Trace inverse algorithms for the general eigenvalue problem

Computation of matrix eigenvalues forms one of the basic problems in numerical linear algebra and is of fundamental importance in applied science and engineering. In the paper, trace inverse algorithms in rational and radical forms are introduced These algorithms are applied for computing the eigenvalues of rank one modification, bordered matrices, and the Hessenberg eigen value problem. Using this approach a sample of extremum eigen value finders are developed. These methods are iterative and can be designed to have convergence of any prescribed order. Generalization to the general nonlinear eigenvalue problem is also presented.