×

zbMATH — the first resource for mathematics

On the Newton-Kantorovich hypothesis for solving equations. (English) Zbl 1055.65066
Author’s summary: The famous Newton-Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton’s method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Here using Lipschitz and center-Lipschitz conditions we show that the Newton-Kantorovich hypothesis can be weakened. The error bounds obtained under our semilocal convergence result are more precise than the corresponding ones given by the dominating Newton-Kantorovich theorem.

MSC:
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
47J25 Iterative procedures involving nonlinear operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Appel, J; DePascale, E; Zabrejko, P.P, On the application of the newton – kantorovich method to nonlinear integral equations of uryson type, Numer. funct. anal. optim., 12, 3 and 4, 271-283, (1991)
[2] Argyros, I.K, Relations between forcing sequences and inexact Newton iterates in Banach space, Computing, 63, 131-144, (1999) · Zbl 0937.65062
[3] Argyros, I.K, Newton methods on Banach spaces with a convergence structure and applications, Comput. math. appl., 40, 1, 37-48, (2000) · Zbl 0957.65047
[4] Argyros, I.K, Advances in the efficiency of computational methods and applications, (2000), World Scientific River Edge, NJ · Zbl 0976.65054
[5] Argyros, I.K, A newton – kantorovich theorem for equations involving m-Fréchet-differentiable operators and applications in radiative transfer, J. comput. appl. math., 131, 1-2, 149-159, (2001) · Zbl 0983.65069
[6] Argyros, I.K; Szidarovszky, F, The theory and applications of iteration methods, (1993), CRC Press Boca Raton, FL · Zbl 0802.65076
[7] A.L. Cauchy, Sur la détermination approximative des racines d’une équation algébrique ou transcendante, in: Lecons sur le Calcul Différentiel, Buré freres, Paris (1829), reprinted in Oeuvres complétes (IV), 2nd series, Gauthier-Villars, Paris, 1899, pp. 573-609.
[8] Dennis, J.E, Toward a unified convergence theory for Newton-like methods, (), 425-472
[9] J.B.J. Fourier, Question d’analyse algebrique, in: Oeuvres complétes (II), Gauthier-Villars, Paris, 1890, pp. 243-253.
[10] Gragg, W.B; Tapia, R.A, Optimal error bounds for the newton – kantorovich theorem, SIAM J. numer. anal., 11, 1, 10-13, (1974) · Zbl 0284.65042
[11] Kantorovich, L.V; Akilov, G.P, Functional analysis, (1982), Pergamon Press Oxford
[12] Miel, G.J, Majorizing sequences and error bounds for iterative methods, Math. comput., 34, 149, 185-202, (1980) · Zbl 0425.65033
[13] Moret, I, A note on Newton-type iterative methods, Computing, 33, 65-73, (1984) · Zbl 0532.65045
[14] Ostrowski, A.M, Solution of equations in Euclidean and Banach spaces, (1973), Academic Press New York · Zbl 0304.65002
[15] Potra, F.A, On Q-order and R-order of convergence, SIAM J. optim. theory appl., 63, 3, 415-431, (1989) · Zbl 0663.65049
[16] Potra, F.A; Ptǎk, V, Sharp error bounds for Newton’s process, Numer. math., 34, 67-72, (1980) · Zbl 0434.65034
[17] Yamamoto, T, A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions, Numer. math., 49, 203-220, (1986) · Zbl 0607.65033
[18] Yau, L; Ben-Israel, A, The Newton and Halley methods for complex roots, Amer. math. monthly, 105, 806-818, (1998) · Zbl 1002.65059
[19] Ypma, T.J, Local convergence of inexact Newton methods, SIAM J. numer. anal., 21, 3, 583-590, (1984) · Zbl 0566.65037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.