Numerical Analysis: Introduction to Interpolation Techniques

This lecture introduces the fundamental concepts of numerical analysis, focusing on interpolation methods. The instructor begins by discussing the relevance of numerical analysis in engineering, particularly in mechanical engineering applications. Examples include laser peening and fluid mechanics simulations, illustrating how numerical methods can replicate complex physical phenomena. The lecture covers the basics of interpolation, defining the problem of finding a polynomial that passes through given data points. The instructor explains the construction of Lagrange polynomials and their significance in interpolation. The mathematical formulation is presented, emphasizing the relationship between the degree of the polynomial and the number of data points. The lecture also addresses potential issues with high-degree polynomials, such as instability and oscillations, and introduces the concept of piecewise interpolation as a solution. The session concludes with a discussion on the importance of understanding the theoretical underpinnings of numerical methods and their practical applications in various fields, including environmental science and aerospace engineering.