Summary

Summary#

This chapter introduces numerical integration (quadrature) as a powerful alternative to analytical integration when closed-form antiderivatives are difficult or impossible to obtain. It builds upon ideas from polynomial interpolation to construct interpolatory quadrature rules, and explores both theoretical and practical aspects.

🔹 Section 3.1 – Introduction and Classical Rules

  • Motivation for numerical quadrature using examples of complicated integrands.

  • Classical rules reviewed: Left endpoint, Right endpoint, Midpoint, Trapezoidal, and Simpson’s rule.

  • All rules are cast in the general quadrature form \(Q[f] = \sum w_i f(x_i)\).

🔹 Section 3.2 – Quadrature from Polynomial Interpolation

  • Derivation of quadrature rules by integrating Lagrange interpolants.

  • Demonstrates how weights are computed as integrals of cardinal basis functions.

  • Examples include revisiting Simpson’s rule and introducing Gauß-Legendre quadrature.

  • A first look at the accuracy of quadrature rules

🔹 Section 3.3 – Degree of Exactness

  • Defines degree of exactness as the highest degree of polynomial a rule integrates exactly.

  • Empirical testing and code examples to compute this degree for various quadrature rules.

  • Notable results: Simpson and Gauss-Legendre have higher precision per point than trapezoidal or midpoint.

🔹 Section 3.4 – Error Estimates

  • Derives general error bounds for quadrature based on interpolation theory.

  • Presents sharper estimates for specific rules:

  • Trapezoidal error: \(\frac{(b-a)^3}{12} f{\prime}{\prime}(\xi)\)

  • Simpson error: \(-\frac{(b-a)^5}{2880} f^{(4)}(\xi)\)

🔹 Section 3.5 – Newton-Cotes Formulas

  • Discusses rules with equispaced nodes.

  • Highlights issues with negative weights for higher-order Newton-Cotes rules, limiting practical use to \(n \leq 6\).

  • Differentiates between closed (includes endpoints) and open (excludes endpoints) Newton-Cotes rules.

🔹 Section 3.6 – Composite Quadrature Rules

  • General construction of quadrature rules and composite quadrature rules starting from a reference interval.

  • Improves accuracy by dividing the domain into subintervals and applying quadrature on each.

  • Derives and implements composite trapezoidal and midpoint rules.

🔹 Section 3.7 – Composite Trapezoidal Rule & Convergence

  • Experiments show error decreases as \(\mathcal{O}(h^2)\).

  • Includes full implementation and analysis of error convergence plots.

  • Defines experimental order of convergence (EOC) and demonstrates how to estimate it numerically.

🎯 Learning Outcomes for Chapter 3

By the end of this chapter, students will be able to:

💡 Conceptual Understanding

  • Explain the motivation for numerical integration and its relevance when antiderivatives are unknown or intractable.

  • Describe classical quadrature rules and identify their strengths and weaknesses.

🔢 Quadrature Design and Implementation

  • Derive numerical quadrature rules from polynomial interpolation using Lagrange polynomials.

  • Compute weights in quadrature rules from cardinal functions.

  • Implement rules such as:

    • Left/right endpoint

    • Midpoint and trapezoidal

    • Simpson’s rule

    • Gauß-Legendre quadrature

  • Implement composite quadrature rules for improved accuracy.

📈 Error Analysis

  • Define and determine the degree of exactness of a quadrature rule.

  • Use interpolation error estimates to bound integration error based on function regularity and number of nodes.

  • Apply error estimates for single interval quadrature rules to derive bounds for composite rules.

🧠 Theory and Practice

  • Discuss Newton-Cotes formulas, including the issues with negative weights and their practical implications.

  • Implement composite quadrature rules for improved accuracy on general intervals.

  • Explain and apply the concept of experimental order of convergence (EOC) using log-log plots.

  • Use composite quadrature rules (e.g., composite trapezoidal) to efficiently and accurately approximate integrals.