Extending the applicability of Newton's method using nondiscrete induction

Ioannis K. Argyros; Saïd Hilout

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 1, page 115-141
  • ISSN: 0011-4642

Abstract

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We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.

How to cite

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Argyros, Ioannis K., and Hilout, Saïd. "Extending the applicability of Newton's method using nondiscrete induction." Czechoslovak Mathematical Journal 63.1 (2013): 115-141. <http://eudml.org/doc/252486>.

@article{Argyros2013,
abstract = {We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.},
author = {Argyros, Ioannis K., Hilout, Saïd},
journal = {Czechoslovak Mathematical Journal},
keywords = {Newton's method; Banach space; rate of convergence; semilocal convergence; nondiscrete mathematical induction; estimate function; Newton's method; Banach space; rate of convergence; semilocal convergence; nondiscrete mathematical induction; estimate function; nonlinear operator equation; center-Lipschitz conditions; error estimates; numerical examples},
language = {eng},
number = {1},
pages = {115-141},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Extending the applicability of Newton's method using nondiscrete induction},
url = {http://eudml.org/doc/252486},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Argyros, Ioannis K.
AU - Hilout, Saïd
TI - Extending the applicability of Newton's method using nondiscrete induction
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 1
SP - 115
EP - 141
AB - We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.
LA - eng
KW - Newton's method; Banach space; rate of convergence; semilocal convergence; nondiscrete mathematical induction; estimate function; Newton's method; Banach space; rate of convergence; semilocal convergence; nondiscrete mathematical induction; estimate function; nonlinear operator equation; center-Lipschitz conditions; error estimates; numerical examples
UR - http://eudml.org/doc/252486
ER -

References

top
  1. Amat, S., Bermúdez, C., Busquier, S., Gretay, J., 10.1216/rmjm/1181068756, Rocky Mt. J. Math. 37 (2007), 359-369. (2007) Zbl1140.65040MR2333375DOI10.1216/rmjm/1181068756
  2. Amat, S., Busquier, S., 10.1016/j.jmaa.2007.02.052, J. Math. Anal. Appl. 336 (2007), 243-261. (2007) Zbl1128.65036MR2348504DOI10.1016/j.jmaa.2007.02.052
  3. Amat, S., Busquier, S., Gutiérrez, J. M., Hernández, M. A., 10.1016/j.cam.2007.07.022, J. Comput. Appl. Math. 220 (2008), 17-21. (2008) Zbl1149.65035MR2444150DOI10.1016/j.cam.2007.07.022
  4. Argyros, I. K., The Theory and Application of Abstract Polynomial Equations, St. Lucie/CRC/Lewis Publ. Mathematics series, Boca Raton, Florida, USA (1998). (1998) 
  5. Argyros, I. K., 10.1016/j.jmaa.2004.04.008, J. Math. Anal. Appl. 298 (2004), 374-397. (2004) Zbl1061.47052MR2086964DOI10.1016/j.jmaa.2004.04.008
  6. Argyros, I. K., 10.1016/j.cam.2004.01.029, J. Comput. Appl. Math. 169 (2004), 315-332. (2004) Zbl1055.65066MR2072881DOI10.1016/j.cam.2004.01.029
  7. Argyros, I. K., 10.1016/j.na.2005.02.113, Nonlinear Anal., Theory Methods Appl. 62 (2005), 179-194. (2005) Zbl1072.65079MR2139363DOI10.1016/j.na.2005.02.113
  8. Argyros, I. K., Approximating solutions of equations using Newton's method with a modified Newton's method iterate as a starting point, Rev. Anal. Numér. Théor. Approx. 36 (2007), 123-137. (2007) Zbl1199.65179MR2498828
  9. Argyros, I. K., Computational Theory of Iterative Methods, Studies in Computational Mathematics 15. Elsevier, Amsterdam (2007). (2007) Zbl1147.65313MR2356038
  10. Argyros, I. K., 10.1016/j.cam.2008.08.042, J. Comput. Appl. Math. 228 (2009), 115-122. (2009) Zbl1168.65349MR2514268DOI10.1016/j.cam.2008.08.042
  11. Argyros, I. K., 10.1090/S0025-5718-2010-02398-1, Math. Comput. 80 (2011), 327-343. (2011) Zbl1211.65057MR2728982DOI10.1090/S0025-5718-2010-02398-1
  12. Argyros, I. K., Hilout, S., Efficient Methods for Solving Equations and Variational Inequalities, Polimetrica Publisher, Milano, Italy (2009). (2009) 
  13. Argyros, I. K., Hilout, S., Enclosing roots of polynomial equations and their applications to iterative processes, Surv. Math. Appl. 4 (2009), 119-132. (2009) Zbl1205.26023MR2558651
  14. Argyros, I. K., Hilout, S., 10.1016/j.cam.2010.04.014, J. Comput. Appl. Math. 234 (2010), 2993-3006. (2010) Zbl1195.65075MR2652146DOI10.1016/j.cam.2010.04.014
  15. Argyros, I. K., Hilout, S., Tabatabai, M. A., Mathematical Modelling with Applications in Biosciences and Engineering, Nova Publishers, New York, 2011. MR2895345
  16. Bi, W., Wu, Q., Ren, H., Convergence ball and error analysis of the Ostrowski-Traub method, Appl. Math., Ser. B (Engl. Ed.) 25 (2010), 374-378. (2010) Zbl1240.65167MR2679357
  17. Cătinaş, E., 10.1090/S0025-5718-04-01646-1, Math. Comput. 74 (2005), 291-301. (2005) Zbl1054.65050MR2085412DOI10.1090/S0025-5718-04-01646-1
  18. Chen, X., Yamamoto, T., 10.1080/01630568908816289, Numer. Funct. Anal. Optimization 10 (1989), 37-48. (1989) Zbl0645.65028MR0978801DOI10.1080/01630568908816289
  19. Deuflhard, P., Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics 35. Springer, Berlin (2004). (2004) Zbl1056.65051MR2063044
  20. Ezquerro, J. A., Gutiérrez, J. M., Hernández, M. A., Romero, N., Rubio, M. J., The Newton method: from Newton to Kantorovich, Spanish Gac. R. Soc. Mat. Esp. 13 (2010), 53-76. (2010) Zbl1195.65001MR2647925
  21. Ezquerro, J. A., Hernández, M. A., 10.1016/j.cam.2005.10.023, J. Comput. Appl. Math. 197 (2006), 53-61. (2006) Zbl1106.65048MR2256051DOI10.1016/j.cam.2005.10.023
  22. Ezquerro, J. A., Hernández, M. A., 10.1090/S0025-5718-09-02193-0, Math. Comput. 78 (2009), 1613-1627. (2009) Zbl1198.65096MR2501066DOI10.1090/S0025-5718-09-02193-0
  23. Ezquerro, J. A., Hernández, M. A., Romero, N., 10.1016/j.amc.2009.03.072, Appl. Math. Comput. 214 (2009), 142-154. (2009) Zbl1173.65032MR2541053DOI10.1016/j.amc.2009.03.072
  24. Gragg, W. B., Tapia, R. A., 10.1137/0711002, SIAM J. Numer. Anal. 11 (1974), 10-13. (1974) Zbl0284.65042MR0343594DOI10.1137/0711002
  25. Hernández, M. A., 10.1016/S0377-0427(01)00393-4, J. Comput. Appl. Math. 137 (2001), 201-205. (2001) Zbl0992.65057MR1865886DOI10.1016/S0377-0427(01)00393-4
  26. Kantorovich, L. V., Akilov, G. P., Functional Analysis. Transl. from the Russian, Pergamon Press, Oxford (1982). (1982) Zbl0484.46003MR0664597
  27. Krishnan, S., Manocha, D., 10.1145/237748.237751, ACM Trans. on Graphics. 16 (1997), 74-106. (1997) DOI10.1145/237748.237751
  28. Lukács, G., The generalized inverse matrix and the surface-surface intersection problem, Theory and Practice of Geometric Modeling, Lect. Conf., Blaubeuren/FRG 1988 167-185 (1989). (1989) Zbl0692.68076MR1042329
  29. Ortega, J. M., Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Computer Science and Applied Mathematics. Academic Press, New York (1970). (1970) Zbl0241.65046MR0273810
  30. Ostrowski, A. M., Sur la convergence et l'estimation des erreurs dans quelques procédés de résolution des équations numériques, French Gedenkwerk D. A. Grave, Moskau 213-234 (1940). (1940) Zbl0023.35302MR0004377
  31. Ostrowski, A. M., La méthode de Newton dans les espaces de Banach. (The Newton method in Banach spaces), French C. R. Acad. Sci., Paris, Sér. A 272 (1971), 1251-1253. (1971) Zbl0228.65041MR0285110
  32. Ostrowski, A. M., Solution of Equations in Euclidean and Banach Spaces. 3rd ed. of solution of equations and systems of equations, Pure and Applied Mathematics, 9. Academic Press, New York (1973). (1973) Zbl0304.65002MR0359306
  33. Păvăloiu, I., Introduction in the Theory of Approximation of Equations Solutions, Dacia Ed. Cluj-Napoca (1976). (1976) 
  34. Potra, F. A., A characterization of the divided differences of an operator which can be represented by Riemann integrals, Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numér. Théor. Approximation 9 (1980), 251-253. (1980) Zbl0523.65043MR0651780
  35. Potra, F. A., An application of the induction method of V. Pták to the study of regula falsi, Apl. Mat. 26 (1981), 111-120. (1981) Zbl0486.65038MR0612668
  36. Potra, F. A., The rate of convergence of a modified Newton's process, Apl. Mat. 26 (1981), 13-17. (1981) Zbl0486.65039MR0602398
  37. Potra, F. A., 10.1007/BF01396443, Numer. Math. 38 (1982), 427-445. (1982) Zbl0465.65033MR0654108DOI10.1007/BF01396443
  38. Potra, F. A., 10.1007/BFb0069378, Iterative solution of nonlinear systems of equations, Proc. Meeting, Oberwolfach 1982, Lect. Notes Math. 953 125-137. Zbl0507.65020MR0678615DOI10.1007/BFb0069378
  39. Potra, F. A., On the a posteriori error estimates for Newton's method, Beitr. Numer. Math. 12 (1984), 125-138. (1984) MR0732159
  40. Potra, F. A., On a class of iterative procedures for solving nonlinear equations in Banach spaces, Computational Mathematics, Banach Cent. Publ. 13 607-621 (1984). (1984) Zbl0569.65042MR0798124
  41. Potra, F. A., Sharp error bounds for a class of Newton-like methods, Libertas Math. 5 (1985), 71-84. (1985) Zbl0581.47050MR0816258
  42. Potra, F. A., Pták, V., 10.1007/BF01463998, Numer. Math. 34 (1980), 63-72. (1980) Zbl0434.65034MR0560794DOI10.1007/BF01463998
  43. Potra, F. A., Pták, V., Nondiscrete induction and a double step secant method, Math. Scand. 46 (1980), 236-250. (1980) Zbl0423.65034MR0591604
  44. Potra, F. A., Pták, V., 10.1080/01630568008816049, Numer. Funct. Anal. Optimization 2 (1980), 107-120. (1980) Zbl0472.65049MR0580387DOI10.1080/01630568008816049
  45. Potra, F. A., Pták, V., 10.1007/BF01396659, Numer. Math. 36 (1981), 333-346. (1981) Zbl0478.65039MR0613073DOI10.1007/BF01396659
  46. Potra, F. A., Pták, V., Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston (1984). (1984) Zbl0549.41001MR0754338
  47. Proinov, P. D., 10.1016/j.jco.2008.05.006, J. Complexity 25 (2009), 38-62. (2009) Zbl1158.65040MR2475307DOI10.1016/j.jco.2008.05.006
  48. Proinov, P. D., 10.1016/j.jco.2009.05.001, J. Complexity 26 (2010), 3-42. (2010) Zbl1185.65095MR2574570DOI10.1016/j.jco.2009.05.001
  49. Pták, V., 10.1007/BF02052841, Math. Ann. 163 (1966), 95-104. (1966) Zbl0138.37602MR0192316DOI10.1007/BF02052841
  50. Pták, V., A quantitative refinement of the closed graph theorem, Czech. Math. J. 24 (1974), 503-506. (1974) Zbl0315.46007MR0348431
  51. Pták, V., 10.1007/BF01411490, Manuscr. Math. 13 (1974), 109-130. (1974) Zbl0286.46008MR0348430DOI10.1007/BF01411490
  52. Pták, V., Deux théoremes de factorisation, C. R. Acad. Sci., Paris, Sér. A 278 (1974), 1091-1094. (1974) Zbl0277.46047MR0341096
  53. Pták, V., Concerning the rate of convergence of Newton's process, Commentat. Math. Univ. Carol. 16 (1975), 699-705. (1975) Zbl0314.65023MR0398092
  54. Pták, V., A modification of Newton's method, Čas. Pěst. Mat. 101 (1976), 188-194. (1976) Zbl0328.46013MR0443326
  55. Pták, V., 10.1016/0024-3795(76)90098-7, Linear Algebra Appl. 13 (1976), 223-238. (1976) Zbl0323.46005MR0394119DOI10.1016/0024-3795(76)90098-7
  56. Pták, V., 10.1007/BF01399416, Numer. Math. 25 (1976), 279-285. (1976) Zbl0304.65037MR0478587DOI10.1007/BF01399416
  57. Pták, V., 10.1007/BFb0068681, Gen. Topol. Relat. mod. Anal. Algebra IV, Proc. 4th Prague topol. Symp. 1976, Part A, Lect. Notes Math. 609 166-178 (1977). (1977) Zbl0367.46007MR0487618DOI10.1007/BFb0068681
  58. Pták, V., What should be a rate of convergence? RAIRO, Anal, Numér. 11 (1977), 279-286. (1977) MR0474799
  59. Pták, V., Stability of exactness, Commentat. math., spec. Vol. II, dedic. L. Orlicz (1979), 283-288. (1979) Zbl0445.46003MR0552012
  60. Pták, V., 10.1080/01630567908816015, Numer. Funct. Anal. Optimization 1 (1979), 255-271. (1979) Zbl0441.46010MR0537831DOI10.1080/01630567908816015
  61. Pták, V., Factorization in Banach algebras, Stud. Math. 65 (1979), 279-285. (1979) Zbl0342.46036MR0567080
  62. Ren, H., Wu, Q., 10.1016/j.amc.2006.09.111, Appl. Math. Comput. 188 (2007), 281-285. (2007) Zbl1118.65044MR2327116DOI10.1016/j.amc.2006.09.111
  63. Rheinboldt, W. C., 10.1137/0705003, SIAM J. Numer. Anal. 5 (1968), 42-63. (1968) Zbl0155.46701MR0225468DOI10.1137/0705003
  64. Tapia, R. A., 10.2307/2316909, Am. Math. Mon. 78 (1971), 389-392. (1971) Zbl0215.27404MR1536290DOI10.2307/2316909
  65. Wu, Q., Ren, H., 10.1016/j.amc.2006.11.043, Appl. Math. Comput. 188 (2007), 1790-1793. (2007) Zbl1121.65052MR2335032DOI10.1016/j.amc.2006.11.043
  66. Yamamoto, T., 10.1007/BF01400355, Numer. Math. 51 (1987), 545-557. (1987) Zbl0633.65049MR0910864DOI10.1007/BF01400355
  67. Zabrejko, P. P., Nguen, D. F., 10.1080/01630568708816254, Numer. Funct. Anal. Optimization 9 (1987), 671-684. (1987) Zbl0627.65069MR0895991DOI10.1080/01630568708816254
  68. Zinčenko, A. I., Some approximate methods of solving equations with non-differentiable operators, Ukrainian Dopovidi Akad. Nauk Ukraïn. RSR (1963), 156-161. (1963) MR0160096

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