Improving the Convergence of Interval Single-Step Method for Simultaneous Approximation of Polynomial Zeros

Abstract

This paper describes the extended method of solving real polynomial zeros problems using the single-step method, namely, the interval trio midpoint symmetric single-step (ITMSS) method, which updates the midpoint at each forward-backward-forward step. The proposed algorithm will constantly update the value of the midpoint of each interval of the previous roots before entering the preceding steps; hence, it always generate intervals that decrease toward the polynomial zeros. Theoretically, the proposed method possesses a superior rate of convergence at 16, while the existing methods are known to have, at most, 9. To validate its efficiency, we perform numerical experiments on 52 polynomials, and the results are presented, using performance profiles. The numerical results indicate that the proposed method surpasses the other three methods by fine-tuning the midpoint, which reduces the final interval width upon convergence with fewer iterations.