(424) Interpolation Formulas صيغ الاستيفاء - Abdel Karim, Abbas I. (معهد التخطيط القومى, 2024)
This study examines the theoretical and mathematical foundations of interpolation formulas as essential quantitative tools used in mathematical, statistical, and economic analysis for estimating unknown values located between known observations within a dataset. The main objective of the study is to present the principal interpolation formulas and evaluate their theoretical and practical characteristics while demonstrating their role in solving estimation and approximation problems. The paper begins by introducing interpolation as a mathematical process intended to construct a functional relationship capable of estimating values of a variable based on previously observed data points. The study emphasizes the importance of interpolation techniques because of their wide range of applications in economics, statistics, engineering, and applied sciences, particularly in situations where data are incomplete or where intermediate values cannot be directly observed. The study further reviews several general interpolation approaches, including interpolation polynomials, the Lagrange interpolation method, and Newton's general interpolation formula. Particular attention is given to the concept of interpolation error and the factors affecting the accuracy and reliability of estimated values. The research explains that the choice of an appropriate interpolation technique depends on the characteristics of the available data, the number of observations, and the regularity of intervals among observations. The paper also discusses special interpolation formulas such as Newton–Gregory formulas, Gaussian interpolation methods, Everett–Laplace formulas, linear interpolation techniques, inverse interpolation procedures, and Aitken's repeated process. These methods are compared in terms of computational efficiency, accuracy, and suitability for various numerical applications. The academic significance of the study lies in its development of a systematic mathematical framework for understanding interpolation techniques and their practical applications. The study therefore contributes to improving quantitative estimation accuracy and supports analytical and decision-making processes across scientific and applied disciplines.