AbstractsMathematics

This dissertation concerns the field of values of the inverse of a matrix. Techniques of approximation of this set are considered for large, sparse matrices, and applications are discussed. A new method is presented that is similar in computational cost to previous methods, but may yield better approximations in practice. Also, a new technique for finding eigenvalue inclusion regions is presented, developed from the relationship between the field of values of the inverse and the eigenvalue extraction technique known as harmonic Rayleigh-Ritz. By intersecting these eigenvalue inclusion regions, a new characterization of the spectrum of a matrix is obtained. The technique for generating these regions can be generalized by replacing the field of values with other eigenvalue inclusion sets, and this is demonstrated using the Geršgorin region of a matrix.