Iterative methods for nonlinear equations and their semilocal convergence

Abstract

We are concerned with the numerical methods for nonlinear equation and their semilocal convergence in this paper. The construction techniques of iterative methods are induced by using linear approximation, integral interpolation, Adomian series decomposition, Taylor expansion, multi-step iteration, etc. The convergent conditions and proof methods, including majorizing sequences and recurrence relations, in semilocal convergence of iterative methods for nonlinear equations are discussed in the theoretical analysis. The majorizing functions, which are used in majorizing sequences, are also discussed in this paper.

Key words： Nonlinear equation; numerical method; semilocal convergence; Newton method; Banach space

References

1 Appell J, De Pascale E, Lysenko J V, Zabrejko P P. New results on Newton-Kantorovich approximations with applications to nonlinear integral equations. Numer Funct Anal Optim 1997; 18(1/2): 1–17

2 Argyros I K. On the approximate solutions of non-linear functional equations under mild differentiability conditions. Acta Math Hungar 1991; 58(1/2): 3–7

3 Argyros I K. On Newton’s method under mild differentiability conditions and applications. Appl Math Comput 1999; 102(2/3): 177–183

4 Argyros I K. A Newton-Kantorovich theorem for equations involving m-Fréchet differentiable operators and applications in radiative transfer. J Comput Appl Math 2001; 131(1/2): 149–159

5 Argyros I K. On the semilocal convergence of a fast two-step Newton method. Revista Colombiana de Matematicás 2008; 42(1): 15–24

6 ArgyrosI K. Convergence and Applications of Newton-type Iterations. New York: Springer-Verlag, 2008

7 Argyros I K, Chen D. On the midpoint method for solving nonlinear operator equations and applications to the solution of integral equations. Revue d’Analyse Numérique et de Théorie de l’Approximation 1994; 23(2): 139–152

8 Argyros I K, Ezquerro A, J J M, Gutiérrez M A, Hernández S. On the semilocal convergence of efficient Chebyshev-Secant-type methods. J Comput Appl Math 2011; 235(10): 3195–3206

9 Argyros I K, Hilout S. On the Gauss-Newton method. J Appl Math Comput 2011; 35(1/2): 537–550

10 Argyros I K, Hilout S. Improved local convergence of Newton’s method under weak majorant condition. J Comput Appl Math 2012; 236(7): 1892–1902

11 Argyros I K, Uko L U. An improved convergence analysis of a one-step intermediate Newton iterative scheme for nonlinear equations. J Appl Math Comput 2012; 38(1/2): 243–256

12 Babajee D K R, Dauhoo M Z. An analysis of the properties of the variants of Newton’s method with third order convergence. Appl Math Comput 2006; 183(1): 659–684

13 Banach S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund Math 1922; 3(1): 133–181

14 BaoA G. The theoretical analysis of some high-order iterative methods for nonlinear equations. Master thesis. Hangzhou: Zhejiang Normal University, 2007 (in Chinese)

15 Bi W H, Ren H M, Wu Q B. A new semilocal convergence theorem of Müller’s method. Appl Math Comput 2008; 199(1): 375–384

16 Biazar J, Ilie M, Khoshkenar A. An improvement to an alternate algorithm for computing Adomian polynomials in special cases. Appl Math Comput 2006; 173(1): 582–592

17 Candela V, Marquína A. Recurrence relations for rational cubic methods I: The Halley method. Computing 1990; 44(2): 169–184

18 Candela V, Marquína A. Recurrence relations for rational cubic methods II: The Chebyshev method. Computing 1990; 45(4): 355–367

19 CauchyA L. Sur la détermination des racines d’une équation algébrique ou transcendente. In: Leçons sur le Calcul Différentiel, Paris: Bure Fréres, 1829. Reprinted in: Euvres Complétes (II), Vol 4. Paris: Gauthier-Villars, 1899, 573–609 (in French)

20 Chen D, Argyros I K. The midpoint method for solving nonlinear operator equations in Banach space. Appl Math Lett 1992; 5(4): 7–9

21 Chen D, Argyros I K, Qian Q S. A note on the Halley method in Banach spaces. Appl Math Comput 1993; 58(2/3): 215–224

22 Chen J H, Li W G. Convergence behaviour of inexact Newton methods under weak Lipschitz condition. J Comput Appl Math 2006; 191(1): 143–164

23 Chen L, Gu C Q, Ma Y F. Semilocal convergence for a fifth-order Newton’s method using recurrence relations in Banach spaces. J Appl Math 2011; 2011: 786306

24 Chen L, Gu C Q, Ma Y F. Recurrence relations for the Harmonic mean Newton’s method in Banach spaces. J Comput Anal Appl 2012; 14(6): 1154–1164

25 Chen L, Ma Y F. A new modified King-Werner method for solving nonlinear equations. Comput Math Appl 2011; 62(10): 3700–3705

26 Chen L, Ma Y F. A fourth-order iterative method for solving nonlinear equations. J Huaibei Normal Univ (Natural Science) 2012; 33(1): 5–10

27 Chen W H, Lu Z Y. An algorithm for Adomian decomposition method. Appl Math Comput 2004; 159(1): 221–235

28 Chun C B. Iterative methods improving Newton’s method by the decomposition method. Comput Math Appl 2005; 50(10/11/12): 1559–1568

29 DennisJ ESchnabel R B. Numerical Methods for Unconstrained Optimisation and Nonlinear Equations. Englewood Cliffs: Prentice Hall, 1983

30 Deuflhard P, Heindl G. Affine invariant convergence theorems for Newton’s method and extensions to related methods. SIAM J Numer Anal 1979; 16(1): 1–10

31 Ezquerro J A, González D, Hernández M A. Majorizing sequences for Newton’s method from initial value problems. J Comput Appl Math 2012; 236(9): 2246–2258

32 Ezquerro J A, Gutiérrez J M, Hernández M A, Salanova M A. The application of an inverse-free Jarratt-type approximation to nonlinear integral equations of Hammerstein-type. Comput Math Appl 1998; 36(4): 9–20

33 Ezquerro J A, Hernández M A. Relaxing convergence conditions for an inverse-free Jarratt-type approximation. J Comput Appl Math 1997; 83(1): 131–135

34 Ezquerro J A, Hernández M A. Avoiding the computation of the second Fréchet-derivative in the convex acceleration of Newton’s method. J Comput Appl Math 1998; 96(1): 1–12

35 Ezquerro J A, Hernández M A. New iterations of R-order four with reduced computational cost. BIT Numer Math 2009; 49(2): 325–342

36 FengG C. Iterative Methods for Systems of Nonlinear Equations. Shanghai: Shanghai Scientific & Technical Publishers, 1989 (in Chinese)

37 Fenyö I, Rényi A. Über die lösung der im Banachschen Raume definierten nichtlinearen gleichungen. Acta Math Acad Sci Hungar 1954; 5(1/2): 85–93

38 Frontini M, Sormani E. Some variant of Newton’s method with third-order convergence. Appl Math Comput 2003; 140(2/3): 419–426

39 Frontini M, Sormani E. Modified Newton’s method with third- order convergence and multiple roots. J Comput Appl Math 2003; 156(2): 345–354

40 Gerlach J. Accelerated convergence in Newton’s method. SIAM Rev 1994; 36(2): 272–276

41 Geum Y H, Kim Y I. A penta-parametric family of fifteenth-order multipoint methods for nonlinear equations. Appl Math Comput 2010; 217(6): 2311–2319

42 Grau M, Noguera M. A variant of Cauchy’s method with accelerated fifth-order convergence. Appl Math Lett 2004; 17(5): 509–517

43 Gutiérrez J M. A new semilocal convergence theorem for Newton’s method. J Comput Appl Math 1997; 79(1): 131–145

44 Gutiérrez J M, Hernández M A. Recurrence relations for the super-Halley method. Comput Math Appl 1998; 36(7): 1–8

45 Gutiérrez . M. and Hernández, M.A., An acceleration of Newton’s method: super-Halley method. Appl Math Comput 2001; 117(2/3): 223–239

46 Halley E. Methodus Nova Accurata & facilis inveniendi Radices Æqùationium quarumcumque generaliter, sine prævia Reductione. Phil Trans Royal Soc London 1694; 18: 136–148

47 Han D F. The convergence on a family of iterations with cubic order. J Comput Math 2001; 19(5): 467–474

48 HasanovV IIvanov I GNedjibovG. A new modification of Newton’s method. In: Applications of Mathematics in Engineering and Economics, Proceedings of the XXVII Summer School. Sofia: Heron Press, 2002, 278–286

49 Hernández M A. Chebyshev’s approximation algorithms and applications. Comput Math Appl 2001; 41(3/4): 433–445

50 Hernández M A, Salanova M A. A family of Chebyshev type methods in Banach spaces. Int J Comput Math 1996; 61(1/2): 145–154

51 Homeier H H H. A modified Newton method for root finding with cubic convergence. J Comput Appl Math 2003; 157(1): 227–230

52 Homeier H H H. A modified Newton method with cubic convergence: the multivariate case. J Comput Appl Math 2004; 169(1): 161–169

53 Homeier H H H. On Newton-type methods with cubic convergence. J Comput Appl Math 2005; 176(2): 425–432

54 Homeier H H H. On Newton-type methods for multiple roots with cubic convergence. J Comput Appl Math 2009; 231(1): 249–254

55 Hosseini M. Adomian decomposition method with Chebyshev polynomials. Appl Math Comput 2006; 175(2): 1685–1693

56 HuangX DZeng Z GMaY N. The Theory and Methods for Nonlinear Numerical Analysis. Wuhan: Wuhan University Press, 2004 (in Chinese)

57 Huang Z D. A note on the Kantorovich theorem for Newton iteration. J Comput Appl Math 1993; 47(2): 211–217

58 KantorovicL. Functional analysis and applied mathematics. Uspehi Mat Nauk (N S), 1948, 3: 89‒185 (in Russian). Translated by Benster C D. National Bureau of Standards, Report No 1509, Washington, 1952

59 Khattri S K, Argyros I K. Sixth order derivative free family of iterative methods. Appl Math Comput 2011; 217(12): 5500–5507

60 Kim Y-I, Chun C B. New twelfth-order modifications of Jarratt’s method for solving nonlinear equations. Studies Nonl Sci 2010; 1(1): 14–18

61 Kou J S. Some variants of Cauchy’s method with accelerated fourth-order convergence. J Comput Appl Math 2008; 213(1): 71–78

62 Kou J S, Li Y T. Modified Chebyshev’s method free from second derivative for non-linear equations. Appl Math Comput 2007; 187(2): 1027–1032

63 Kou J S, Li Y T, Wang X H. A variant of super-Halley method with accelerated fourth-order convergence. Appl Math Comput 2007; 186(1): 535–539

64 Kou J S, Wang X H, Li Y T. Some eighth-order root-finding three-step methods. Commun Nonl Sci Numer Simulat 2010; 15(3): 536–544

65 Layeni O. Remark on modifications of Adomian decomposition method. Appl Math Comput 2008; 197(1): 167–171

66 LiQ YMo ZQiL Q. Numerical Methods for Systems of Nonlinear Equations. Beijing: Science Press, 1987 (in Chinese)

67 Li X W, Mu C L, Ma J W, Wang C. Sixteenth-order method for nonlinear equations. Appl Math Comput 2010; 215(10): 3754–3758

68 LiuJ. The Theoretical Analysis of High-Order Iterative Methods for Non-Linear Equations. Master thesis, Hangzhou: Zhejiang University, 2004

69 Luo X G. A note on the new iteration method for solving algebraic equations. Appl Math Comput 2005; 171(2): 1177–1183

70 Melman A. Geometry and convergence of Euler’s and Halley’s methods. SIAM Rev 1997; 39(4): 728–735

71 MiaoH. The theoretical analysis of some iterative methods for nonlinear equations. Master thesis, Hangzhou: Zhejiang University, 2006

72 Noor M A, Khan W A, Noor K I, Al-said E. Higher-order iterative methods free from second derivative for solving nonlinear equations. Internat J Phys Sci 2011; 6(8): 1887–1893

73 Noor K I, Noor M A. Predictor-corrector Halley method for non-linear equations. Appl Math Comput 2007; 188(2): 1587–1591

74 OrtegaJ MRheinboldt W C. Iterative Solution of NonlinearEquations in Several Variables. Philadelphia: SIAM, 1970

75 OstrowskiA. Solution of Equations and Systems of Equations. New York: Academic Press, 1973

76 Özban A Y. Some new variants of Newton’s method. Appl Math Lett 2004; 17(6): 677–682

77 Ren H M, Argyros I K. A new semilocal convergence theorem for a fast iterative method with nondifferentiable operators. J Appl Math Comput 2010; 34(1/2): 39–46

78 Ren H M, Wu Q B, Bi W H. On convergence of a new secant-like method for solving nonlinear equations. Appl Math Comput 2010; 217(2): 583–589

79 Rokne J. Newton’s method under mild differentiability conditions with error analysis. Numer Math 1971; 18(5): 401–412

80 Sargolzaei P, Soleymani F. Accurate fourteenth-order methods for solving nonlinear equations. Numer Algor 2011; 58(4): 513–527

81 Scavo T R, Thoo J B. On the geometry of Halley’s method. Amer Math Monthly 1995; 102(5): 417–426

82 Sharma J R, Guha R K, Gupta P. Some efficient derivative free methods with memory for solving nonlinear equations. Appl Math Comput 2012; 219(2): 699–707

83 SmaleS. Newton’s method estimates from data at one point. In: The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics. New York: Springer-Verlag, 1986, 185–196

84 SmaleS. Algorithms for solving equations. In: Proceedings of the International Congress of Mathematicians. Providence: AMS, 1987, 172–195

85 Soleymani F. Letter to the editor regarding the article by Khattri: derivative free algorithm for solving nonlinear equations. Computing 2013; 95(2): 159–162

86 Tatari M, Dehghan M. On the convergence of He’s variational iteration method. J Comput Appl Math 2007; 207(1): 121–128

87 Thukral R. Introduction to a Newton-type method for solving nonlinear equations. Appl Math Comput 2008; 195(2): 663–668

88 TraubJ F. Iterative Methods for the Solution of Equations. Englewood Cliffs: Prentice-Hall, 1964

89 WangD R. Numerical Methods for Nonlinear Equations and Optimization. Beijing: People’s Education Press, 1979

90 Wang X H. Convergence of a family of Halley’s method under weak condition. Chinese Sci Bull 1997; 42(2): 119–121

91 Wang X H. Convergence of the iteration of Halley’s family and Smale operator class in Banach space. Sci China Ser A 1998; 41(7): 700–709

92 Wang X H. Convergence of Newton’s method and inverse function theorem in Banach space. Math Comp 1999; 68(225): 169–186

93 Wang X H. Convergence of Newton’s method and uniqueness of thesolution of equations in Banach space. IMA J Numer Anal 2000; 20(1): 123–134

94 Wang X H, Dzunić J, Zhang T. On an efficient family of derivative free three-point methods for solving nonlinear equations. Appl Math Comput 2012; 219(4): 1749–1760

95 Wang X H, Han D F. Criterion α and Newton’s method in the weak conditions. Math Numer Sin 1997; 19(1): 103–112

96 Wang X H, Kou J S. Semilocal convergence and R-order for modified Chebyshev-Halley methods. Numer Algor 2013; 64(1): 105–126

97 Wang X H, Kou J S, Li Y T. Modified Jarratt method with sixth-order convergence. Appl Math Lett 2009; 22(12): 1798–1802

98 Wang X H, Li C. Local and global behaviors for algorithms of solving equations. Chinese Sci Bull 2001; 46(6): 444–451

99 Wang X H, Li C. On the united theory of the family of Euler-Halley type methods with cubical convergence in Banach spaces. J Comput Math 2003; 21(2): 195–200

100 Wang X H, Li C. Convergence of Newton’s method and uniqueness of the solution of equations in Banach spaces II. Acta Math Sin Engl Ser 2003; 19(2): 405–412

101 Weerakoon S, Fernando T G I. A variant of Newton’s method with accelerated third-order convergence. Appl Math Lett 2000; 13(8): 87–93

102 Wu M. A convergence theorem for the Newton-like methods under some kind of weak Lipschitz conditions. J Math Anal Appl 2008; 339(2): 1425–1431

103 WuM. Some Theories About Iterative Methods for Solving Nonlinear Equations. Ph D thesis. Hangzhou: Zhejiang University, 2008

104 WuP. Iterative Methods with Higher-Order Convergence for Solving Nonlinear Equations and Analysis Convergence. Ph D thesis. Hangzhou: Zhejiang University, 2008

105 Ypma T J. Affine invariant convergence results for Newton’s method. BIT Numer Math 1982; 22(1): 108–118

106 Zhang H, Li D S, Liu Y Z. A new method of secant-like for nonlinear equations. Commun Nonl Sci Numer Simulat 2009; 14(7): 2923–2927

107 Zheng L, Gu C Q. Fourth-order convergence theorem by using majorizing functions for super-Halley method in Banach spaces. Int J Comput Math 2012; 90(2): 423–434

108 Zheng L, Gu C Q. Semilocal convergence of a sixth-order method in Banach spaces. Numer Algor 2012; 61(3): 413–427