Approximation of the vth Root of N
We present a class of functions gK (w), K ≥ 2, for which the recursive sequences w n + 1 = gK (wn ) converge to N 1/v with relative error . Newton's method results when K = 2. The coefficients of the gK (w) form a triangle, which is Pascal's for v = 2. In this case, if w 1 = x 1/y 1, where x 1, y 1 is the first positive solution of Pell's equation x 2 − Ny 2 = 1, then w n + 1 = x n + 1/y n + 1 is the Knpth or 2Knpth convergent of the continued fraction for , its period p being even or odd.
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Document Type: Research Article
Affiliations: 1: University of California 2: Los Alamos Scientific Laboratory
Publication date: June 1, 1971