Perturbation analysis of some matrix factorizations

Matrix factorizations are among the most important and basic tools in numerical linear algebra. Perturbation analyses of matrix factorizations are not only important in their own right, but also useful in many applications, e.g. in estimation, control and statistics. The aim of such analyses is to show what effects changes in the data will have on the factors. This thesis is concerned with developing new general purpose perturbation analyses, and applying them to the Cholesky, QR and LU factorizations, and the Cholesky downdating problem.; We develop a new approach, the so called 'matrix-vector equation' approach, to obtain sharp results and true condition numbers for the above problems. Our perturbation bounds give significant improvements on previous results, and could not be sharper. Also we use the so called 'matrix equation' approach originated by G. W. Stewart to derive perturbation bounds that are usually weaker but easier to interpret. This approach allows efficient computation of satisfactory estimates for the true condition numbers derived by our approach. The combination of these two approaches gives a powerful understanding of these problems. Although first-order perturbation bounds are satisfactory for all but the most delicate work, we also give some rigorous perturbation bounds for some factorizations.; We show that the condition of many such factorizations is significantly improved by the standard pivoting strategies (except the L factor in the LU factorization), and provide firmly based theoretical explanations as to why this is so. This extremely important information is very useful for designing more reliable matrix algorithms.; Our approach is a powerful general tool, and appears to be applicable to the perturbation analysis of any matrix factorization.