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《Journal of Jiangxi Normal University (Natural Sciences Edition)》 2005-04
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A New Proveing of Newton- Decline Method's Convergence

LI Yang(School of Mathematics and Physics, Liaoning University of Petroleum & Chemical Technology,Fushun 113001,China)  
Newton- Decline method x_(n+1)=x_n-ω_nf ′~(-1)(x_n)f(x_n) is a traditional iterative method for solving nonlinear equation f(x)=0.It is necessary to research it's convergent conditions to keep it's big range of convergence. To make it more meaningful in general, by using dominating sequence method and choosing a common decline factor sequence {ω_n} under a more common condition.This paper proves the convergence of Newton- Decline method.The condition can be expressed as ‖f ′~(-1)(x_0)f(x_0)‖≤β,‖f ′~(-1)(x_0)f ″(x_0)‖≤γ,‖f ′~(-1)(x_0)(f ″(x)-f ″(y))‖≤∫~(‖x-y‖)_0L(u+‖x-x_0‖)du.While the condition has more common quality than traditional Kantorovich-kind conditions, mainly lying on the flexibility of the no reducible and positive function L(u),and it can adapt to much more environments.
【Fund】: 辽宁省自然科学基金资助项目(001084).
【CateGory Index】: O241
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