The computational complexity of algebraic numbers

Let {xi} be a sequence approximating an algebraic number α 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 M¯ 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 Ē({xi}) ≤ 1 or equivalently, [equation]. In this paper we show that (i) [equation]; (ii) if E({xi}) = 1 then α is a rational number; (iii) if Ē({xi}) = 1 then α is a rational or quadratic irrational number. This settles the question of when the multiplication efficiency E({xi}) or Ē({xi}) achieves its optimal value of unity.