Krylov subspace methods for the solution of linear Toeplitz systems

Abstract

In this thesis we study the preconditioning of square, non-symmetric and real Toeplitz systems. We prove theoretical results, which constitute sufficient conditions for the efficiency of the proposed preconditioners and the fast convergence to the solution of the system, by the Preconditioned Generalized Minimal Residual method (PGMRES) as well as by the Preconditioned Conjugate Gradient method applied to the system of Normal Equations (PCGN). As introduction, in the first chapter, we give the basic definitions and theorems/lemmas that we use to prove the theoretical results of the thesis. These are dealing with the clustering of the eigenvalues, as well as of the singular values, which is a criterion for the efficiency of the preconditioner. In the second chapter we construct a band Toeplitz preconditioner for wellconditioned, as well as for ill-conditioned systems. The preconditioning technique is based on the elimination of the roots of the generating function (if there exist), by a trigonometric polynomial, and on a further approximation. The clustering of the eigenvalues and the singular values of the preconditioned system has been proven. In the next chapter we construct a circulant preconditioner dealing with well-conditioned Toeplitz systems and a band-times-circulant preconditioner for ill-conditioned ones. We prove analogous theoretical results and we give a comparison with the preconditioner proposed previously at the numerical results of the last section. In the fourth and last chapter of the thesis we study Toeplitz systems, having an unknown generating function. We adapt the preconditioners constructed at the previous chapters. After estimating the generating function, its roots and the multiplicities of them, we construct the corresponding preconditioners.