AMSC 460 Notes, Fall 2016
As they become available, Class Notes can be downloaded here.
They are pdf files.

Floating Point Representation
and the IEEE Standard

Basic Quadrature Methods (30 October version)
 Riemann Sums:
Uniform Subintervals
 LeftHand and RightHand 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:
LeftHand and RightHand Rules, Midpoint Rule,
Trapezoidal Rule, Simpson Rule.

Gauss Quadrature Methods (18 November 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

Numerical Methods to Solve InitialValue Problems (18 November version)
 InitialValue Problems for FirstOrder Systems:
Normal Form, Notion of Solution, Existence and Uniqueness
of Solutions.
 Recasting HigherOrder Problems as FirstOrder Systems:
Quadrature Points.
 Numerical Approximation:
Definition of OneStep Methods, Step Size.
 Explicit and Implicit Euler Methods:
Forward and Backward Difference Derivations.
 Explicit OneStep Methods Based on Taylor Approximation:
Explicit Euler, SecondOrder Taylor, and ThirdOrder Taylor
Methods.
 Explicit OneStep Methods Based on Quadrature:
Explicit Euler, RungeTrapezoidal, RungeMidpoint,
and Classical RungeKutta Methods.
 General Explicit RungeKutta Methods:
Multistage Methods, Heun SecondOrder Method,
Heun ThirdOrder Method, Kutta ThirdOrder Method,
Embedded Methods, ode45.