Lagrange Interpolation Algorithm Implementation with Example Validation

This article explores the principles, implementation, and applications of the Lagrange interpolation algorithm. The Lagrange interpolation method is a fundamental numerical analysis technique that constructs a polynomial function passing through a given set of data points. This polynomial can estimate values at new data points or approximate unknown functions. To illustrate the algorithm's implementation, we demonstrate how to compute the Lagrange basis polynomials using nested loops, where each basis polynomial L_i(x) is constructed by excluding the i-th data point and performing product operations on the differences (x - x_j)/(x_i - x_j) for all j ≠ i. The final interpolation polynomial is obtained by summing the products of each basis polynomial with its corresponding y_i value. We validate the algorithm's correctness through a practical example where we apply Lagrange interpolation to fit a curve predicting temperature variations in a city over upcoming days. The implementation involves defining data points as coordinate pairs, calculating intermediate terms using multiplicative accumulation, and combining results through weighted summation. Through this article, readers will gain comprehensive understanding of the algorithm's mathematical foundation, practical applications, and implementation methodology, enabling them to effectively apply this technique to solve real-world problems involving data approximation and prediction.