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By J.F. Traub

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Then Δ. 3, we know that even if f is bounded above or bounded below, we still cannot do better than binary search in the worst case sense. 4. If f; [a,b] R satisfies the following proper­ ties: Μ ^ f'(x) ^ m > 0 for all χ € [a^bl. and f(a) < 0, f(b) > 0 m,2/,,i+l. then Δ. ^ ( b - a ) [ ( l . 4, Micchelli and Miranker [75] showed that i Δ, ^^(b-a)(l3)2 . Hence m 3 their algorithm is better than binary search when — ^ ^. 4, we know that the problem cannot be solved superlinearly even when f' is known to be , 42 COMPLEXITY OF STARTING POINTS bounded above and below by some constants.

Hence a root can be located within a ball of radius 10"^r by 2 6 0 Newton steps. 5. SUMMARY AND CONCLUSIONS The search and iteration phases should be studied togeth­ er. A methodology for studying the worst case complexity of the two phases is proposed. Results based on the methodology are global and non-asymptotic (see Theorems 4 . 2 and 4 . 3 ) , while the usual results in analytic computational complexity are local and asymptotic. Optimal time for switching from the search phase to the iteration phase can also be determined from the methodology.

Then the time t needed to approximate a root to within β > 0 is the smallest i such that φ 6 ^ ε, and an algorithm is said to be optimal for approximating a root to within e > 0 if sup δ(φ,f) = Δ . f€F ^ Hence the complexity of the problem is determined by the sequence ίΔ^}. We say the problem is solvable if [Δ^,} converg­ es to zero, otherwise it is unsolvable. We are interested in solvable problems. }, which is defined to be ρ = lim(|log Δ ^ ) ^ / \ provided the limit exists. , ρ > 1, we say the problem can be solved superlinearly.

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