Numerical Analysis is concerned with the development of numerical methods to solve mathematical problems in a reliable and efficient way. The ability to compute numerical solutions to mathematical problems has always been an important part of mathematics. For instance, an effective method for the evaluation of the square root of a number was discovered over 3600 years ago. Nowadays, numerical methods are under continuous research and development and are widely used in science, engineering, finance and other areas, to formulate theories, to interpret data, and to make predictions.

This module provides an introduction to computational methods for the approximation of functions on an interval of the real line. We begin with the study of methods and errors associated with the solution of systems of linear equations. We then study the interpolation of functions by polynomials of a given degree, and use these techniques for the derivation of numerical integration rules and their error analysis. In particular, we shall describe the use of orthogonal polynomials for the construction of polynomials of best approximation as well as their relevance in deriving Gauss-type numerical integration rules. We shall also turn our attention to interpolation by piecewise polynomials of low degree, known as splines. These are the building blocks for the construction of more advanced numerical methods, and require only minimal prerequisites in differential and integral calculus, differential equations, and linear algebra. Experience of numerical computation is essential for a true understanding of the applications and limitations of numerical methods, and numerous examples will be presented to explain and interpret the theoretical results.

Recommended Modules: MA0123 Analysis I, MA0126 Analysis II

• Solve (large) systems of linear equations directly by Gaussian elimination, or iteratively by Jacobi’s and Gauss-Seidel methods;
• Understand the use of matrix and vector norms when solving a perturbed system of equations;
• Perform approximation of a function by polynomials on an interval, or by piecewise polynomials on a subdivision of an interval, and derive estimates for the approximation error;
• Know the definitions and basic properties of Chebychev polynomials;
• Be able to derive the general form of a Gaussian quadrature formula, perform the associated error analysis, derive formulae of a specific order, and apply them to examples.

27 - 50 minute lectures

Lecture notes will generally be provided via Learning Central, but students will be expected to take notes during lectures where more detailed information will be presented on the board.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Ability to relate mathematical theory to explicit computation. 

Ability to solve large systems of linear equations.

Ability to approximate functions and their integrals.

Formative assessment is carried out by means of regular coursework exercises.  Feedback to students on their solutions and their progress towards the learning outcomes is provided during lectures. 

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above-average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• Solution of Systems of Linear Equations
  • Elementary Linear Algebra Prerequisites
  • Direct Methods
    • Triangular Systems
    • Gaussian Elimination, Pivoting and Row Scaling
    • LU Decomposition
    • Cholesky’s Method
    • Banded Systems
    • Vector and Matrix Norms, Condition Numbers
  • Classical Iterative Methods
    • Jacobi’s Method
    • Gauss-Seidel Method
• Interpolation of Functions
  • Lagrange Interpolation
  • Hermite Interpolation
  • Chebyshev Polynomials
• Best Approximation of Functions
  • Least Squares Polynomial Approximation
  • Orthogonal Polynomials
  • Interpolating Splines
• Numerical Differentiation and Integration
  • Finite Differences
  • Newton-Cotes Quadrature
  • Gauss Quadrature

Not applicable.

An Introduction to Numerical Analysis, Suli, E. & Mayers, D. F., Cambridge University Press, 2006.