Topic Guide (for end-of-semester revision)

The following list provides an overview of the topics (not in chronological order) to be explored during the semester.

Modelling and Computation

1. Computer-assisted problem solving; methodology.
2. Algorithms and their representation: sequence, selection, iteration.
3. Conceptual model of a computer.
4. Graphical representation of data.
5. Python as a calculator.
6. Program tracing.

Numerical Methods

1. Numerical methods and types of error: truncation and round-off error.
2. Representation of numbers and floating-point arithmetic.
3. Propagation of error, machine precision.
4. Solution of a nonlinear equation: bisection, fixed-point iteration
   Newton-Raphson and secant methods.
   Rates of convergence.
5. Modelling dynamic systems with ordinary differential equations.
   Solving first-order ODEs using Euler, Huen and Runge-Kutta methods.
   Taylor series and graphical descriptions.
   Local and global errors, stability and convergence.
6. Solution of systems of linear equations.
   Direct methods (Gauss-Jordan elimination) and iterative methods (Jacobi and Gauss-Siedel).
   Sensitivity and condition number.
7. Interpolation of data using the monomial basis and the Lagrange basis.
   Ill-conditioning and the selection of sample points.
8. Numerical differentiation via interpolation and via Taylor series expansion.
9. Quadrature using Newton-Cotes methods.
   Adaption of panel-size based on a recursive implementation.
10. Data modelling via least-squares curve fitting.
    Formulation with monomial and more general basis functions.
11. Optimisation: one-dimensional search using Golden section,
    mutlidimenesional search using the (nonlinear) simplex method.
    Application to function-solving, curve-fitting, design, etc.

Programming in Python

1. Lexical structure of Python:
   • lines and indentation: physical lines, logical lines,
   • comments,
   • statements: simple and compound,
   • tokens, identifiers, keywords, operators, delimiters and literals.
2. Simple data types: integer, floating-point and complex numbers.
3. Sequence data types: strings, tuples, lists, dictionaries.
4. Variables, assignments, namespaces, objects and object-space.
   (This topic is also part of discussion on functions and modules.)
5. Expressions and operations on data:
   arithmetic, comparison, logical, indexing, slicing.
6. Simple output: print statement.
7. String interpolation: % operator and conversion specifiers.
8. Simple input: raw_input function and conversion from strings to other types.
9. Control structures for program flow: if, while, break, continue and pass statements.
   The for statement and the range function.
10. Defining your own functions.
    • Passing information to and returning data from a function.
    • Multiple return values and default parameters for functions
    • Recursive functions.
    • Namespaces and scope.
11. Program modules to help organise your program; import statement.
12. Reading and writing of files.
13. Documentation: doc strings in modules and functions.
14. Debugging.
15. The use of classes to organise your program (optional, advanced).
16. Matrices and vectors in Python: lists-of-lists and the Numeric extension.