No description available.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.

We consider here two classes of non-linear systems giving different degrees of non-linearity. In both cases the systems arise from finite difference discretisations of non-linear elliptic partial differential equations. Our solution methods can also fit into two categories - linearisation and non-linearisation techniques - and in our studies we have pursued three main objectives. 1) For mildly non-linear systems we generalise certain iterative techniques from the solution of linear systems to the solution of nonlinear systems with symmetric Jacobians. We are especially concerned with the effect of preconditioning of the equations here. 2) We consider the use of bidiagonalisation on non-linear systems, using preconditioning in two ways and considering both classes of nonlinear problem. 3) We solve the laminar flow problem and assess the effects of multigrid acceleration on non-linear S.I.P. techniques.

This thesis has three sections. Section one contains two chapters, the first describing those techniques used later, principally multidimensional scaling, procrustes fitting and cluster analysis. Least squares scaling, preprocessing the dissimilarity matrix and clustering by maximum likelihood partition are less known. The second chapter reviews simulation studies previously published in multidimensional scaling literature. Section two contains one chapter detailing four simulation studies in multidimensional scaling. The first considers the robustness of classical scaling in the presence of error in the dissimilarity matrix. Four probabilistic models generating euclidean-distance-like dissimilarity functions are proposed, which reflect some of the ways dissimilarities actually arise, and allow dependence between dissimilarities to be studied. Next we compare how well various scaling methods reconstruct specific configurations, given the same dissimilarity matrix. Properties of preprocessing the matrix and least squares scaling are demonstrated. Thirdly we describe a study, designed to measure the redundancy in a dissimilarity matrix, which justifies subsequent use of scaling with missing data. Finally we determine the robustness of approximations to procrustes statistics obtained from perturbational analysis of classical scaling by Sibson (1979). Section three contains four applications chapters. Firstly multidimensional scaling is applied to data concerning the voting behaviour of M.P.s in 1861. This large data set requires special handling, some dissimilarity values being best treated as unknown. The results identify both unusual and regular voting behaviour. The second application is in ethnology. Dissimilarity values derived from phonetic differences between languages are used to derive their genetic origin. The techniques, especially clustering by maximum likelihood partition, reproduce known relationships satisfactorily and suggest others. The third example uses morphological and meristic parameters to generate dissimilarities between specimens of the fish species Colisa. Here the aim is taxonomic. Finally we consider dietary changes across Britain through time to identify regional and temporal differences.