**AMA2004 - Numerical Analysis (2**^{nd} semester)

*Pre-requisite:* MTH1001 and MTH1002

*Lecturer:* Dr M Gruening

**Introduction**

Numerical Analysis is concerned with devising methods for finding approximate, numerical solutions to mathematically expressed problems. The methods are analysed for their accuracy, efficiency and robustness. As a simple example, consider solving the equation *f*(*x*) = 0, where *f*(*x*) is a specified function. If *f*(*x*) is of even moderate complexity, we will not be able to solve this equation analytically. In Numerical Analysis we develop different procedures, or algorithms, to solve this problem. Finding the most suitable requires an appreciation of the methods. For example, some will guarantee convergence to a solution, but may require much effort, while other methods may converge quickly with less effort, but may also diverge, depending on the function and initial guess. Faced with such differing behaviour of the methods, what is the ‘best’ strategy to adopt? In AMA2004 we cover the basic introductory material of Numerical Analysis. We investigate the solution of equations, interpolation, function approximation, differentiation, integration and the solution of ordinary differential equations.

An important element of this module are the practical classes, in which algorithms developed during the lectures are implemented using MATLAB, which is an interactive and programmable software package. The practical work culminates in a numerical project, in which a MATLAB program is written to implement a particular algorithm, or to investigate the behaviour of a method. Although prior familiarity with computers is helpful, it is not essential.

A third element of the module is the oral presentation. You are required to give a 5-minute presentation to a small group of students and staff.

**Contents**

*Introduction and basic properties of errors:* Introduction; Review of basic calculus; Taylor's theorem and truncation error; Storage of non-integers; Round-off error; Machine accuracy; Absolute and relative errors; Richardson's extrapolation.
*Solution of equations in one variable:* Bisection method; False-position method; Secant method; Newton-Raphson method; Fixed point and one-point iteration; Aitken's δ^{2} process; Roots of polynomials.
*Solution of linear equations:* Gaussian elimination; Pivoting strategies; Calculating the inverse; LU decomposition; Norms; Condition number; Ill-conditioned linear equations; Iterative refinement; Iterative methods.
*Interpolation and polynomial approximation:* Why use polynomials? Lagrangian interpolation; Neville's algorithm; Other methods.
*Approximation theory:* Norms; Least-squares approximation; Linear least-squares; Orthogonal polynomials; Error term; Discrete least-squares; Generating orthogonal polynomials.
*Numerical quadrature:* Newton-Cotes formulae; Composite quadrature; Romberg integration; Adaptive quadrature.
*Numerical solution of ordinary differential equations:* Boundary-value problems; Finite-difference formulae for first and second derivatives; Initial-value problems; Errors; Taylor-series methods; Runge-Kutta methods.

**Assessment**

Exam 50% Project (long) 25% Project (short) 15% Presentation 10%

All assessment components are compulsory