This segment reveals a critical issue in rational function interpolation: the potential for division by zero. The speaker explains how seemingly innocent mathematical manipulations can lead to unexpected and problematic results, such as undefined function values at certain points. The example illustrates the importance of carefully considering the implications of division by zero and the potential for unexpected behavior. This segment demonstrates the process of solving a linear system of equations to find a rational function that interpolates three given data points. The speaker highlights the challenges and potential pitfalls of this approach, such as scaling issues and the possibility of non-trivial solutions. The example showcases the complexities involved in finding a suitable rational function and the need for additional constraints. Polynomial interpolation is widely used, but rational interpolation, while offering more flexibility, is prone to division-by-zero errors. Piecewise polynomial interpolation avoids global effects of data point changes, using functions with compact support (zero outside a specific range). Multivariable interpolation methods like nearest neighbor and barycentric interpolation address higher-dimensional data, with barycentric interpolation offering smoother results but requiring more computation. Bilinear interpolation is a common technique for gridded data. This segment discusses the global impact of local changes in polynomial interpolation. The speaker explains how modifying a single data point can significantly alter the entire interpolated function, highlighting the lack of local control in polynomial interpolation. This segment emphasizes the limitations of polynomial interpolation and motivates the need for alternative methods with better local control. This segment introduces the concept of compact support in basis functions used for interpolation. The speaker explains that functions with compact support have limited influence, affecting only a small region around a given data point. This contrasts with functions without compact support, such as polynomials, whose influence extends globally. The discussion highlights the advantages of compact support in improving local control and reducing computational complexity. This segment explores the use of piecewise polynomials in interpolation and the potential for the Gibbs phenomenon. The speaker discusses the trade-offs between using high-degree polynomials and the risk of introducing unwanted oscillations and discontinuities. The discussion highlights the importance of choosing the appropriate degree of polynomial based on the specific application and the nature of the data. The Gibbs phenomenon is explained as a consequence of overfitting the data. This segment explains nearest neighbor interpolation, a straightforward method for estimating function values at arbitrary points based on the closest known data point. The explanation includes a visual description of Voronoi cells, highlighting their role in assigning values within the interpolated function. The simplicity and ease of implementation of this method are emphasized. This segment details barycentric interpolation, a method that expresses a point as a weighted average of known data points. The explanation focuses on the geometric interpretation of weights as proportional to the areas of sub-triangles formed by connecting the point to the vertices of a larger triangle. The segment clarifies how this method provides a smooth interpolation within the triangle's boundaries.
