Math 151a Notes, Fall 2014

You may download some notes that supplement the text here. They are pdf files.

1. Basic Quadrature Methods (4 December version)
   • Riemann Sums: Uniform Subintervals
   • Left-Hand and Right-Hand Rules: Error Bounds for Monotonic Integrands, Asymptotic Error.
   • Midpoint and Trapezoidal Rules: Error Bounds for Convex and Concave Integrands, Asymptotic Error.
   • Simpson Rule: Relation to Midpoint and Trapezoidal Rules, Asymptotic Error.
   • Error Estimates for Quadrature Methods: Left-Hand and Right-Hand Rules, Midpoint Rule, Trapezoidal Rule, Simpson Rule.

2. Gauss Quadrature Methods (8 December version)
   • Introduction: Posing the Question of Maximum Precision.
   • Quadrature Weights: Formulas for Quadrature Weights in Terms of Quadrature Points.
   • Maximum Possible Precision: Expected Maximum Precision, Upper Bound on Precision.
   • Orthogonality Condition: Characterization of Precision.
   • Orthogonal Polynomials: Construction of Orthogonal Polynomials, Simple Roots.
   • Gaussian Quadrature Sets: Recipe from Orthogonal Polynomials, Positivity of Gaussian Quadrature Weights, Three Examples