5.7 Numerical Integration (Quadrature)

The speaker introduces the topic of numerical integration, emphasizing its practical applications in applied mathematics. They highlight the difference between numerical and analytical integration, noting that the former is more straightforward. The lecture aims to empower students to integrate a wide variety of functions numerically and to deepen their understanding of the concept of integration. The speaker also introduces the term 'quadrature', relating it to the historical method of counting squares under a curve to approximate an integral's value.

This paragraph delves into the specifics of numerical integration methods, starting with the simple act of 'counting boxes' to approximate integrals. It explains the importance of accurately treating the top of these boxes to improve integration rules. The speaker references the Riemann definition of an integral, which is the foundation for numerical integration algorithms. They describe how these algorithms work by summing the product of function values at certain points and weights, which are typically fractions of the box height.

The speaker presents the trapezoid rule, a numerical integration method that approximates the function with straight lines. They discuss how to calculate the area under a trapezoid by averaging the heights at the ends and multiplying by the width. The paragraph explains the process of summing these areas for multiple trapezoids to approximate the integral over an interval, emphasizing the simplicity and uniform step size of this method.

Simpson's rule is introduced as an improvement over the trapezoid rule, using parabolic fits instead of straight lines. The speaker explains the requirement for an even number of intervals and an odd number of points for this method. They detail the process of fitting parabolas to the function across two intervals and calculating the area under each parabola, then summing these areas with specific weights to approximate the integral.

Gaussian quadrature is introduced as an advanced and highly effective integration rule. The speaker explains that it involves fitting a high-degree polynomial to the function using the points and weights determined by Gauss. These points are the zeros of the Legendre polynomials, and the weights are related to the derivative of the Legendre polynomial at these zeros. The method is praised for its accuracy and is commonly used by working scientists.

The speaker discusses the concept of integration errors, defining relative error and its significance in computational science. They explain how to analyze errors by taking the logarithm of the error to determine the number of significant figures. The paragraph includes a discussion of approximation error and round-off error, showing how these errors can be visualized and analyzed using a log-log plot. The speaker emphasizes the importance of understanding when to stop increasing the number of integration points to balance between computational efficiency and accuracy.

This paragraph focuses on determining the optimal number of points for numerical integration, balancing between the decreasing algorithmic error and the increasing round-off error. The speaker provides a method to set the two types of errors equal to each other and solve for the best value of n, the number of points. They illustrate this with examples from the trapezoid and Simpson rules, showing how the Simpson rule achieves higher precision with fewer points due to less opportunity for round-off error to accumulate.

The speaker encourages students to conduct their empirical error analysis using the integration rules learned, such as the trapezoid, Simpson, and Gaussian quadrature. They suggest integrating functions like e^(-x) and x^2 * e^(-x) from 0 to 1 to understand the error behavior. The paragraph guides students to plot the error against the number of points on a log-log scale, deduce the slope, and compare it with theoretical predictions. The speaker also challenges students to try more complex cases, like integrating a sine wave or a function with a rapidly oscillating component.

In conclusion, the speaker highlights the importance of practical application of numerical integration techniques and the challenges associated with integrating functions that result in zero or near-zero values. They caution about the difficulty of accurately calculating such integrals and the limitations of even the most powerful computers. The speaker ends with an invitation for students to explore these challenges further, encouraging them to enjoy the process of computation and numerical integration.