Course co-ordinator(s): Dr Lyonell Boulton (Edinburgh).
Aims:
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.
Detailed Information
Pre-requisite course(s): F17CA Calculus A & F17CB Calculus B .
Location: Edinburgh.
Semester: 2.
Syllabus:
Solving general non-linear scalar equations: The bisection method. Convergence of
the bisection method. The regula falsi methods. Absolute, relative and percentage errors.
(3 lectures)
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.
(3 lectures)
Analysis of orders of covergence for fixed point iteration: How to terminate fixed point
iteration. Multiple roots of a functional equation. The Newton method for multiple roots.
(3 lectures)
Interpolation: Polynomial interpolation. Direct method. Lagrange polynomials. Newton
polynomials. Interpolation error and Chevyshev points. (3 lectures)
Numerical Integration: Simple trapezoidal, midpoint and Simpson’s rules. Derivation of
the simple integration rules. The composite trapezoidal, midpoint and Simpson’s rules.
(3 lectures)
Convergence of the numerical integration methods: Convergence of the simple rules.
Convergence of the composite rules. Minimal number of quadrature points for accuracy.
(1 lecture)
Numerical differentiation: Forward, backward, central and five-point differentiation. Error
formulas for differentiation methods. Higher order derivatives. (2 lectures)
Computer arithmetics: Number bases. Floating point numbers. Round off errors. Floating
point arithmetic. (3 lectures)
Basics of Matlab Programming: Command prompt, M-files, M-functions, variables and
internal Matlab functions. Debugging, error messages and using the editor to change
existing files. Bisection method algorithm. “If” and “while” conditionals. Bisection and
regula falsi algorithms with different stopping criteria. (3 lectures)
Iteration techniques in Matlab: “For” loops. Newton’s method. Text strings and vectors
in Matlab. Fixed points and the logistic map. Interpolation in Matlab. (3 lectures)
Integration and differentiation in Matlab: Matlab integration with applications to solving
different practical problems. Order of magnitude of the error in computing integrals.
Differentiating in Matlab and applications. (3 lectures)
Assessment Methods:
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.
Credits: 15.