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
- 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.
-
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 Initial-Value Problems (18 November version)
- Initial-Value Problems for First-Order Systems:
Normal Form, Notion of Solution, Existence and Uniqueness
of Solutions.
- Recasting Higher-Order Problems as First-Order Systems:
Quadrature Points.
- Numerical Approximation:
Definition of One-Step Methods, Step Size.
- Explicit and Implicit Euler Methods:
Forward and Backward Difference Derivations.
- Explicit One-Step Methods Based on Taylor Approximation:
Explicit Euler, Second-Order Taylor, and Third-Order Taylor
Methods.
- Explicit One-Step Methods Based on Quadrature:
Explicit Euler, Runge-Trapezoidal, Runge-Midpoint,
and Classical Runge-Kutta Methods.
- General Explicit Runge-Kutta Methods:
Multistage Methods, Heun Second-Order Method,
Heun Third-Order Method, Kutta Third-Order Method,
Embedded Methods, ode45.