MATH2722 Numerical Analysis

This topic includes: Representation of numbers, computer numbers, rounding, computer arithmetic; sources of error, their classification and analysis. Iterative algorithms, tolerance, stopping conditions. Introduction to Matlab. Order and rage of convergence. Solving f(x) = 0 (bisection, secant, regular falsi, Newton). Interpolation (Lagrange, Hermite polynomials, divided differences, cubic splines). Numerical differentiation. Numerical integration, trapezoidal and Simpson formulas, error estimates, Legendre polynomials, Gaussian quadrature, double and triple integrals. Gaussian Elimination Algorithm and its applications, pivoting strategies, complexity, LU factorisation. Special matrices and GEA (diagonally dominant, self-adjoint, positive definite, unitary, banded). Matrix norms, spectral radius. Iterative methods (Jacobi, Gauss-Seidel, SOR). Eigenvalues and eigenvectors, basic spectral mapping theorem, power, symmetric power, and inverse power methods. Jacobi matrix and its properties, Newton's method for nonlinear systems. Numerical methods for ordinary differential equations (ODE's) and systems of ODE's (Euler, Taylor and Runge-Kutta methods), stiff ODE's.

This topic aims to provide:

• An understanding of the relationship between mathematical analysis of problems and the computation of numerical solutions
• An understanding of the sources of errors introduced by the use of computers in implementing mathematical descriptions of solutions
• An understanding of the role and methods for approximation
• An understanding of the methods for, and the limitations of, finding numerical solutions to problems
• Experience in scientific computing
• Experience in integrating mathematical derivations, numerical computations, and figures in presenting solutions to problems

1. Understand the sources of errors introduced by the use of computers to perform computations
2. Reformulate expressions so as to facilitate accurate computation
3. Understand techniques for arriving at numerical solutions to many classes of problems including linear and non-linear systems of equations, integration, and differential equations
4. Present solutions to problems by integrating mathematical derivations, numerical implementation, figures, and written text
5. Understand both the breadth of problems to which the techniques can be applied and the limitations of solutions in individual circumstances
6. Have improved their ability to understand and implement numerical methods on their own