This project is aimed at providing a foundation for the algorithms that are employed to approximate the eigenvalues of a real nonsymmetric matrix. Current software in the EISPACK and LAPACK subroutine libraries rely on variations of the shifted QR algorithm. The primary objective is to obtain an understanding of the global convergence properties of these algorithms and determine whether they succeed for a set of problems of full measure. Computation of eigenvalues is required in the study of vibrations and other problems involving differential equations. For many years the EISPACK subroutine library has been an effective resource for approximation of eigenvalues. While experience has shown the algorithms to be both reliable and fast, there is no theoretical basis for their success. This project involves the application of techniques from dynamical systems and complexity to analyze the established numerical software. The principal goal is to either establish the validity of the current methods or to identify areas in which they are inadequate and devise a more efficient modification.