Taxes & Accounting

A variant of super-Halley method with accelerated fourth-order convergence

Description
A variant of super-Halley method with accelerated fourth-order convergence
Published
of 5
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  A variant of super-Halley method with acceleratedfourth-order convergence Jisheng Kou  a,* , Yitian Li  a , Xiuhua Wang  b a State Key Laboratory of Water Resources and Hydropower Engineering Sciences, Wuhan University, Wuhan 430072, China b School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Abstract In this paper, we present a new variant of super-Halley method for solving non-linear equations. Analysis of conver-gence shows that the method has the order of convergence four. Per iteration the new method requires the same evalua-tions of the function, the first derivative and second derivative as its classical predecessor, super-Halley method. Numericalresults show that the method has definite practical utility.   2006 Elsevier Inc. All rights reserved. Keywords:  Super-Halley method; Newton’s method; Non-linear equations; Iterative method 1. Introduction Solving non-linear equations is one of the most important problems in numerical analysis. In this paper, weconsider iterative methods to find a simple root of a non-linear equation  f  ( x ) = 0, where  f   :  D  R ! R for anopen interval  D  is a scalar function.Newton’s method for a single non-linear equation is written as  x n þ 1  ¼  x n   f  ð  x n Þ  f  0 ð  x n Þ :  ð 1 Þ This is an important and basic method [1], which converges quadratically.An acceleration of Newton’s method, called super-Halley method, is introduced in [2,3]  x n þ 1  ¼  x n   1 þ 12  L  f  ð  x n Þ 1   L  f  ð  x n Þ    f  ð  x n Þ  f  0 ð  x n Þ ;  ð 2 Þ where  L  f   ( x n ) =  f  00 ( x n )  f  ( x n )/  f  0 ( x n ) 2 . This method is, in general, an iterative process with order of convergencethree although the method converges with fourth order when it is applied to quadratic equations [4]. From 0096-3003/$ - see front matter    2006 Elsevier Inc. All rights reserved.doi:10.1016/j.amc.2006.07.118 * Corresponding author. E-mail address:  koujisheng@yahoo.com.cn (J. Kou).Applied Mathematics and Computation 186 (2007) 535–539 www.elsevier.com/locate/amc  a practical point of view, it is interesting and expected to research higher-order variants of super-Halley meth-od for general non-linear equations.In this paper, we present a new variant of super-Halley method with fourth-order convergence. A detailedconvergence analysis of the new method is supplied. The new method requires one evaluation of the function,one of its first derivative and one of its second derivative, which are the same as super-Halley method. Con-sequently, the new method could be of practical interest, as we show in some examples. 2. The method and analysis of convergence Here, we define  y  n  ¼  x n   f  ð  x n Þ  f  0 ð  x n Þ ;  ð 3 Þ  z  n  ¼  x n þ h ð  y  n   x n Þ ;  h 2 R :  ð 4 Þ We modify super-Halley method by using the second derivative  f  00  at  z n  in (2) instead of   x n  and obtain  x n þ 1  ¼  x n   1 þ 12 e  L  f  ð  x n Þ 1  e  L  f  ð  x n Þ  !  f  ð  x n Þ  f  0 ð  x n Þ ;  ð 5 Þ where  e  L  f  ð  x n Þ¼  f  00 ð  z  n Þ  f  ð  x n Þ =  f  0 ð  x n Þ 2 . Obviously, when we take  h  = 0, the super-Halley method is obtained. For(5), we have Theorem 1.  Assume that the function  f   :  D  R ! R  for an open interval D has a simple root x * 2 D. Let f(x) besufficiently smooth in the neighborhood of the root x * , then the order of convergence of the method defined by ( 5 )is four if   h  = 1/3. Proof.  Let  e n  =  x n  x * and  d  n  =  z n  x n , where  z n  is defined by (4). Using Taylor expansion, we have  f  ð  x  Þ¼  f  ð  x n Þ  f  0 ð  x n Þ e n þ 12  f  00 ð  x n Þ e 2 n  16  f  ð 3 Þ ð  x n Þ e 3 n þ  124  f  ð 4 Þ ð  x n Þ e 4 n þ O  e 5 n   : Taking into account  f  ( x * ) = 0, we have [5]  f  ð  x n Þ¼  f  0 ð  x n Þ  e n  c 2 e 2 n þ c 3 e 3 n  c 4 e 4 n þ O ð e 5 n Þ   ;  ð 6 Þ where  c k   = (1/ k  !)  f  ( k  ) ( x n )/  f  0 ( x n ),  k   = 2,3, . . .  Furthermore, we have  f  ð  x n Þ  f  0 ð  x n Þ¼ e n  c 2 e 2 n þ c 3 e 3 n  c 4 e 4 n þ O ð e 5 n Þ ;  ð 7 Þ d  n  ¼ h  f  ð  x n Þ  f  0 ð  x n Þ¼ h  e n  c 2 e 2 n þ O ð e 3 n Þ   :  ð 8 Þ Expanding  f  00 ( z n ) about  x n , we have  f  00 ð  z  n Þ¼  f  0 ð  x n Þ  2 c 2 þ 6 c 3 d  n þ 12 c 4 d  2 n þ O ð d  3 n Þ   ; and then from (8), we have  f  00 ð  z  n Þ¼  f  0 ð  x n Þ  2 c 2  6 h c 3 e n þ  12 h 2 c 4 þ 6 h c 2 c 3   e 2 n þ O ð e 3 n Þ   :  ð 9 Þ From (7) and (9), we have e  L  f  ð  x n Þ¼ 2 c 2 e n   6 h c 3 þ 2 c 22   e 2 n þ  12 h 2 c 4 þð 2 þ 12 h Þ c 2 c 3   e 3 n þ O ð e 4 n Þ :  ð 10 Þ Thus, we have12 e  L  f  ð  x n Þ 1  e  L  f  ð  x n Þ¼ c 2 e n þ  c 22  3 h c 3   e 2 n þ  6 h 2 c 4 þð 1  6 h Þ c 2 c 3   e 3 n þ O ð e 4 n Þ :  ð 11 Þ 536  J. Kou et al. / Applied Mathematics and Computation 186 (2007) 535–539  From (7) and (11), we have12 e  L  f  ð  x n Þ 1  e  L  f  ð  x n Þ  f  ð  x n Þ  f  0 ð  x n Þ¼ c 2 e 2 n  3 h c 3 e 3 n þ  6 h 2 c 4  c 32 þð 2  3 h Þ c 2 c 3   e 4 n þ O  e 5 n   :  ð 12 Þ Since from (5) we have e n þ 1  ¼ e n   1 þ 12 e  L  f  ð  x n Þ 1  e  L  f  ð  x n Þ  !  f  ð  x n Þ  f  0 ð  x n Þ ; from (7) and (12), we have e n þ 1  ¼ e n   e n  c 2 e 2 n þ c 3 e 3 n  c 4 e 4 n þ O ð e 5 n Þ     c 2 e 2 n  3 h c 3 e 3 n þ  6 h 2 c 4  c 32 þð 2  3 h Þ c 2 c 3   e 4 n þ O ð e 5 n Þ   ¼ð 3 h  1 Þ c 3 e 3 n þ  c 32 þð 3 h  2 Þ c 2 c 3 ð 6 h 2  1 Þ c 4   e 4 n þ O ð e 5 n Þ :  ð 13 Þ If the order is four, we have3 h  1 ¼ 0 ; which implies h ¼ 13 :  ð 14 Þ Using (14) in (13), we have e n þ 1  ¼  c 32  c 2 c 3 þ 13 c 4   e 4 n þ O ð e 5 n Þ :  ð 15 Þ Therefore, we havelim n !þ1 e n þ 1 e 4 n ¼ 18  f  00 ð  x  Þ 3  f  0 ð  x  Þ 3    112  f  00 ð  x  Þ  f  ð 3 Þ ð  x  Þ  f  0 ð  x  Þ 2  þ  172  f  ð 4 Þ ð  x  Þ  f  0 ð  x  Þ  :  ð 16 Þ This ends the proof.  h From Theorem 1, when we take  h  = 1/3 in (5), we can obtain a new variant of super-Halley method withfourth-order convergence  x n þ 1  ¼  x n   1 þ 12  L  f  ð  x n Þ 1   L  f  ð  x n Þ    f  ð  x n Þ  f  0 ð  x n Þ ;  ð 17 Þ where  L  f  ð  x n Þ¼  f  00 ð  x n   f  ð  x n Þ = ð 3  f  0 ð  x n ÞÞÞ  f  ð  x n Þ  f  0 ð  x n Þ 2  : Per iteration the present method requires one evaluation of the function, one of its first derivative and oneof its second derivative. We consider the definition of efficiency index [6] as  p  1 w , where  p  is the order of themethod and  w  is the number of function evaluations per iteration required by the method. If we assume thatall the evaluations have the same cost as function one, we have that the present method has the efficiency indexequal to  ffiffiffi 4 3 p  ’ 1 : 587, which is better than the ones of the super-Halley method  ffiffiffi 3 3 p  ’ 1 : 442 and Newton’smethod  ffiffiffi 2 p  ’ 1 : 414. 3. Numerical results Now, we employ a new variant of super-Halley method (VSHM), Eq. (17), obtained in this paper to solvesome non-linear equations and compare it with Newton’s method (NM), Eq. (1), and the super-Halley method J. Kou et al. / Applied Mathematics and Computation 186 (2007) 535–539  537  (SHM), Eq. (2). Displayed in Table 1 are the number of iterations ( n ) and the number of function evaluations(NFE) required such that  j  f  ( x n ) j < 1.E  15.The results in Table 1 show that the present method VSHM improves the computational efficiency of itsclassical method SHM. As far as the results we consider, in general, VSHM requires the less NFEs as com-pared to NM and SHM. Thus, the present method VSHM can compete with NM and SHM.We use the following functions, which are the same as in [7,8] respectively.  f  1 ð  x Þ¼  x 3 þ 4  x 2  10 ;  x   ¼ 1 : 3652300134140969 :  f  2 ð  x Þ¼  x 2  e  x  3  x þ 2 ;  x   ¼ 0 : 25753028543986084 :  f  3 ð  x Þ¼  x exp ð  x 2 Þ sin 2 ð  x Þþ 3cos ð  x Þþ 5 ;  x   ¼ 1 : 20764782713091893 :  f  4 ð  x Þ¼ cos ð  x Þ  x ;  x   ¼ 0 : 73908513321516067 :  f  5 ð  x Þ¼ sin 2 ð  x Þ  x 2 þ 1 ;  x   ¼ 1 : 4044916482153411 :  f  6 ð  x Þ¼  x 2 þ sin ð  x = 5 Þ 1 = 4 ;  x   ¼ 0 : 4099920179891371 :  f  7 ð  x Þ¼ e   x þ cos ð  x Þ ;  x   ¼ 1 : 7461395304080124 :  f  8 ð  x Þ¼ e  x  4  x 2 ;  x   ¼ 0 : 7148059123627778 : Acknowledgement Work supported by National Natural Science Foundation of China (50379038). References [1] A.M. Ostrowski, Solution of Equations in Eucilidean and Banach Space, third ed., Academic Press, New York, 1973.[2] J.M. Gutie´rrez, M.A. Herna´ndez, An acceleration of Newton’s method: super-Halley method, Appl. Math. Comput. 117 (2001) 223– 239.[3] D. Chen, I.K. Argyros, Q.S. Qian, A local convergence theorem for the Super-Halley method in a Banach space, Appl. Math. Lett. 7(5) (1994) 49–52.[4] J.A. Ezquerro, J.M. Gutie´rrez, M.A. Herna´ndez, M.A. Salanova, Solving nonlinear integral equations arising in radiative transfer,Numerical Functional Analysis and Optimization 20 (7–8) (1999) 661–673.Table 1Comparison of various iterative methods  f  ( x )  x 0  n  NFENM SHM VSHM NM SHM VSHM  f  1   0.5 97 60 14 194 180 421 5 3 3 10 9 91.5 4 3 2 8 9 6  f  2  0 4 3 2 8 9 60.5 4 3 2 8 9 6  f  3   0.5 10 5 4 20 15 12  1.5 6 4 3 12 12 9  f  4  0 5 3 3 10 9 91 4 3 2 8 9 6  f  5  1 6 3 3 12 9 93 6 4 3 12 12 9  f  6  0 7 4 4 14 12 120.5 4 2 2 8 6 6  f  7  1.5 4 3 2 8 9 63 4 3 2 8 9 6  f  8  0.3 6 4 3 12 12 92 6 4 3 12 12 9538  J. Kou et al. / Applied Mathematics and Computation 186 (2007) 535–539  [5] M. Frontini, E. Sormani, Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math.Comput. 149 (2004) 771–782.[6] W. Gautschi, Numerical Analysis: An Introduction, Birkha¨user, 1997.[7] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000)87–93.[8] M. Grau, J.L. Dı´az-Barrero, An improvement to Ostrowski root-finding method, Appl. Math. Comput. 173 (2006) 450–456. J. Kou et al. / Applied Mathematics and Computation 186 (2007) 535–539  539
Search
Similar documents
View more...
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks