# Theorem: Convergence of Newton’s Method

May 19th, 2015
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F ϵ C2 [a,b], Ǝ r ϵ [a,b], for which f(r)=0, and f’(r)≠0; then Ǝ δ>0, such that ɏ r0 ϵ [r-δ,r+δ] The Newton’s sequence will converge to r..Secant Method From r = r_0 – (f(r_0))/(f’(r_0))

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Newtons IterationAlgorithm:While |r1-r0| > er0 r1r1 r0 f(r0)/f(r0)EndNeeds:f is C2f has to be availabler0 starting point: difficult to findTheorem: Convergence of Newtons MethodF C2 [a,b], r [a,b], for which f(r)=0, and f(r)0; then >0, such that r0 [r-,r+]The Newtons sequence will converge to rSecant MethodFrom Algorithm:Input: f C1 [a,b], tolerance Initialization: r0, r1 in [a,b]While |r1-r0| > r2 r1 f(r1) r1-r0 / f(r1) f(r0)r0 r1r1 r2Order of Convergence (of iterative methods)Assume

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