# Mathematics

## Advanced Numerical Analysis

Module code: 852G1
Level 7 (Masters)
15 credits in autumn semester
Teaching method: Lecture
Assessment modes: Coursework, Unseen examination

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

### Module learning outcomes

• Analyse in depth the convergence properties of advanced iterative methods
• Implement and apply advanced iterative methods to solve linear and nonlinear problems
• Analyse in depth the convergence and stability properties of advanced time-stepping methods
• Conduct thorough error analysis
• Implement and apply time-stepping methods to solve ODE's and time-dependent PDE's including boundary value problems