620-381 Computational Mathematics

Any one of 620-122, 620-142, [05]620-192, [05]620-194, 620-211; together with one of 620-113, 620-123, 620-143, [05]620-193; and one of computer science 433-171 or 433-151 or 620-131 or other evidence of competence in C, C⁺⁺, Fortran, Pascal, or similar languages.

This subject introduces the underlying basis for numerical techniques to solve a variety of problems; and the role of various kinds of numerical error and how algorithms are designed to minimise this error; and develops basic algorithms in the areas of root-finding, linear systems, interpolation, quadrature and solution of differential equations. Students should acquire skills in implementing the above algorithms in well-constructed computer programs and interpreting the results obtained from the programs. This subject demonstrates the difficulties and possible pitfalls in numerical computation. It also shows where to find sources of reliable numerical software.

Topics include errors, roundoff, truncation error and stability; root-finding, iteration, bisection, Newton's method and secant method; linear systems, Gauss elimination, pivoting, LU factorisation, tridiagonal systems, condition number; interpolation, polynomial and spline; data fitting and least squares methods; quadrature, Newton-Cotes, Gaussian quadrature, adaptive quadrature and improper integrals; and differential equations and initial value problems: Euler, Runge-Kutta, predictor-corrector and stiff problems.

Computational assignments of up to 75 pages in total due during the semester (50%); a 2-hour written examination in the examination period (50%).