 (x) The Mthemtic code is P [] = G[x ]:= /(/m + (m )/msqrt[ m/(m )P [x]d[p [x], {x, 2}]/D[P [x], x] 2]) T 6[x ]:=x P [x]/d[p [x], x]g[x] T 6[x]/.x Simplify[D[T 6[x], x]/.x ] Simplify[D[T 6[x], {x, 2}]/.x ] Simplify[D[T 6[x], {x, 3}]/.x ] 3( 2+m)P [] 2 4( +m)p []P (3) [] 4( +m)p [] 2

7 Modified Newton methods through computer lgebr 23 As expected, we found tht the convergence order of the method is 3. References [] Crtn H.: Clcul différentiel. Ed. Hermnn, Pris, 967. [2] Frontini M., Sormnie E.: Some vrints of Newtons method with third-order convergence. Appl. Mth. Comput., 4 (23), [3] Homeier H.H.H.: A modified Newton method for rootfinding with cubic convergence. J. Comput. Appl. Mth., 57 (23), [4] Homeier H.H.H.: On Newton-type methods with cubic convergence. J. Comput. Appl. Mth., 76 (25), [5] Mishr B., Pny A.K., Dutt S.: A New Modified Newton Method use of Hr Wwelet for Solving Nonliner Equtions. rxiv:7.468v, 26. [6] Möller H., Convergence nd visuliztion of Lguerre s rootfinfing lgorithm. rxiv: 5.268v, 25. [7] Özbn A.Y.: Some new vrints of Newtons method. Appl. Mth. Lett., 7 (24), [8] Press, W.H., Teukolsky S.A., Vetterling W.T., Flnnery B.P., Numericl Recipes: The Art of Scientific Computing (3rd ed.). New York: Cmbridge University Press, 27. [9] Scheiber E., On the Convergence Order for Some Methods of Simultneously Root Finding for Polynomils Using Computer Algebr System. Bulletin of the Trnsilvni University of Brşov, 4(53) (2), no., [] Wng P.: A Third-Order Fmily of Newton-Like Itertion Methods for Solving Nonliner Equtions. J. Numericl Mth. Stochstics, 3 (2), no., 3-9. [] Weerkoon S., Fernndo T.G.I.: A vrint of Newtons method with ccelerted third-order convergence, Appl. Mth. Lett., 3 (2), [2] * * *, [3] * * *,

8 24 Ernest Scheiber