University of Macau Abstract Boundary Value Method with skew-circulant Preconditioner for ODEs by Yeung Choi-In Thesis Supervisor: Prof. Jin Xiao Qing Department of Mathematics In this thesis, we consider the solution of ordinary differential equations (ODEs) by using boundary value methods (BVMs). These methods require the solution of one or more unsymmetric, large and sparse linear systems. In chapter 1, we introduce some important matrices, and their basic properties. We give a formula for the eigenvalues of a skew-circulant matrix in section 1.1. In section 1.2, we study the Generalized Minimal Residual (GMRES) method and its algorithm. In chapter 2, we introduce Boundary Value Methods (BVMs). We use the GMRES method with skew-circulant preconditioner proposed for solving the linear system of ODEs. We prove that if the BVM satisfies certain stability condition then our preconditioner is invertible. Some numerical examples are given in section 2.3.