An efficient method for the solution of lower rank extracted systems and analysis of the dynamics of a repeated impact oscillator

This dissertation consists of two independent parts: The first part proposes efficient numerical solutions to a class of symmetric positive definite linear systems Ax = b, called Lower Rank Extracted Systems (LRES). Such systems appear in numerical modelling of convolution type integral equations defined on arbitrary domains. We compute their solution using a recursive method called the Preconditioned Conjugate Gradient Method (PCGM).;First, we consider integral equations defined on one dimensional domains. The corresponding coefficient matrix, A is shown to be a principal submatrix of an N x N Toeplitz matrix, A. The preconditioner we pro pose is provided in terms of the inverse of a 2N x 2N circulant matrix constructed from the elements of A. The preconditioner is shown to yield clustering in the spectrum of preconditioned matrix. Our analysis further demonstrates that the computational expense to solve LRES is reduced from O(N2) to O(N log N) operations.;We generalize this approach to solve convolution type integral equations defined on a class of p-dimensional domains and achieve similar substantial reduction in the computational expense to solve p-dimensional LRES.;The second part of the dissertation models and studies the dynamics of a mass attached to a spring undergoing repeated impacts with a massive, sinusoidally oscillating table. The dissipation of energy is modelled by coefficient of restitution. First, we study the case in which the impacts are assumed to be inelastic. We prove the existence of an invariant Cantor set on which the dynamics are equivalent to the chaotic "shift map". A similar study is done for the case of plastic impacts where we show the existence of a Smale horseshoe. We also perform bifurcation studies with respect to the frequency and amplitude of the oscillations of the table; and study the dynamics of the system with "soft" springs and the case when the oscillation frequency of the table the natural frequency of the mass are equal.