Further Numerical Methods with Optimization at the University of Zimbabwe

HMTH333 Further Numerical Methods with Optimization

Lecturer:

Duration :2 semester

Prerequisites :HMTH212, 221

48 lectures

Aim:

To cover the main areas of numerical methods and optimization as useful tools in applied mathematics. New methods for the topics covered in the courses HMTH212 and HMTH221 are developed, and new areas, such as computing the eigen-systems of matrices and integer programming, are introduced. It is intended that a careful understanding of the methods and their limitations will be given, and this will be achieved through examining one or more significant techniques in each area thoroughly and illustrating others, using suitable software (e.g. Matlab or Maple).

Course Outline:

Brief revision of methods for the solutions of linear algebraic equations; solution of systems of non-linear equations. Norms, errors and the condition number of a matrix. 7

The eigenvalue problem: One method in detail, say Given's method for symmetric equations, eigenvectors by inverse iteration. General methods. 7

Further curve fitting. Orthogonal polynomials and Chebyshev economisation. 5

Brief revision of the Newton-Cotes methods for numerical integration; adaptive quadrature. Gaussian quadrature. 5

Runge-Kutta and predictor-corrector methods for ordinary differential equations. Stability. 4

Optimization. A brief revision of the methods of HMTH221. The SUMT method of Fiacco and McCormic. 6

Integer programming. 8

Some special methods for non-linear constrained and unconstrained optimization. Method of conjugate gradients. Method of multipliers. 6

If time permits: outline of genetic algorithms.

Recommended Texts:

T A Akai, Numerical Methods (John Wiley, 1994).
M Minoux, Programming Theory and Algorithms (John Wiley, 1986).

Additional Reading:

R Burden, J D Faires & A C Reynolds , Numerical Analysis (Prindle, Weber, Schmidt, 1981).
W H Press, B P Flannery, S A Teukolsky, W T Vetterling, Numerical Recipes (Cambridge, 1989).
J L Buchanan & P R Turner, Numerical Methods and Analysis (McGraw-Hill, 1992).
P Henrici, Essentials of Numerical Analysis: with pocket calculator demonstrations (Wiley, 1982).
S D Conte & C de Boor, Elementary Numerical Analysis (McGraw-Hill 1972).
T A Cirani & R C Lechman, Optimization in Industry (Wiley 1993).
S S Rao, Optimization: Theory and Application (Wiley, 1995).
Back to undergraduate courses
Back to main maths department page

Brief revision of methods for the solutions of linear algebraic equations; solution of systems of non-linear equations. Norms, errors and the condition number of a matrix.	7
The eigenvalue problem: One method in detail, say Given's method for symmetric equations, eigenvectors by inverse iteration. General methods.	7
Further curve fitting. Orthogonal polynomials and Chebyshev economisation.	5
Brief revision of the Newton-Cotes methods for numerical integration; adaptive quadrature. Gaussian quadrature.	5
Runge-Kutta and predictor-corrector methods for ordinary differential equations. Stability.	4
Optimization. A brief revision of the methods of HMTH221. The SUMT method of Fiacco and McCormic.	6
Integer programming.	8
Some special methods for non-linear constrained and unconstrained optimization. Method of conjugate gradients. Method of multipliers.	6
If time permits: outline of genetic algorithms.