In this paper we discuss a number of modified eigenvalue problems that arise in a variety of applications. The problems that we consider here are of interest when the matrices are large and sparse and hence require special algorithmic methods for estimating a few of the eigenvalues. In particular we will consider problems where a given matrix, which is symmetric and positive definite is modified by a matrix of rank one, and we shall also consider eigenvalue problems where there is a homogeneous constraint to be satisfied. For many of these problems, one wishes to estimate the smallest eigenvalue for the unmodified problem as well as the modified problem. For the problems we have in mind we will be able to estimate both sets of eigenvalues. We develop algorithms which are applicable in a parallel environment. The basic theory is dependent upon the theory of moments and Gaussian quadrature.