Perturbation Stability in Evaluating Modified Stieltjes Continued Fractions for Modern Applications

Abstract

Due to the demands of modern applications, such as forecasting using ordinal continued fraction regression models in economics, the study of structural stability to data perturbations has become an urgent need. While continued fractions demonstrate superior effectiveness over traditional polynomial models due to their inherent structural stability, their sensitivity to coefficient perturbations can lead to accuracy loss under finite-precision machine arithmetic. This paper investigates the stability to perturbations of the coefficients of modified Stieltjes continued fractions (modified S-fractions). We develop a rigorous theoretical framework based on a fundamental recursive formula for the relative error of the nth approximant, expressing error propagation directly in terms of coefficient perturbations. Using this geometric and analytical approach, we establish three distinct uniform stability sets in the complex plane: the half-plane ΩC, the angular set Ωα,C under weaker geometric conditions, and the exterior of the disk ΩR, which extends verified stability directly to points on and near the imaginary axis. Furthermore, we derive explicit, a priori error bounds for each stability set and prove pointwise stability on a significantly expanded domain, demonstrating that the modified S-fraction remains stable on R∖{0} for any bounded coefficients. These findings establish explicit boundaries for safe computation using standard double-precision arithmetic, providing a mathematical guarantee of algorithmic stability for high-demand contemporary applied models.