Numerical methods are necessary in science and engineering, because most problems of practical interest are just too difficult to be solved in “closed form”. While many important problems, such as the motion of a mass on a spring etc., have “exact” solutions in terms of known functions such as sin or cos, these are nevertheless “simple” problems. Making these problems just a little more complicated leads very quickly to equations that have no easy solutions. Numerical techniques are therefore needed.

Often the approach that is needed to solve a problem numerically is quite unlike the way that you would solve the same problem by hand (eigenvalues are a good example of this). In addition, the numerical solution of some problems can have its own behaviour, which may not be an accurate reflection of the behaviour of the true solution. For that reason, it is important to keep a check on how accurate a method is, and whether it converges (in some sense) to the true solution.

This unit gives an introduction to using numerical methods to solve some of the key problems in science and engineering. We first consider how computers represent numbers and functions, then consider how to solve algebraic equations. A problem of major importance concerns the solution of linear (matrix) equations. Eigenvalues of a matrix are very important in applications, since they tell us the vibrational frequencies of a structure for example, and we will consider how to calculate the eigenvalues and eigenvectors of a matrix. Approximate methods for integrating and differentiating functions will be discussed. Finally, we will look at solving ordinary differential equations using the computer.

Numerical principles will be demonstrated and explored with the scientific software Matlab.