Convergence on a deformed Newton method1

摘要

For the iteration which was independently proposed by King, [Numer. Math. 18 (1972) 298–304] and Werner [Numer. Math. 32 (1979) 333–342) and has the convergent order 1+2 to solve the operator equation f(x)=0 in Banach space, we prove that it converges when the criterion proposed by Smale, [in: The Merging of Disciplines; New Directions in Pure, Applied and Computational Mathematics, Springer, New York, 1986, pp. 185–196] α=βγ⩽3−22. But in the paper we only assume that ‖f′(x0)−1f(x0)‖⩽β,‖f′(x0)−1f″(x0)‖⩽2γ and ‖f′(x0)−1f‴(x)‖⩽6γ2(1−γ‖x−x0‖)−3, where γ‖x−x0‖⩽1−1/2.