Spring semester.

Unit(s) of Assessment	Weighting Towards Module Mark( %)
Coursework: Consisting of a class test and two other pieces of work, one involving Maple exercises and the other based on some numerical methods.	40%
Exam: Written examination (1½ hours, unseen)	60%
Qualifying Condition(s)

MAT1017 Proof Probabilty and Experiment

The objectives of the course are to introduce numerical methods for the solution of mathematical problems and to explore more advanced techniques of computer algebra.

At the end of the course, the students should know how approximate solutions can be obtained to a range of problems which have no analytic solution using appropriate numerical methods. They should have a thorough understanding of the derivation of these methods work, and of their error analysis and rates of convergence. They should also be competent at solving mathematically demanding problems using Maple.

A number of topics in computer algebra are studied including series expansions, linear algebra, graphics and writing simple procedures.

Five topics in numerical methods are studied:

Solution of Systems of Linear Equations:

• Gaussian elimination and its relation to LU decomposition.
• Derivation and convergence results of iterative methods.
  

Solution of Non-linear Equations:

• Derivation of Newton 's method and quasi-Newton methods for solving systems of non-linear equations.
• Analysis of rates of convergence.
  

Polynomial Interpolation:

• Derivation of the Lagrange and divided difference forms of the interpolating polynomial and their error analysis, cubic Hermite interpolants and error analysis
  

Numerical Differentiation and Integration:

• Derivation of finite difference formulae for approximating first and second order derivatives and their error analysis
• Techniques for numerical integration:
• Newton-Cotes rules
• Gaussian quadrature and their error analysis
• composite rules
• extrapolation methods
• adaptive integration.
  

Numerical Solution of Ordinary Differential Equations:

• One step methods for solving initial value problems, local and global error analysis
• finite difference methods for boundary value problems.

Teaching is by lectures, tutorials and practical computing sessions. Learning takes place through lectures, tutorials, exercises, practicals, coursework and background reading.

3 lecture/tutorial hours per week for 10 weeks, including 1 supervised computing lab per week for the first 4 weeks.

Detailed notes are supplied by the convener. 

Recommended Texts:

J.D. Faires and R.L. Burden, Numerical Methods, PWS-Kent, (1993). 

P. Deuflhard. Numerical Analysis – A First Course in Scientific Computation. W de-Gruyter, 1995. 

W.H. Press, S.A. Teucholsky, W.T Vetterling and B.P. Flannery, Numerical Recipes: The Art of Scientific Computing (Second Edition), Cambridge University Press, (1992). 

A. Neumaier, Introduction to Numerical Analysis, Cambridge , (2001). 

M. Heath, Scientific Computing - An introductory Survey (Second Edition), McGraw-Hill, (2002). 

J.F. Epperson, An Introduction to Numerical Methods and Analysis, John Wiley and Sons, (2002). 

S.D. Conte and C. de Boor, Elementary Numerical Analysis (Third Edition), McGraw-Hill, (1980). 

C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis (Fifth Edition), Addison-Wesley, (1994). 

L.W. Johnson and R.D. Riess, Numerical Analysis (Second Edition), Addison-Wesley, (1982).

16th July 2007