Document Type : Original Article

Authors

• Sadia Kumail Awad ¹
• Ali Tavakoli ²

¹ Polytechnic College/Al-Qadisiyah, Al-Furat Al-Awsat Technical University, Iraq

² University of Mazandaran

Abstract

Polynomial interpolation is a fundamental tool in numerical analysis, with its accuracy classically characterized by the Lagrange remainder term. This term involves an unknown mean value point , which depends on  and the function . Consequently, while the formula provides an exact error representation, its practical utility for a priori error estimation is limited, as determining a sharp, computable bound for high-order derivatives is often challenging. This paper introduces a novel probabilistic framework to address this longstanding limitation. Instead of treating  as an unknown deterministic value, we model its location probabilistically. By considering the interpolation nodes as random variables or analyzing the distribution of  for a fixed , we derive a statistical estimate for its expected value. This approach allows us to propose a specific, computable value for  that provides a highly accurate estimate of the actual interpolation error. The theoretical findings are substantiated with several numerical examples. These experiments demonstrate that our probabilistic estimate of the remainder term consistently aligns with the true error, offering a practical and powerful alternative to traditional worst-case error bounds. This method provides a new perspective on error analysis in approximation theory, bridging deterministic numerical methods with probabilistic techniques.

Keywords

• Polynomial interpolation
• Chebyshev nodes
• Trapezoidal rule