Newton Interpolation

This article provides a detailed exploration of Newton Interpolation method, a widely used interpolation technique in mathematics and computer science. Newton Interpolation constructs an interpolation polynomial through known data points to estimate unknown values. The method's accuracy significantly depends on both the number of data points and their spacing, requiring careful selection of these parameters during implementation. From a computational perspective, the algorithm typically involves calculating divided differences using a recursive approach, where each new data point adds a term to the polynomial without recalculating previous coefficients. Beyond basic interpolation, Newton's method can be extended to calculate derivatives and integrals through polynomial differentiation and integration techniques. This article will not only examine practical applications of Newton Interpolation but also discuss strategies for optimizing data point selection and demonstrate how to implement derivative and integral calculations using the method. Key implementation considerations include handling unequal spacing through divided difference tables and ensuring numerical stability when working with high-order polynomials.