Krylov subspace methods for signals, systems and control

Krylov subspace iterative methods have become the methods of choice for the solution of large-scale problems in science and engineering. We describe how, under suitable conditions, GMRES-type Krylov subspace methods will converge to a least-squares solution of large singular, possibly inconsistent linear systems. We also establish error bounds for the computed iterates.;Ill-posed problems arise in many important applications. The solution to an ill-posed problem is typically highly sensitive to perturbations in the data. The process of modifying an ill-posed problem in order to compute a solution that is less sensitive to perturbations in the data is usually referred to as regularization. We show that the GMRES and BiCG Krylov subspace iterative methods can produce meaningful approximate solutions to ill-posed problems by stopping the iteration according to a suitable stopping criterion. Specifically, we show that these Krylov subspace methods are regularization methods for certain linear ill-posed problems in Hilbert space by analyzing the projection of the problems onto the Krylov subspaces associated with the methods.;The stopping criterion is crucial to the success of the methods when applied to ill-posed problems. We propose a new graphical tool for determining a termination index that is based on the condition number of projections of the problem onto the Krylov subspaces. Under certain conditions on the linear system, the termination index corresponds to the "vertex" of an L-shaped curve.;The location of the eigenvalues of very large matrices play an important role in many problems in control theory. We show that Krylov subspace methods can be applied to the solution of the partial eigenvalue assignment problem for single-input, continuous time-invariant linear control systems. We present new algorithms for the large-scale partial eigenvalue assignment problem based on the implicitly restarted Arnoldi process. Our algorithms are demonstrated to be computationally efficient in terms of storage and arithmetic.;The theoretical results established in this thesis are aimed at shedding light on various aspects of Krylov subspace methods not previously well-studied, and at showing the feasibility of their use for real-world applications.