MAPL/CMSC 667: Review Outline - Bo Li
MAPL/CMSC 667: Numerical Analysis (II)
Spring, 2001
Instructor: Bo Li
Review Outline
- Part 1. Iterative Methods for Nonlinear Systems of Equations and Optimization
(Reference: Chapter 5 of the textbook.)
- concept of order of convergence, linear and superlinear convergence
- fixed point iteration, contractive mapping, related convergence results
- Newton's method: derivation and convergence
- the method of regula falsi, Muller's method
- steepest descent method, line search (Lemma 5.4.1.2 and Theorem 5.4.1.4)
- quasi-Newton methods: ideas, sufficient conditions for convergence
- Part 2. Numerical Solution of Ordinary Differential Equations
(Reference: class notes.)
- linear multi-step methods: basic concept, derivation, examples
- linear multi-step methods: conditions for consistency and
order-m local truncation error
- linear multi-step methods: root condition, convergence and stability theory
- predictor-corrector methods
- single-step methods, basic concept
- derivation of single-step methods by finite Taylor series
and numerical quadrature, Runge-Kutta methods
- stiff systems, region of absolute stability
- finite difference methods for boundary value problems
- variational methods for BVPs: weak formulation, error estimates
- piecewise finite element discretization, interpolation, error estimates