Lecture 15: Interpolation (part I) | Highlights and Annotations by Gistr.

This lecture covers polynomial interpolation. Three methods—Vandermonde, Lagrange, and Newton—are presented for finding a polynomial fitting given data points. Lagrange offers fast evaluation but suffers numerical instability; Vandermonde is stable but slow; Newton provides a compromise. The homework involves implementing these methods in MATLAB to solve a graph layout problem using conjugate gradient and gradient descent. The instructor describes a graph layout problem where the goal is to find the positions of vertices in a graph to minimize the total length of edges. The problem involves solving a linear system of equations, and the instructor demonstrates how to prove the existence of a solution. This segment highlights the practical application of linear algebra concepts.The instructor explains the graph layout problem in detail, emphasizing the use of energy minimization and the connection to classical mechanics. The task involves coding in MATLAB to solve the system using conjugate gradients, providing a practical coding exercise.This segment compares conjugate gradients and gradient descent methods for solving the graph layout problem. The instructor explains how these iterative methods update estimates of the solution and visually demonstrates the movement of vertices using a MATLAB simulation. The instructor introduces the concept of interpolation, highlighting its importance in situations where function values are known only at sparse points. The instructor uses examples from image processing and machine learning to illustrate the practical relevance of interpolation. The instructor frames interpolation as a minimization problem, focusing on finding a function that fits given data points. The challenge of working with an uncountably infinite set of functions is addressed by restricting the search to a set of basis functions, transforming the problem into a linear algebra problem. This segment introduces the Newton basis as a compromise between the Vandermonde and Lagrange methods. It explains the structure of the Newton basis functions and shows how they lead to a lower triangular system of equations, which can be solved efficiently in n² time. The segment highlights the numerical stability and computational efficiency advantages of the Newton basis compared to the other two methods. This segment explains the Lagrange basis polynomials, highlighting their convenient property for interpolation. It demonstrates how to use these basis functions to construct an interpolating polynomial by summing weighted y-values, where the weights are the Lagrange basis functions evaluated at the x-values. The explanation includes the Kronecker delta notation for simplifying the expression. This segment compares the Vandermonde system and Lagrange basis for polynomial interpolation. It discusses the advantages and disadvantages of each method, including computational complexity (Vandermonde being slower but numerically stable, Lagrange being faster but prone to numerical instability when x-values are close). The discussion sets the stage for introducing a compromise method.