We present an extension of a Theorem by Wang (see Math. Comp. 68, 169–186 1999) for Smale’s α-theory [16, 17] in order to approximate a locally unique solution of a nonlinear equation using Newton’s method. The advantages of our approach over the studies such as Argyros (Atti. Semin. Mat. Fis. Univ. Modena Reggio Emilia 56, 31–40 2008/09), Cianciaruso (Numer. Funct. Anal. Optim. 28, 631–645 2007), Rheinboldt (Appl. Math. Lett. 1, 3–42 1988), Shen and Li (J. Math. Anal. Appl. 369, 29–42 2010), [16, 17], Wang and Zhao (J. Comput. Appl. Math. 60, 253–269 1995), Wang (Math. Comp. 68, 169–186 1999) under the same computational cost are: weaker sufficient convergence condition; more precise error estimates on the distances involved and an at least as precise information on the location of the solution. These advantages are obtained by introducing the notion of the center γ0-condition and using this condition in combination with the γ-condition in the convergence analysis of Newton’s method. Numerical examples and applications are also provided to show that the older convergence criteria are not satisfied but the new convergence criteria are satisfied.
Newton’s method Banach space Semi-local convergence Smale’s α-theory Fréchet-derivative
AMS Subject Classification Codes
65J15 65H05 90C30 47H17 49M15
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