Residual norm estimates are derived for a general class of methods based on projection techniques on subspaces of the form K m + W, where K m is the standard Krylov subspace associated with the original linear system, and W is some other subspace. Thesèaugmented Krylov subspace methods' include eigenvalue deeation techniques as well as block-Krylov methods. Residual bounds are established which...

Krylov subspace recycling is a powerful tool when solving long series of large, sparse linear systems that change only slowly over time. In PDE constrained shape optimization, these appear naturally, as typically hundreds or thousands optimization steps are needed with small changes in the geometry. this setting, however, applying can be difficult task. As geometry evolves, general, so does fin...

INCORPORATING KRYLOV SUBSPACE METHODS IN THE ETDRK4 SCHEME by Jeffrey H. Allen The University of Wisconsin–Milwaukee, 2014 Under the Supervision of Professor Bruce Wade A modification of the (2, 2)-Padé algorithm developed by Wade et al. for implementing the exponential time differencing fourth order Runge-Kutta (ETDRK4) method is introduced. The main computational difficulty in implementing th...

Krylov subspace methods have proved effective for many non-Hermitian eigenvalue problems, yet the analysis of such algorithms is involved. Convergence can be characterized by the angle the approximating subspace forms with a desired invariant subspace, resulting in a geometric framework that is robust to eigenvalue ill-conditioning. This paper describes a new bound on this angle that handles th...

Let Mn(R) and Sn(R) be the spaces of n × n real matrices and real symmetric matrices respectively. We continue to study d(n, n − 2,R): the minimal number such that every -dimensional subspace of Sn(R) contains a nonzero matrix of rank n−2 or less. We show that d(4, 2,R) = 5 and obtain some upper bounds and monotonicity properties of d(n, n − 2,R). We give upper bounds for the dimensions of n − ...