A bound on the multiplication efficiency of iteration
STOC '72 Proceedings of the fourth annual ACM symposium on Theory of computing
A canonical form for piecewise defined functions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
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Let {xi} be a sequence approximating an algebraic number &agr; of degree r, and let [equation], for some rational function @@@@ with integral coefficients. Let M denote the number of multiplications or divisions needed to compute @@@@ and let &Mmarc; denote the number of multiplications or divisions, except by constants, needed to compute @@@@. Define the multiplication efficiency measure of {xi} as [equation] or as [equation], where p is the order of convergence of {xi}. Kung [1] showed that &Emarc;({xi}) ≤ 1 or equivalently, [equation]. In this paper we show that (i) [equation]; (ii) if E({xi}) &equil; 1 then &agr; is a rational number; (iii) if &Emarc;({xi}) &equil; 1 then &agr; is a rational or quadratic irrational number. This settles the question of when the multiplication efficiency E({xi}) or &Emarc;({xi}) achieves its optimal value of unity.