An Approximation to the Least Root of a Cubic Equation with Application to the Determination of Units in Pure Cubic Fields

Authors

T. A. Pierce, University of Nebraska - Lincoln

Date of this Version

1926

Comments

Published in Bull. Amer. Math. Soc. 32 (1926) 263-268.

Abstract

Approximation to the Least Root of a Cubic Bernoulli's method of approximating the largest root of an equation
(1) x³ = ax² + bx + c,
with real coefficients, is to use (1) as a scale of relation for the recursion formula A_n = aA_n-1 + bA_n-2 + cA_n-3. Successive A’s are calculated starting from any initial values. Then A_n+i/A_n for increasing values of n approximates that root of (1) which has the greatest absolute value if that root is real. The method here given for approximating the least root of (1) is similar to Bernoulli's.