## MTH/HMTH212 Numerical Methods

#### Core Course for Major, minor

24 lectures

Aim:

The course introduces simple techniques in each of the basic areas of numerical mathematics, with the exception of "Optimization" which is treated separately in the course HMT221. The aim is to give an understanding, intuitive, geometric, of numerical methods together with some analysis in order to explain relative merits. All the methods are computer based, so part of the course provides an introduction to computing and experience of a simple programming language. It is intended that as a result of the course students can apply numerical methods to problems which they meet in Mathematics and other subjects in their degree studies. The MATLAB software is chosen for the course as it has a fairly standard language structure and incorporates powerful mathematical functions and good graphics. Some tutorials and assignments are set aside to develop programming skills.

Course Outline:

 Computers: programming and errors. Programming with MATLAB. Rounding errors and their effect. Stability 4 Numerical solution of a system of linear equations: Gaussian elimination with partial pivoting. Gauss-Jordan and a consideration of efficiency. L-U decomposition and the inverse matrix. 4 Roots of non-linear equations: The bisection, Newton's and the secant method for a single equation. Newton's method for set of non-linear equations. 4 Curve fitting: Least squares fitting of polynomials and general functions. Polynomial interpolation with the error formula. Splines as design curves. 4 Numerical differentiation and integration: Numerical differentiation with the truncation and rounding errors The trapezoidal and Simpson's rules extended to composite form. 4 Numerical solution of ordinary differential equations: First order problems, Euler, Taylor and Runge-Kutta methods. Systems of first order equations and second order initial value problems. The shooting method and finite differences for boundary value problems. 4
There is an "in-house" text written for the course.

Recommended Texts:

• D J Henwood, Numerical Methods, available from the Mathematics Department office.
Numerous texts give a good covererage of the syllabus, such as:-