ON SHANKS' ALGORITHM FOR COMPUTING THE CONTINUED FRACTION OF log b a
Terrence R. Jackson, Keith R. Matthews · 2002
Abstract. We give a more practical variant of Shanks ’ 1954 algorithm for computing the continued fraction of logb a, for integers a> b> 1, using the floor and ceiling functions and an integer parameter c> 1. The variant, when repeated for a few values of c = 10r, enables one to guess if logb a is rational and to find approximately r partial quotients. 1. Shanks ’ algorithm In his article [1], Shanks gave an algorithm for computing the partial quotients of logb a, where a> b are positive integers greater than 1. Construct two sequences a0 = a, a1 = b, a2,... and n0, n1, n2,..., where the ai are positive rationals and the ni are positive integers, by the following rule: If i ≥ 1 and ai−1> ai> 1, then a ni−1 i ≤ ai−1 ai+1 ≥ 1. Also (1.1) implies ai ≤ a1/ni−1i−1 for i ≥ 1 and hence by induction on i ≥ 0, ai+1 ≤ a1/n0···ni0.(1.3) Also by induction on j ≥ 0, a2j = ar0/a