Much of today’s science and engineering depends on large-scale calculations performed with computers. These calculations find solutions or approximate solutions to mathematical models and enable scientists and engineers to predict behaviours of interest. A prime example is the weather: PDE models are used to predict the weather based on recent observations. The accuracy of the predictions depend on many things, including the accuracy of the numerical solution to the PDEs. In this unit, we explore the theoretical basis for these numerical methods, especially their reliability and efficiency. The name given to this subject is Numerical Analysis. Numerical analysis is half theory and half practice. We want to prove that algorithms work with rigorous mathematical analysis as well as implement them. An essential part of the course will be for you to implement and use the methods yourself. Topics covered in this unit include: • Interpolation: Linear, polynomial, Newton’s divided-difference formulae. • Numerical integration: Newton-Cotes rules, Gaussian quadrature. • Solving nonlinear equations: Fixed point iterations, Newton's method. • Solving differential equations: Euler's method, convergence analysis. • Solving linear system of equations: Iterative methods, condition number.
Lecture notes (pdf, html) Problem sheets can be found below. Solutions will be posted on Moodle. Problem Sheet 1 (pdf, html) Problem Sheet 2 (pdf, html) Problem Sheet 3 (pdf, html) Problem Sheet 4 (pdf, html) Problem Sheet 5 (pdf, html) Problem Sheet 6 (pdf, html) Problem Sheet 7 (pdf, html) Problem Sheet 8 (pdf, html) Problem Sheet 9 (pdf, html) Problem Sheet 10 (pdf, html)