Course co-ordinator(s): David Bourne (Edinburgh).
When solving problems in science, engineering or economics, a real-life situation is first converted into a mathematical model. This is often called the formulation of the problem and it is given in terms of mathematical equations. Only a handful of model equations can be solved in a neat analytical form. Hence we need numerical analysis, comprising a set of techniques for finding approximate solutions of these equations. This course provides an introduction to very basic methods in numerical analysis both from a theoretical and a practical perspective. It also provides an introduction to programming the scientific computing package Matlab.
Computer arithmetics: Number bases. Floating point numbers. Significant figures and digits. Round off errors. Absolute, relative and percentage errors. Floating point arith- metics.
The elements of Matlab: The different Matlab environments. M-files and M-functions. Variables, data types and internal Matlab functions. Text strings. The "If" conditional and logical pipelines. "For" loops.
The basics of Matlab Programming: The bisection method. Convergence of the bisection method. The regula falsi methods. Bisection and regula falsi algorithms with different stopping criteria in Matlab.
Solving algebraic equations numerically: The bisection method. Convergence of the bisection method. The regula falsi methods. Bisection and regula falsi algorithms with different stopping criteria in Matlab.
Solving smooth non-linear scalar equations: Newton's method. Fixed point iteration. Taylor's Theorem. Order of convergence. Order of convergence for fixed point iteration.
Analysis of orders of covergence for fixed point iteration: How to terminate fixed point iterations in practice. Multiple roots of a functional equation. The Newton method for multiple roots. The Newton method in Matlab. Fixed point iterations in Matlab and the logistic map.
Interpolation: Polynomial interpolation. Direct method. Lagrange polynomials. Newton polynomials. Interpolation error and Chevyshev points. Interpolation in Matlab.
Numerical Integration: Simple trapezoidal, midpoint and Simpson's rules. Derivation of the simple integration rules. The composite trapezoidal, midpoint and Simpson's rules. Matlab integration with applications to solving different practical problems.
Convergence of the numerical integration methods: Convergence of the simple rules. Convergence of the composite rules. Minimal number of quadrature points for accuracy. Order of magnitude of the error in computing integrals.
Numerical differentiation: Forward, backward, central and five-point differentiation. Error formulas for differentiation methods. Higher order derivatives. Differentiating in Matlab and applications.
30% by class tests or other continuous assessment
70% by end of module 2-hour exam
Resit Type: resit exam in semester 3
Contact Hours: 3 lectures + 1 tutorial or lab session per weeks.
SCQF Level: 8.