The main problem considered is the effect due to changes in the shape of the region on the convergence rate of the Gauss-Seidel iterative method for solving the Dirichlet Difference Problem. Experimentally, it is found that as a rule the number of iterations required to attain convergence decreases as the perimeter of the region is increased. The ensuing theoretical investigation leads to the examination of the corresponding iteration matrices and a qualitative theory results which predicts that the number of iterations should increase with the number of nonzero off - diagonal elements in the matrix of the linear system. Further experiments indicate that the latter relationship is no more precise than the former; the lack of rigour in the theory is undoubtedly to blame. Better results, are obtained in the sub-problem of estimating the number of iterations necessary to satisfy a suitable convergence criterion, given a good estimate of the spectral radius of the iteration matrix corresponding to the region under study. / Science, Faculty of / Mathematics, Department of / Graduate

We consider the entire function [formula omitted] whose set of zeros includes the zeros of [formula omitted](s), expand it in an everywhere converging Maclauring series [formula omitted] Then we determine analytic expressions for the coefficients a[formula omitted] which will enable us to proceed with the numerical evaluation of some of these coefficients. To achieve this, we define an operator D[formula omitted] acting on a restricted class of power series and which we call the zeta operator. Using the operator D[formula omitted], we are able to express the coefficients a[formula omitted] as infinite n-dimensional integrals. Numerical values for the coefficients a₀ and a₁ are easily determined. For a₂ and a₃, we transform the multidimensional integrals into products of single integrals and obtain infinite series expressions for these coefficients. Although our method can also be used on the following coefficients, it turns out that the work involved to obtain an expression leading to a practical numerical evaluation of a₄, a₅, …,seems prohibitive at this stage. We then proceed with the numerical computation of a₂ and a₃ and we use these coefficients to calculate the sums of reciprocals of the zeros of [formula omitted](s) in the critical strip. Finally, assuming Riemann hypothesis, we calculate a few other quantities which may prove to be of interest. / Science, Faculty of / Computer Science, Department of / Graduate