Lecture 16: Numerical integration and differentiation (part I)

This lecture covers numerical integration (quadrature) and differentiation. It discusses methods like Newton-Cotes (midpoint, trapezoidal, Simpson's rules) and their composite forms, emphasizing the trade-offs between accuracy and computational cost. A homework assignment involving MATLAB (or Octave/Python) is assigned, focusing on implementing and comparing these methods. This segment builds upon the previous lecture on interpolation, transitioning the discussion to the calculation of derivatives and integrals. It emphasizes the relative ease of extending single-variable methods to multivariable scenarios and briefly motivates the need for these calculations in various applications. This segment introduces the core concept of numerical integration and differentiation, explaining the challenge of approximating derivatives and integrals of functions when only sample points are available. It highlights the fundamental role these approximations play in various applications and sets the stage for the lecture's focus. This segment provides crucial information regarding the upcoming homework assignment, emphasizing the importance of starting early and mastering the initial problem. It also details the software requirements (MATLAB or Octave) and offers advice on efficient problem-solving and code implementation. This segment showcases the practical applications of numerical integration in computer graphics and image processing. It explains how rendering techniques and image blurring (convolution) rely on integral calculations, demonstrating the relevance of the discussed methods in real-world scenarios. This segment explains the core concepts of Newton-Cotes quadrature, differentiating between closed and open methods. It details how the choice of including or excluding endpoints in the integration scheme impacts the selection of sample points and the resulting numerical method, providing a clear visual representation of each approach. This segment introduces composite quadrature rules, which improve accuracy by dividing the integration interval into smaller subintervals and applying simpler rules to each subinterval. It focuses on the composite Simpson's rule, highlighting its unique approach and potential drawbacks compared to other methods, such as the lack of guaranteed tangent continuity between parabolas fitted to adjacent subintervals. This segment compares and contrasts the trapezoidal rule and Simpson's rule, two common Newton-Cotes methods. It illustrates how these methods approximate integrals using different geometric shapes (trapezoids and parabolas) and discusses the trade-offs between simplicity and accuracy. The discussion also touches upon the limitations of higher-degree polynomial approximations in Newton-Cotes methods.