MATH3701 Numerical Analysis

2017

4.5

1 x 120-minute lecture-2 weekly
1 x 60-minute lecture-1 weekly
1 x 50-minute tutorial weekly
1 x 120-minute computer lab weekly

1 of MATH2702, MATH2121

1 of MATH2041, MATH2722, MATH8701 has been successfully completed

Representation of numbers, computer arithmetic, overflow and underflow, rounding error, cancellation error, truncation error, stopping conditions, solving f(x) = 0 (bisection, secant, Newton),interpolation (Lagrange polynomials, divided differences, splines), solving Ax = b (pivoting strategies, LU decomposition, complexity, special matrices (diagonally dominant, positive definite, banded), iterative methods (Jacobi, Gauss-Seidel, SOR, conjugate gradient) ), nonlinear systems (Newton, steepest descent, line search), approximation theory (normal equations, SVD, Legendre polynomials, Chebyshev polynomials), numerical integration (simple quadrature rules, composite rules, error estimates, Gaussian quadrature, improper integrals), numerical differentiation, numerical solutions to initial value ODE (Runge-Kutta, predictor-corrector, adaptive techniques, stiff ODE).

This topic provides

1. An understanding of the relationship between mathematical analysis of problems and the computation of numerical solutions
2. An understanding of the sources of errors introduced by the use of computers in implementing mathematical descriptions of solutions
3. An understanding of the role and methods for approximation
4. An understanding of the methods for, and the limitations of, finding numerical solutions to problems
5. Experience in scientific computing
6. Experience in integrating mathematical derivations, numerical computations, figures and written text in presenting comprehensive solutions to problems

At the completion of the topic, students are expected to be able to:

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 breath 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