Numerical methods in Python (MAGIC099)
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The aim of this course is to introduce the students to the practical application of a relatively wide spectrum of numerical techniques and familiarise the students with numerical coding. Often in mathematics, it is possible to prove the existence of a solution to a given problem, but it is not possible to "find it". For example, there are general theorems to prove the existence and uniqueness of an initial value problem for an ordinary differential equation. However, it is in general impossible to find an analytical expression for the solution. In cases like these numerical methods can provide an answer, albeit limited: for example, there are numerical procedures (called algorithms) that, given an initial value problem, will compute its solution. This module is designed to cover four key areas: linear equations, quadratures (i.e. the evaluation of definite integrals) and the solution of Ordinary and Partial Differential Equations. The nature of the module is eminently practical: we will cover relatively little of the mathematical background of the numerical techniques that we will study. On the other hand students will be required to do a reasonable amount of programming in Python; the assessment will test their ability to code in Matlab or Python and to put into practice the theoretical methods studied at lectures. Computer laboratory sessions are associated to this module and will complement the lectures.
The assessment is through coursework:
(A) One shorter test of Python knowledge
(B) Two longer coursework assignment to test programming skills and numerical problem solving.
Autumn 2017 (Monday, October 9 to Friday, December 15)
No prerequisites information is available yet.
Linear Systems Linear systems, direct methods (Gaussian and LU decomposition), indirect methods (Jacobi, Gauss- Seidel). Quadratures Polynomial interpolation methods and adaptive methods. Initial Value for Ordinary Differential Equations Basic theory, one-step methods (Euler, Runge-Kutta), predictor-corrector methods, multi-step-methods (Adam-Bashforth, Adam-Moulton). Higher order ODEs and systems of ODEs. Boundary Value Problems for ODEs Shooting, finite differences. Partial Differential Equations Basic theory, simple PDEs (Poisson, Heat, Wave). Finite difference algorithms for parabolic, hyperbolic and elliptic PDEs. Non-Linear Equations Bisection method. Contraction mappings and Newton’s method for functions of one or more variables. Python Introduction, commands to solve quadratures and integrate ordinary and partial differential equations. Basic programming techniques.
No bibliography has been specified for this course.
No assessment information is available yet.
No assignments have been set for this course.
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