Rational approximation formulas for computing the positive zeros of J0(x)

This paper develops four near-minimax rational approximation formulas for computing the positive zeros js, s = 1,2,… of the Bessel function J0(x). Discrete best approximation techniques are used in each case to numerically determine the coefficients for the rational function that minimizes the maximum relative error over the discrete domain s = 1,2,…. These coefficients are then replaced by nearby rational numbers to obtain a convenient near-minimax approximation formula. Each of the resulting four approximations can be used to accurately compute j, for any s ⩾ 1. The maximum relative error for the four rational functions we develop range from 2.2 × 10−4 for the first approximation, down to 1.6 × 10−15 for the fourth approximation.