BSc Computer Science and Mathematics

Numerical analysis is an important branch of applied mathematics that is concerned with finding numerical (or approximate) 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 field, focusing on the three core topics of: interpolation, quadrature and iterative methods.

The unit starts with a study of interpolation schemes, methods for approximating functions (of one variable) by polynomials, and then quadrature schemes, numerical methods for approximating integrals (of functions of one variable). The second half of the unit looks at solving systems of linear equations and finding roots of non-linear equations via iterative techniques. In the case of linear systems, examples will be drawn from the numerical solution of differential equations via finite difference schemes.

Students will learn about practical and theoretical aspects of all the algorithms introduced in the unit. Insight into the algorithms will be given through demonstrations in MATLAB as well as via computational exercises.  

Computer Science and Maths students can take this without taking MATH24420

This course unit aims to introduce students to theoretical analysis and practical aspects of numerical methods associated with (i) the approximation of functions by polynomials, (ii) the approximation of integrals via quadrature schemes, and (iii) the numerical solution of linear and non-linear equations. 

On completion of this unit successful students will be able to:

• demonstrate, and mitigate for the effect of, non-exact arithmetic on the approximation of simple mathematical calculations,
• quantify the computational cost of simple numerical algorithms, and apply Horner’s rule for the efficient evaluation of polynomials
• construct, apply theorems regarding, and analyse the error and precision of polynomial interpolation schemes
• construct, apply theorems regarding, and analyse the error and precision of quadrature schemes 
• derive and implement iterative methods for the approximate solution of linear and nonlinear systems of equations
• apply and recall proofs of theorems regarding the convergence of iterative methods for linear and nonlinear systems, and associated error bounds  
• apply the results of, and methods used in, proofs of theorems in the course in familiar as well as unseen settings
• implement selected methods from the course in MATLAB

1. Introduction to Numerical Analysis. Essential background theorems from analysis. Floating point arithmetic. Catastrophic cancellation and the quadratic equation formula. Efficiency and Horner's method. [1.5 weeks]  

2. Interpolation. Lagrange interpolation in one dimension. Uniqueness and existence of interpolants. Error bounds. Divided difference form of interpolant. Runge's phenomenon. [1.5 weeks]

3. Integration and Quadrature. Trapezium rule. Error bounds. Simpson’s rule. Runge’s phenomenon. Composite integration rules. [2 weeks]

4. Iterative Methods for Linear Systems. Examples of linear systems associated with finite difference discretisations of differential equations. Jacobi and Gauss-Seidel methods. Vector and matrix norms, spectral radius and matrix condition number. Theoretical results on error and convergence. Practical implementation. [4 weeks]

5. Iterative Methods for Non-linear Equations. Approximation of roots of non-linear equations (in one variable) by the bisection method, fixed point iteration, and Newton's method. Fixed point theorem. Rates of convergence. [2 weeks] 

The material will be delivered in blended format. Students will watch a number of videos covering the new material each week (supplemented by notes, Examples Sheets, computer demonstrations and short STACK or Blackboard quizzes). There will be 1 Review Class per week (in person), and one Tutorial (in person) per fortnight (6 in total).   

• Formative Online Test; Weighting within unit 0%
• End of semester examination; Weighting within unit 100%

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.

• 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.

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

• You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
• You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
• There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
• In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.