Lecture 16: Numerical integration and differentiation (part III)

This segment details Richardson extrapolation, a clever method to enhance the accuracy of derivative estimations. It demonstrates how evaluating a function at two different points and solving a linear system yields a higher-order approximation, transforming a first-order approximation into a second-order one. The presenter emphasizes the method's ingenuity and broad applicability within a larger class of sequence acceleration methods. This segment explains a straightforward method for deriving various numerical differentiation schemes using Taylor series expansion, focusing on canceling terms to achieve the desired order of approximation. The presenter highlights the ease and efficiency of this approach, making it valuable for understanding numerical methods. Numerical differentiation approximates derivatives using Taylor series. Adding forward and backward difference approximations yields a centered difference formula for the second derivative. Richardson extrapolation improves accuracy by combining derivative estimates at different step sizes (h), achieving higher-order approximations. However, choosing an appropriate step size (h) is crucial to avoid numerical instability from excessively small values.