MATH8701 Numerical Analysis GE

2017

4.5

3 x 50-minute lectures weekly
1 x 50-minute tutorial weekly
2 x 50-minute computer labs weekly

1 of MATH2722, MATH3701 has been successfully completed

Working knowledge of first year mathematics and familiarity with basic linear algebra and differential equations.

Representation of numbers, computer arithmetic, overflow and underflow, rounding error, cancellation error, truncation error, algorithms, stopping conditions. Solving f(x) = 0, bisection, secant, Newton methods. Interpolation, Lagrange polynomials, Hermite polynomials, divided differences, splines. Solving Ax = b, Gaussian elimination algorithm, pivoting strategies, LU decomposition, complexity, special matrices (diagonally dominant, positive definite, banded). Iterative methods, Jacobi, Gauss-Seidel, SOR methods. Eigenvalues and eigenvectors, power method, Householder's method, QR-algorithm. Non-linear systems, Newton's and steepest descent methods. Approximation theory, orthogonal polynomials, Legendre polynomials, Chebyshev polynomials. Numerical differentiation. Numerical integration, simple and composite rules, error estimates, Gaussian quadrature, adaptive quadrature methods, improper integrals, multiple integrals. Numerical methods in ODE, Runge-Kutta, stiff ODE's, predictor-corrector and adaptive techniques. Numerical solutions to partial differential equations.

At the completion of this 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 breadth of problems to which the techniques can be applied and the limitations of solutions in individual circumstances
6. Understand and implement numerical methods on their own