This thesis aims at building a numerical linear algebra library with appropriate software engineering abstractions. Three areas of knowledge, namely, Numerical Linear Algebra (NLA), Software Engineering and Compiler Optimisation Techniques, are involved. Numerical simulation is widely used in a large number of distinct disciplines to help scientists understand and discover the world. The solutions to frequently occurring numerical problems have been implemented in subroutines, which were then grouped together to form libraries for ease of use. The design, implementation and maintenance of a NLA library require a great deal of work so that the other two topics, namely, software engineering and compiler optimisation techniques have emerged. Generally speaking, these both try to divide the system into smaller and controllable concerns, and allow the programmer to deal with fewer concerns at one time. Band matrix operation, as a new level of abstraction, is proposed for simplifying library implementation and enhancing extensibility for future functionality upgrades. Iteration Space Partitioning (ISP) is applied, in order to make the performance of this generalised implementation for band matrices comparable to that of the specialised implementations for dense and triangular matrices. The optimisation of ISP can be either programmed using the pointcut-advice model of Aspect-Oriented Programming, or integrated as part of a compiler. This naturally leads to a comparison of these two different techniques for resolving one fundamental problem. The thesis shows that software engineering properties of a library, such as modularity and extensibility, can be improved by the use of the appropriate level of abstraction, while performance is either not sacrificed at all, or at least the loss of performance is limited. In other words, the perceived trade-off between the use of high-level abstraction and fast execution is made less significant than previously assumed.

In this paper we investigate (d + 1)-point D-optimal designs for d-th degree polynomial regression with weight function w(x) > 0 on the interval [a, b]. Suppose that w'(x)/w(x) is a rational function and the information of whether the optimal support contains the boundary points a and b is available. Then the problem of constructing (d + 1)-point D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix with k auxiliary unknown constants. We characterize the weight functions corresponding to the cases when k= 0 and k= 1. Then, we can solve (d + 1)-point D-optimal designs directly from differential equation (k = 0) or via eigenvalue problems (k = 1). The numerical results show us an interesting relationship between optimal designs and ordered eigenvalues.