Applied Numerical Analysis (L.7) (852G1)
15 credits, Level 7 (Masters)
Autumn teaching
This module will cover topics including:
- Iterative methods for linear systems: Jacobi and Gauss-Seidel, conjugate gradient, GMRES and Krylov methods
- Iterative methods for nonlinear systems: fixed point iteration, Newton's method and Inexact Newton
- Optimisation: simplex methods, descent methods, convex optimisation and non-convenx optimisation
- Eigenvalue problems: power method, Von Mises method, Jacobi iteration and special matrices
- Numerical methods for ordinary differential equations: existence of solutions for ODE's, Euler's method, Lindelöf-Picard method, continuous dependence and stability of ODE's
- Basic methods: forward and backward Euler, stability, convergence, midpoint and trapezoidal methods (order of convergence, truncation error, stability convergence, absolute stability and A-stability)
- Runge-Kutta methods: one step methods, predictor-corrector methods, explicit RK2 and RK4 as basic examples, and general theory of RK methods such as truncation, consitency, stability and convergence
- Linear multistep methods: multistep methods, truncation, consistency, stability, convergence, difference equaitons, Dahlquist's barriers, Adams family and backward difference formulas
- Boundary value problems in 1d, shooting methods, finite difference methods, convergence analysis, Galerkin methods and convergence analysis