University of Mines and Technology(UMaT), Tarkwa. Ghana

20.5.2    Masters Programme in Computational Mathematics

a)    Core and Compulsory Modules

The MSc. coursework comprises four (4) core/compulsory modules namely:
•    Research Methods (MA 501).
•    Computer Programming (MA 507).
•    Numerical Methods for Linear and Non Linear Equations (MA 517).
•    Operations Research (MA 505).

In addition, a minimum of three other modules must be selected by the candidate in consultation with his/her Supervisor(s). Applicants without adequate Mathematics or Statistics background will be required to register for the module in Optimisation Techniques and Computer Applications.

b)    Content of Modules

First Semester

Second Semester

FIRST SEMESTER

MA 275 Numerical Methods
Credits: 0

Sources and types of error; round-off errors, truncation error, Basic error analysis. Evaluation of functions. Numerical solution of non-linear algebraic equation; one-point methods; simple iteration, secant and Newton-Raphson methods. Acceleration and relaxation. Bracketing methods; Bisection and false-position methods. Numerical solution of sets of linear algebraic equations: elimination back substitution. Matrix inversion. Instabilities and pivoting. Gaussian elimination. Iterative methods for linear systems: Gauss-Jacobi, Gauss-Siedel and successive over relaxation (SOR). Convergence and error analysis. Order of an iterative process. Use of computer essential. Conjugate Gradient.

Methods for first-order differential equations: Taylor’s method, Euler methods, Runge-Kutta methods, multi-step methods. Methods for higher-order differential equations: Taylor’s, Euler and Runge-Kutta methods.

MA 501 Research Methods
Credits: 3

Introduction to research: Research project formulation/management, the research process, literature review and organization. Epistemology and its implications for research methodology and design. Theoretical framework (variable definition and generation of hypothesis). Scientific research design (differences between qualitative and quantitative methodology, measurement issues: reliability and validity). Qualitative data collection (e.g. in-depth interviews, focus groups, observations). Analysis of qualitative data. Principles of quantitative data analysis (descriptive statistics). Quantitative methods ( hypothesis testing, inferential statistics). Sampling, questionnaire design and methods for pre-testing. Research proposal for competitive research grant. Research presentation (formatting dissertation). Case studies.

MA 505    Operations Research
Credits: 3

Introduction to Deterministic methods for Optimization, with focus on mathematical programming (linear, nonlinear, integer) and network methods. Introduction to probabilistic methods for modeling and analyzing the performance of complex systems. Topics include Markov chains, queuing, forecasting, discrete event simulation and inventory modeling.


MA 507    Computer Programming
Credits: 3

Input and output procedures. Elementary mathematical functions . User defined functions. Relational and logical operators. Conditional statements . Looping and the switch structure. Solution of Linear and non linear algebraic equations. Application to differential equations. Symbolic processing with MATLAB.

MA 517    Numerical Methods for Linear and Non Linear Equations
Credits: 3

Solutions of algebraic equations; Direct methods for linear equations, orthogonal factorization, sparse matrix techniques. Markowitz criterion, Nested dissection, applications. Solutions of non – linear equations; one point iterative methods, Newton’s and Brain methods, convergence of these methods; Multi – step iteration formulae, secant methods, gradient methods, Bracketing methods, convergence and stability of these methods; special methods, applications.

MA 519    Application of Numerical Analysis to Ordinary Differential Equations (ODEs)
Credits: 3

Initial and Boundary value problems in ODEs, Numerical approximation of solutions, Higher order one step methods, Taylor series, Runge-Kutta (R-K) methods, convergence and stability of these methods, Multistep methods. Topics in approximation; Chebyshev polynomial approximation, least – squares approximation. Approximation by series, Rational approximation.

MA 521     Computational Methods in Optimization
Credits: 3

Unconstrained continuous Optimal Control Problems. Fletcher – Reeves Algorithm. Polak – Ribiere algorithm and its application to equality – constrained control problems. Unconstrained Discrete Optimal control problems and methods of solution.

MA 523     Programming in Higher Level Language
Credits: 3

Training in the FORTRAN, C++, C and C sharps programming language for persons with no previous programming experience. Basic concepts and properties of algorithms for solution of numerical and non-numerical problems, including running of programmes on a computer.

SECOND SEMESTER

MA 500 Thesis
Credits: 12

The thesis must be an embodiment of independent research work under the guidance of Supervisor(s) on a topic of the student’s area of specialization. A thesis embodying the results of the research will be presented to the Department for assessment. A panel of examiners will assess the thesis.


MA 512    Applications of Numerical Analysis to Partial Differential Equations (PDEs)
Credits: 3

Partial differential equations, Classification, Parabolic equation; solution techniques by explicit methods, Fourier stability methods, matrix methods, stability and convergence analysis. Elliptic equations, solution techniques by finite difference methods, SOR, methods, convergence and stability of these methods. Hyperbolic equation, solution techniques by methods of characteristics, Explicit methods, Hybrid methods, Hopscotch methods, convergence and stability analysis

MA 514     Further Numerical Methods
Credits: 3

Weighted Residual methods, allocation methods, orthogonal allocation, Ritz Galerking methods, Nagume’s Lemma, application; introduction to finite elements, applications.

MA 516     Computational Methods for Optimal Control Problems
Credits: 3

Optimization Problems. Examples of Optimization problems. The Optimization in one dimension. Iterative methods of Optimization. Least squares procedures for solving equations, contraction mapping theorem. Newton’s methods. Steepest Descent Methods, Conjugate direction Methods in R, Conjugate Gradient Method Algorithm, Projection Methods.

MA 518     Further Computational Methods in Optimization
Credits: 3

Equality and Inequality Constraints, unconstrained minimization, Pontryain’s principles. Hamiltonian principles. Extremization of Integrals. Sensitivity analysis. Penalty methods; function space Algorithm. Projection Methods and applications to Optional Control Problems.

MA 520     Update Methods in Computational Optimization
Credits: 3

Rate of convergence of Conjugate Gradient Method, Function space algorithms, Linear Convergence, Superlinear convergence: Quasi- Methods, Super – Linear convergence: Conjugate Gradient Methods Super- linear convergence Variable Metric algorithm.

MA 530 Postgraduate Seminar
Credits: 3

Students will be required to make at least two presentations on the progress and research underway in their areas of specialisation. This will be assessed by a Departmental Panel. Postgraduate students are required to attend the seminar(s).