(406) Notes on Interpolation Formulae ملاحظات حول صيغ الاستيفاء - Hamid, S.; Omar, Fayza H. (معهد التخطيط القومى, 2024)
The work entitled “Notes on Interpolation Formulae” examines interpolation as one of the fundamental mathematical and statistical techniques used to estimate unknown values located between known observations within a data set. The importance of interpolation stems from its role in dealing with incomplete data and generating approximate estimates that support statistical analysis, economic research, scientific studies, and engineering applications. The study aims to provide a theoretical and methodological overview of different interpolation formulae and explain their use in estimating intermediate values based on available observations. It also seeks to clarify the mathematical principles underlying interpolation methods and identify the conditions under which each formula can be appropriately applied according to the characteristics and distribution of data. The analysis relies on concepts from numerical analysis and mathematical statistics, with emphasis on widely used interpolation techniques such as Newton’s interpolation formula, Lagrange interpolation, finite-difference methods, and related mathematical approaches employed in estimating unknown values. The study further investigates the relationship between estimation accuracy, the number of observations used, and the regularity of data distribution. The study indicates that selecting an appropriate interpolation method depends on the nature of the available data and the required level of precision. Increasing the degree of polynomial functions or adding a larger number of data points does not necessarily improve estimation quality. The analysis also suggests that interpolation errors may arise from irregular data patterns or from the use of unsuitable mathematical models. The significance of the study lies in providing a theoretical and practical framework that assists students and researchers in understanding mathematical estimation techniques and improving the use of statistical data in analytical and forecasting processes. Furthermore, the work contributes to strengthening quantitative applications in economics, statistics, and various applied sciences.