Numerical Analysis 1
|Unit level:||Level 2|
|Teaching period(s):||Semester 2|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
The course unit unit aims to introduce students to theoretical and practical aspects of the numerical solution of linear and nonlinear equations, the approximation of functions by polynomials and the approximation of integrals via quadrature schemes.
Numerical analysis is concerned with finding numerical solutions to problems for which analytical solutions either do not exist or are not readily or cheaply obtainable. This course provides an introduction to the subject, focusing on the three core topics of iteration, interpolation and quadrature.
The module starts with 'interpolation schemes', methods for approximating functions by polynomials, and 'quadrature schemes', numerical methods for approximating integrals, will then be explored in turn. The second half of the module looks at solving systems of linear and nonlinear equations via iterative techniques. In the case of linear systems, examples will be drawn from the numerical solution of differential equations.
Students will learn about practical and theoretical aspects of all the algorithms. Insight into the algorithms will be given through MATLAB illustrations, but the course does not require any programming.
On completion of this unit successful students will have:
- demonstrate, and mitigate for the effect of, non-exact arithmetic on the approximation of simple mathematical calculations,
- construct, apply theorems regarding, and analyse the error of, interpolating polynomials which pass through a given set of coordinates,
- construct, apply theorems regarding, and analyse the error of, quadrature schemes for the numerical approximation of integrals,
- construct interpolation and quadrature schemes which are accurate up to a given degree of precision, and justify using theoretical results,
- implement iterative methods for the approximation of solutions to linear and non-linear equations,
- apply theorems regarding the convergence of such iterative schemes to given examples,
- apply the results of, and methods used in, proofs from theorems in the course in familiar and unseen settings.
- Other - 20%
- Written exam - 80%
Assessment Further Information
- Coursework; Weighting within unit 20%
- 2 hours end of semester examination; Weighting within unit 80%
1.Introduction to numerical analysis. Floating point arithmetic. Catastrophic cancellation and the quadratic equation formula. Efficiency and Horner's method. [3 lectures]
2.Approximation. Lagrange interpolation. Uniqueness and existence of interpolants. Error estimates. Runge's example. Divided difference form of interpolant. Application to quadrature. 
3.Linear Algebra. PDE example to introduce sparse matrices. Iterative vs direct methods. Examples of iterative methods (Jacobi, Gauss-Seidel). Vector Norms. Eigenvalues, eigenvectors, spectral radius. Convergence criteria. Error bounds, matrix norms, and condition number. 
4.Solving nonlinear equations. Solution of nonlinear equations by the bisection method, fixed point iteration, and Newton's method. Discussion in one and two dimensions. 
- Endre Suli and David Mayers, An Introduction to Numerical Analysis, Cambridge University Press 2003.
- Richard L. Burden and J. Douglas Faires, Numerical Analysis, Brookes Cole 2004.
- Desmond J. Higham and Nicholas J. Higham, MATLAB Guide, Second edition, SIAM 2005.
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
- Lectures - 22 hours
- Tutorials - 11 hours
- Independent study hours - 67 hours