Computational Science

Typically, problems in Applied Mathematics are modelled using a set of equations that can be written down but
cannot be solved analytically. In this module we examine numerical methods that can be used to solve such
problems on a desktop computer. Practical computer lab sessions will cover the implementation of these methods
using mathematical software (Matlab). No previous knowledge of computing is assumed. Topics and techniques
discussed include but are not limited to the following list. [Computer architecture:] The Von Neumann model
of a computer, memory hierarchies, the compiler. [Floating-point representation:] Binary and decimal notation,
floating-point arithmetic, the IEEE double precision standard, rounding error. [Elementary programming
constructions:] Loops, logical statements, precedence, array operations, vectorization. [Root-finding for
single-variable functions:] Bracketing and Bisection, Newton?Raphson method. Error and reliability analyses
for the Newton?Raphson method. [Numerical integration:] Midpoint, Trapezoidal and methods.Error analysis.
[Solving ordinary differential equations (ODEs):] Euler Method, Runge?Kutta method. Stability and accuracy
for the Euler method. [Linear systems of equations:] Gaussian elimination, partial pivoting. The condition
number of a matrix: quantifying the idea that amatrix can be ?almost? singular, investigating the consequences
of this idea for the robustness of numerical solutions of linear systems. [Fitting data to polynomials using
the method of least squares.] [Random-number generation using the linear congruential method.