This project involves the application of principles and techniques from dynamical systems to the problem of approximating the eigenvalues of a matrix. Rayleigh quotient iteration and the QR algorithm are two iterative numerical algorithms for the approximation of eigenvalues and eigenvectors. The principal investigator will study global convergence of these algorithms and quantify their average speed and likelihood of success. He will either determine that the present QR algorithm is optimal or produce a more efficient modification. For many years, numerical linear algebra's favorite scheme for finding the eigenvalues of a matrix has been the QR algorithm. Examples were known where this scheme failed but these were thought to be rare. The principal investigator discovered an entire pocket of matrices where a related scheme, called Rayleigh quotient iteration, fails. He will now determine whether the QR algorithm fails as well.
