By J.F. Traub

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W. Thatcher, Plenum Press, New York, 1972. 41-52. 33 J. F. TRAUB AND H. WOtNIAKOWSKI Traub [64] Traub t J. F. t Iterative Methods for the Solution of Eguations t Prentice-Hall, 1964. Traub [72] Traub, J. , 1972 t 167-179. WoZ'niakowski [75a] WoZ'niakowski, H. t "Generalized Information and Maximal Order of Iteration for Operator Equations," SIAM J. Numer. AnaL llt 1975, 121-135. WoZ'niakowski [75b] Wozniakowski, H. t "Maximal Order of Multipoint Iterations Using n Evaluations," these proceedings.

2. 2 as input and produces a good starting point for ap proximating a root 1. of f (x) = 0. Set h^ ^ P^KTIq, Xq Xq and i ^ 0. Pick any number δ in (0,-). 2. 2 terminates. 3. 20) η^^^ - (ΐ-λ^)η^. 4. 22) ||[f;(x^^l)]"^fi(Xi^l)|| ^min(f-ßη^^^, ^ ^ r - 2βη^, and ) 5. _^^ - β^κη^+ι6. Set i i+1 and return to step 2. 1. -xJI ^ |P^^-xJ|. + l^^-xj 50 COMPLEXITY OF S T A R T I N G POINTS ^ (r - 2βη^) + 2βη^ = r, Vi. It is clear that ^ Γ " 2' ^0 point for approximating ot. Now suppose ^ starting ^ ? ^, a zero of f^.

The modified algorithm. 2, appears in the proof of the following theorem. In the theorem and the rest of the paper, a Newton step means the computation of χ - [f' (χ) ]"''f (χ), given x, f and f . Hence a Newton step involves one evaluation of f, one of f' at χ and the computation of χ - [ f ' ( χ ) f ( x ) from x, f(x), f'(x). 2. 1 are satisfied and ||f'(x)-f'(y)|| ^K|x-y||. x,y € 5^^(χ^). Then a good starting point for approximating the root of f(x) = 0 by Newton's method can be obtained in N(5) Newton steps, where δ is any number in (0,~) and N(6) is defined as follows.