Module Content

The module will present a range of numerical methods, and applies them with consideration to the convergence, error bounds, and stability of these methods. This is a “tool-kit” numerical methods module with programming skills to enable the student to continue their journey into mathematical computing. 

Preliminaries – Motivation for numerical analysis; Sources and types of errors in numerical analysis; Propagation of errors; Floating-point arithmetic; Well-posed and well-conditioned problems; algorithms. 

Solving Algebraic Equations – Bisection method; Newton’s method; Fixed-point iterations; what are the conditions for convergence and error bounds for these methods. 

Interpolation – Piecewise linear interpolation; Lagrange interpolation; Hermite interpolation; The trouble with interpolating polynomials (instability); Introduction to spline interpolation. 

Differentiation and Integration – Numerical differentiation, with error estimates; Richardson extrapolation for improved accuracy; Degree of precision in quadrature; The trapezoidal method; Simpson’s method; Error bounds for these methods; Composite methods. 

Introduction to Linear Algebra – Algorithmic Gaussian elimination; pivoting strategies. LU and Crout factorization. 

Approximation theory – Discrete least-square approximation; Orthogonal polynomials; Chebyshev polynomials; Trigonometric polynomial approximation. 

ODEs: Initial Value Problems (IVPs) – The Lipschitz condition; Existence and uniqueness of solutions; Euler’s method; General Taylor-series methods; Runge-Kutta type methods; An introduction to multistep methods; Stability considerations; Treatment of higher-order equations/systems of equations; Stiff equations. 

ODEs: Boundary Value Problems – Existence and uniqueness of solutions; Methods based on IVPs, e.g shooting methods; Finite difference methods. 

Teaching Methods

The module will be delivered on campus, with weekly lecture and computer lab tutorials. 

Printed notes will be given ahead of time for each section of the course, to support and enhance students’ preparation and engagement during class sessions. Lectures will follow the notes, with discussions of the main theoretical topics, and study of examples of the applications of the theory. There will be a strong emphasis on student involvement in discussions in lectures, to encourage a more active approach to learning the material, and to allow the delivery to be tailored to build on the students’ current understanding. Computer lab tutorials will allow students to explore numerical algorithms which are too cumbersome to compute by hand, as well as to explore real- world implementation issues. 

Regular formative work will allow development of problem-solving skills, exposure to implementation of real-world numerical algorithms, and practice at mathematical computer programming. This formative work will also feed back into the delivery of lectures and tutorials.

Assessment Methods

The module is assessed through a portfolio of exercises and a presentation.