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A short survey on Kantorovich-like theorems for Newton’s method. (English) Zbl 1365.65155


MSC:

65J15 Numerical solutions to equations with nonlinear operators
65H05 Numerical computation of solutions to single equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
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[1] F. Alvarez, J. Bolte, and J. Munier. A unifying local convergence result for Newton’s method in Riemannian manifolds.Found. Comput. Math., 8(2):197–226, 2008. · Zbl 1147.58008
[2] S. Amat, C. Bermúdez, S. Busquier, and S. Plaza. On a third-order Newton-type method free of bilinear operators.Numer. Linear Algebra Appl., 17(4):639–653, 2010. · Zbl 1240.49046
[3] I. K. Argyros. A new Kantorovich-type theorem for Newton’s method.Appl. Math. (Warsaw), 26(2):151–157, 1999. · Zbl 0998.65059
[4] I. K. Argyros. A new semilocal convergence theorem for Newton’s method in Banach space using hypotheses on the second Fréchet-derivative.J. Comput. Appl. Math., 130(1-2):369–373, 2001. · Zbl 1010.65025
[5] I. K. Argyros and D. González. Extending the applicability of Newton’s method by improving a local result due to Dennis and Schnabel.SEMA J., 63:53–63, 2014. · Zbl 1312.65084
[6] I. K. Argyros and S. K. Khattri. Weaker Kantorovich type criteria for inexact Newton methods.J. Comput. Appl. Math., 261:103–117, 2014. · Zbl 1278.65064
[7] Ioannis K. Argyros. Concerning the convergence of inexact Newton methods.J. Comput. Appl. Math., 79(2):235–247, 1997. · Zbl 0881.65048
[8] Ioannis K. Argyros.Convergence and applications of Newton-type iterations. Springer, New York, 2008. · Zbl 1153.65057
[9] Ioannis K. Argyros, Yeol Je Cho, and Saïd Hilout. On the semilocal convergence of the Halley method using recurrent functions.J. Appl. Math. Comput., 37(1-2):221–246, 2011. · Zbl 1291.65156
[10] Ioannis K. Argyros and Saïd Hilout. Newton’s method for approximating zeros of vector fields on Riemannian manifolds.J. Appl. Math. Comput., 29(1-2):417–427, 2009.
[11] Ioannis K. Argyros and Saïd Hilout. Extending the applicability of the Gauss-Newton method under average Lipschitz-type conditions.Numer. Algorithms, 58(1):23–52, 2011.
[12] Ioannis K. Argyros and Saïd Hilout. Newton-Kantorovich approximations under weak continuity conditions.J. Appl. Math. Comput., 37(1-2):361–375, 2011.
[13] Ioannis K. Argyros and Saïd Hilout.Computational methods in nonlinear analysis. Efficient algorithms, fixed point theory and applications. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
[14] Ioannis K. Argyros, Saïd Hilout, and Sanjay K. Khattri. Expanding the applicability of Newton’s method using Smale’s α-theory.J. Comput. Appl. Math., 261:183–200, 2014. · Zbl 1291.65167
[15] Ioannis K. Argyros and Ángel Alberto Magreñán Ruiz. General convergence conditions of Newton’s method form-Fréchet differentiable operators.J. Appl. Math. Comput., 43(1-2):491–506, 2013.
[16] Ioannis K. Argyros and Hongmin Ren. Ball convergence theorems for Halley’s method in Banach space.J. Appl. Math. Comput., 38(1-2):453–465, 2012. · Zbl 1295.65064
[17] Ioannis K. Argyros and Livinus U. Uko. A semilocal convergence analysis of an inexact Newton method using recurrent relations.Punjab Univ. J. Math. (Lahore), 45:25–32, 2013. · Zbl 1291.65168
[18] Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale.Complexity and real computation. Springer-Verlag, New York, 1998. · Zbl 0872.68036
[19] Filomena Cianciaruso. A further journey in the ”terra incognita” of the Newton-Kantorovich method.Nonlinear Funct. Anal. Appl., 15(2):173–183, 2010. · Zbl 1242.65113
[20] Philippe G. Ciarlet and Cristinel Mardare. On the Newton-Kantorovich theorem.Anal. Appl. (Singap.), 10(3):249–269, 2012. · Zbl 1308.65079
[21] Jean-Pierre Dedieu.Points fixes, zéros et la méthode de Newton, volume 54 ofMathématiques & Applications (Berlin). Springer, Berlin, 2006.
[22] J. E. Dennis, Jr. On the Kantorovich hypothesis for Newton’s method.SIAM J. Numer. Anal., 6:493–507, 1969. · Zbl 0221.65098
[23] J. E. Dennis, Jr. and Robert B. Schnabel.Numerical methods for unconstrained optimization and nonlinear equations, volume 16 ofClassics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. Corrected reprint of the 1983 original. · Zbl 0847.65038
[24] J. A. Ezquerro, D. González, and M. A. Hernández. A modification of the classic conditions of Newton-Kantorovich for Newton’s method.Math. Comput. Modelling, 57(3-4):584–594, 2013.
[25] J. A. Ezquerro, D. González, and M. A. Hernández. On the local convergence of Newton’s method under generalized conditions of Kantorovich.Appl. Math. Lett., 26(5):566–570, 2013.
[26] J. A. Ezquerro, D. González, and M. A. Hernández-Verón. A semilocal convergence result for Newton’s method under generalized conditions of Kantorovich.J. Complexity, 30(3):309–324, 2014.
[27] J. A. Ezquerro, J. M. Gutiérrez, and M. A. Hernández. A construction procedure of iterative methods with cubical convergence. II. Another convergence approach.Appl. Math. Comput., 92(1):59–68, 1998.
[28] José Antonio Ezquerro, Daniel González, and Miguel Ángel Hernández. A general semilocal convergence result for Newton’s method under centered conditions for the second derivative.ESAIM Math. Model. Numer. Anal., 47(1):149–167, 2013.
[29] O. P. Ferreira and B. F. Svaiter. Kantorovich’s theorem on Newton’s method in Riemannian manifolds.J. Complexity, 18(1):304–329, 2002. · Zbl 1003.65057
[30] O. P. Ferreira and B. F. Svaiter. Kantorovich’s majorants principle for Newton’s method.Comput. Optim. Appl., 42(2):213–229, 2009. · Zbl 1191.90095
[31] O. P. Ferreira and B. F. Svaiter. A robust Kantorovich’s theorem on the inexact Newton method with relative residual error tolerance.J. Complexity, 28(3):346–363, 2012. · Zbl 1245.65060
[32] M. Giusti, G. Lecerf, B. Salvy, and J.-C. Yakoubsohn. On location and approximation of clusters of zeros: case of embedding dimension one.Found. Comput. Math., 7(1):1–49, 2007. · Zbl 1124.65047
[33] W. B. Gragg and R. A. Tapia. Optimal error bounds for the Newton-Kantorovich theorem.SIAM J. Numer. Anal., 11:10–13, 1974. · Zbl 0284.65042
[34] Xue-Ping Guo and Iain S. Duff. Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations.Numer. Linear Algebra Appl., 18(3):299–315, 2011. · Zbl 1249.65116
[35] José M. Gutiérrez. A new semilocal convergence theorem for Newton’s method.J. Comput. Appl. Math., 79(1):131–145, 1997. · Zbl 0872.65045
[36] W. M. Häussler. A Kantorovich-type convergence analysis for the Gauss-Newton-method.Numer. Math., 48(1):119–125, 1986.
[37] Zheng Da Huang. A note on the Kantorovich theorem for Newton iteration.J. Comput. Appl. Math., 47(2):211–217, 1993. · Zbl 0782.65071
[38] L. V. Kantorovič. Functional analysis and applied mathematics.Uspehi Matem. Nauk (N.S.), 3(6(28)):89–185, 1948.
[39] L. V. Kantorovich and G. P. Akilov.Functional analysis in normed spaces. Translated from the Russian by D. E. Brown. Edited by A. P. Robertson. International Series of Monographs in Pure and Applied Mathematics, Vol. 46. The Macmillan Co., New York, 1964. · Zbl 0127.06104
[40] Myong-Hi Kim.Computational Complexity of the Euler Type Algorithms for the Roots of Complex Polynomials. PhD thesis, City University of New York, 1985.
[41] Myong-Hi Kim. On approximate zeros and rootfinding algorithms for a complex polynomial.Math. Comp., 51(184):707–719, 1988. · Zbl 0699.30031
[42] S. Kim. A Kantorovich-type convergence analysis for the quasi-Gauss-Newton method.J. Korean Math. Soc., 33(4):865–878, 1996. · Zbl 0872.65044
[43] Chong Li, Nuchun Hu, and Jinhua Wang. Convergence behavior of Gauss-Newton’s method and extensions of the Smale point estimate theory.J. Complexity, 26(3):268–295, 2010. · Zbl 1192.65057
[44] Rongfei Lin, Yueqing Zhao, Zdeněk Šmarda, Yasir Khan, and Qingbiao Wu. Newton-Kantorovich and Smale uniform type convergence theorem for a deformed Newton method in Banach spaces.Abstr. Appl. Anal., pages Art. ID 923898, 8, 2013.
[45] Rongfei Lin, Yueqing Zhao, Zdeněk Šmarda, Qingbiao Wu, and Yasir Khan. Newton-Kantorovich convergence theorem of a new modified Halley’s method family in a Banach space.Adv. Difference Equ., pages 2013:325, 11, 2013. · Zbl 1444.65024
[46] Rongfei Lin, Yueqing Zhao, Qingbiao Wu, and Jueliang Hu. Convergence theorem for a family of new modified Halley’s method in Banach space.J. Appl. Math., pages Art. ID 468694, 8, 2014.
[47] G. J. Miel. The Kantorovich theorem with optimal error bounds.Amer. Math. Monthly, 86(3):212–215, 1979. · Zbl 0407.65023
[48] Igor Moret. A Kantorovich-type theorem for inexact Newton methods.Numer. Funct. Anal. Optim., 10(3-4):351–365, 1989. · Zbl 0653.65044
[49] J. M. Ortega and W. C. Rheinboldt.Iterative solution of nonlinear equations in several variables, volume 30 ofClassics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. Reprint of the 1970 original. · Zbl 0949.65053
[50] James M. Ortega. The Newton-Kantorovich theorem.Amer. Math. Monthly, 75(6):658–660, 1968. · Zbl 0183.43004
[51] A. M. Ostrowski.Solution of equations and systems of equations. Second edition. Pure and Applied Mathematics, Vol. 9. Academic Press, New York-London, 1966. · Zbl 0222.65070
[52] Alexandre Ostrowski. La méthode de Newton dans les espaces de Banach.C. R. Acad. Sci. Paris Sér. A-B, 272:A1251–A1253, 1971. · Zbl 0228.65041
[53] B. T. Polyak. Newton-Kantorovich method and its global convergence.Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 312(Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 11):256–274, 316, 2004.
[54] Florian-A. Potra and Vlastimil Pták. Sharp error bounds for Newton’s process.Numer. Math., 34(1):63–72, 1980.
[55] Weiping Shen and Chong Li. Kantorovich-type convergence criterion for inexact Newton methods.Appl. Numer. Math., 59(7):1599–1611, 2009. · Zbl 1165.65354
[56] M. Shub and S. Smale. Computational complexity: on the geometry of polynomials and a theory of cost. II.SIAM J. Comput., 15(1):145–161, 1986. · Zbl 0625.65036
[57] Mike Shub and Steven Smale. Computational complexity. On the geometry of polynomials and a theory of cost. I.Ann. Sci. École Norm. Sup. (4), 18(1):107–142, 1985. · Zbl 0603.65027
[58] Steve Smale. The fundamental theorem of algebra and complexity theory.Bull. Amer. Math. Soc. (N.S.), 4(1):1–36, 1981. · Zbl 0456.12012
[59] Steve Smale. Newton’s method estimates from data at one point. InThe merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985), pages 185–196. Springer, New York, 1986.
[60] R. A. Tapia. Classroom Notes: The Kantorovich Theorem for Newton’s Method.Amer. Math. Monthly, 78(4):389–392, 1971. · Zbl 0215.27404
[61] Livinus U. Uko and Ioannis K. Argyros. A generalized Kantorovich theorem on the solvability of nonlinear equations.Aequationes Math., 77(1-2):99–105, 2009. · Zbl 1215.65099
[62] J. H. Wang. Convergence of Newton’s method for sections on Riemannian manifolds.J. Optim. Theory Appl., 148(1):125–145, 2011. · Zbl 1228.90155
[63] Xing Hua Wang. Some results relevant to Smale’s reports. InFrom Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), pages 456–465. Springer, New York, 1993.
[64] Xing Hua Wang and Dan Fu Han. On dominating sequence method in the point estimate and Smale theorem.Sci. China Ser. A, 33(2):135–144, 1990.
[65] Xinghua Wang. Convergence of Newton’s method and inverse function theorem in Banach space.Math. Comp., 68(225):169–186, 1999. · Zbl 0923.65028
[66] Xinghua Wang, Chong Li, and Ming-Jun Lai. A unified convergence theory for Newton-type methods for zeros of nonlinear operators in Banach spaces.BIT, 42(1):206–213, 2002. · Zbl 0998.65057
[67] Zhengyu Wang and Xinyuan Wu. A semi-local convergence theorem for a robust revised Newton’s method.Comput. Math. Appl., 58(7):1320–1327, 2009. · Zbl 1189.65096
[68] Xiubin Xu, Yuan Xiao, and Tao Liu. Semilocal convergence analysis for inexact Newton method under weak condition.Abstr. Appl. Anal., pages Art. ID 982925, 13, 2012. · Zbl 1246.90146
[69] T. Yamamoto. A unified derivation of several error bounds for Newton’s process. InProceedings of the international conference on computational and applied mathematics (Leuven, 1984), volume 12/13, pages 179–191, 1985. · Zbl 0582.65047 · doi:10.1016/0377-0427(85)90015-9
[70] Tetsuro Yamamoto. A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions.Numer. Math., 49(2-3):203–220, 1986. · Zbl 0607.65033
[71] Tetsuro Yamamoto. A convergence theorem for Newton-like methods in Banach spaces.Numer. Math., 51(5):545–557, 1987. · Zbl 0633.65049
[72] Tetsuro Yamamoto and Xiao Jun Chen. Ball-convergence theorems and error estimates for certain iterative methods for nonlinear equations.Japan J. Appl. Math., 7(1):131–143, 1990. · Zbl 0699.65042
[73] P. P. Zabrejko and D. F. Nguen. The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates.Numer. Funct. Anal. Optim., 9(5-6):671–684, 1987. · Zbl 0627.65069
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