By J.F. Traub
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Extra info for Analytic Computational Complexity
Example, that h^ = 9 or h^ ^ 9 . 27), Suppose, for Then by Algorithm 4 . 2 with we know that the search phase can be done in 2 5 5 Newton steps and the iteration phase in 5 Newton steps. 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 .
Z(V3) 3 = ~(f) + Z- and 11 can gain only another n 1 Thus comparing this with z(V *) shows we n ~(f). Let ~ denote the class of all multipoint iterations for which wu ~ 2. Then comp(~) S; ~ J C(f)lg(1+t)/( + 170 • \. (c(f» • We can obtain a lower bound on the complexity of the class of multipoint iterations by using an upper bound on the maximal order of any multipoint iteration and a lower bound on the combinatorial complexity. Kung and Traub [74a] con- jecture that any iteration without memory which uses n pieces 31 J.
Will not converge l. We plan to analyze these more realistic models in the future. We also intend to investigate additional basic properties of complexity. Our various results will be used to analyze the complexity of important problems in science and engineering. 32 STRICT LOWER AND UPPER BOUNDS ACKNOWLEDGMENT We thank H. T. Kung for his comments on this paper. REFERENCES Borodin and Munro  Borodin, A. , The Computational Complexity of Algebraic and Numeric Problems, American Elsevier, 1975.
Analytic Computational Complexity by J.F. Traub