Approximating extrema of quadratic forms using Krylov subspaces

Abstract: In this thesis approximations to quadratic forms using Krylov subspaces are presented. These quantities are approximations of the norm and logarithmic norm of a matrix. In order to find these quantities an eigenvalue problem has to be solved, but because of limited storage and time, this is not feasible in practice with large matrices. Instead one can project this problem down to a problem of smaller size with a Krylov subspace and solve it there instead. To test the quality of these projections, different matrices that arise in practice are tested and their norms approximated. The test matrices have known norms and behaviour so the result can be interpreted. Overall the results show that one can obtain two digit accuracy with a low dimension of the subspace, even for matrices with large dimensions, which is truly promising.