7.1 Introduction

In this chapter we describe numerical methods for solving eigenvalue problems, which arise in many branches of science and engineering and seem to be a very fundamental part of the structure of the universe. Eigenvalue problems are important in a less direct manner in numerical applications. For example, to discover the condition factor in the solution of a set of linear algebraic equations involves finding the ratio of the largest to the smallest eigenvalue values of the underlying matrix. Also, the eigenvalue problem is involved when establishing the stiffness of ordinary differential equation problems. In solving eigenvalue problems, mainly we are concerned with the task of finding the values of parameter ? and vector x, which satisfy a set of equations of the form

Linear equation (7.1) represents the eigenvalue problem where A is an n n coefficient matrix, also called the system matrix, x is an unknown column vector, and ? is an unknown scalar. If the set of equations has a zero on the right-hand side, then a very important special case arises. For such case, one solution of (7.1) for a real square matrix A is the trivial solution x=0. However, there is a set of values for the parameter ? for which non-trivial solutions for the vector x exist. These non-trivial solutions are called eigenvectors, and characteristic vectors or latent vectors of a matrix A and the corresponding values of...