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1) in Section 4. Summary and conclusions of the paper are given in the last section. 2. METHODOLOGY AND A USEFUL LEMMA FOR PROVING LOWER BOUNDS Let φ be an algorithm for finding a root α of f (x) = 0 and X the approximation to oi computed by φ. Denote the error of the approximation χ by 6(cp,f) = ||x-a||, where ||· || is a suitable norm. Consider the problem of solv- ing f(x) = 0 where the function (or operator^ f satisf I P S - « Ο Π Ι Ρ property (or some conditions). Since algorithms based on the 37 Η.

Kung and Traub [74b] Kung, H. T. and Traub, J. , "Computational Complexity of One-Point and Multi-Point Iteration," Complexity of Computation, edited by R. Karp, American Mathematical Society, 1974, 149160. , "Optimal Use of Information in Certain Iterative Processes," these proceedings. Also available as a CMU Computer Science Department report. Paterson [72J Paterson, M. , "Efficient Iterations for Algebraic Numbers," in Complexity of Computer Computations, edited by R. Miller and J. W. Thatcher, Plenum Press, New York, 1972.

SUMMARY AND EXTENS IONS TO THE MODEL We have constructed a non-asymptotic theory of iterative computational complexity with strict lower and upper bounds. In order to make the complexity ideas as accessible as possible we have limited ourselves to scalar non-linear problems. The natural setting for this work is in a Banach space of finite or infinite dimension and we shall do our analysis in this setting in a future paper. We have focussed on the sim- = Aei_1. More realistic models include some i of the folloWing features; plified model e 1.

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