Aims & Learning Objectives:
Aims: To revise and develop elementary MATLAB programming techniques. To teach those aspects of Numerical Analysis which are most relevant to a general mathematical training, and to lay the foundations for the more advanced courses in later years.
Students should have some facility with MATLAB programming. They should know simple methods for the approximation of functions and integrals, solution of initial and boundary value problems for ordinary differential equations and the solution of linear systems. They should also know basic methods for the analysis of the errors made by these methods, and be aware of some of the relevant practical issues involved in their implementation.
MATLAB Programming: handling matrices; M-files; graphics.
Concepts of Convergence and Accuracy: Order of convergence, extrapolation and error estimation.
Approximation of Functions: Polynomial Interpolation, error term.
Quadrature and Numerical Differentiation: Newton-Cotes formulae. Gauss quadrature. Composite formulae. Error terms.
Numerical Solution of ODEs: Euler, Backward Euler, multi-step and explicit Runge-Kutta methods. Stability. Consistency and convergence for one step methods. Error estimation and control.
Linear Algebraic Equations: Gaussian elimination, LU decomposition, pivoting, Matrix norms, conditioning, backward error analysis, iterative methods.