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MA934 - Numerical Methods (12 Cats)

Lecturer: Colm Connaughton (Mathematics and Complexity)

Module Aims

This is one of 4 core modules for the new MSc in Mathematics of Systems. The module aims to introduce the most basic topics in the subjects of simulation of complex systems and numerical solution of associated mathematical problems. The first part will introduce the students to the simple and intuitive ideas behind numerical differentiation and integration, interpolation, extremal finding and the solution of ordinary differential equations. The second half will briefly discuss fundamental algorithms for sorting and online big-data analysis, before introducing the idea of Monte Carlo simulations, and illustrating the simplest cases of network construction and sampling. Learning outcomes include understanding the main ideas behind numerical solution of mathematical problems, such as differentiating, integrating and interpolating functions, finding minima and maxima, and solving ODEs. The students will have an appreciation of the principles of equilibrium Monte Carlo simulations, and of the simplest method and models of graph generation and sampling.

Syllabus

  • Basic numerical methods: derivation, interpolation and extrapolation, integration, extremal finding, ordinary differential equations
  • Useful algorithms and data structures: sorting, stacks, queues and linked lists, binary trees and search
  • Numerical solution of linear systems of equations
  • Introduction to Monte Carlo principles: Monte Carlo integration, ergodicity, detailed balance
  • Construction and sampling of networks: Erdős-Rényi random graphs, small-world networks, preferential attachment models.

Illustrative Bibliography

W. H. Press et al., “Numerical recipes in C”, Cambridge University Press

Teaching

See main calendar for timetable

  • Per week: 2 x 2 hours of lectures, 2 x 2 hours of classwork
  • Duration: 5 weeks (second half of term 1)

Assessment

For deadlines see Module Resources page

  • Written homework assignments (25%)
  • Class test (25%) and
  • Oral examination (50%)