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In this thesis we study the preconditioning of square, non-symmetric and realToeplitz systems. We prove theoretical results, which constitute sufficientconditions for the efficiency of the proposed preconditioners and the fastconvergence to the solution of the system, by the Preconditioned GeneralizedMinimal Residual method (PGMRES) as well as by the Preconditioned ConjugateGradient method applied to the system of Normal Equations (PCGN). Asintroduction, in the first chapter, we give the basic definitions andtheorems/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 thesingular values, which is a criterion for the efficiency of the preconditioner.In the second chapter we construct a band Toeplitz preconditioner forwellconditioned, as well as for ill-conditioned systems. The preconditioningtechnique is based on the elimination of the roots of the generating function(if there exist), by a trigonometric polynomial, and on a furtherapproximation. The clustering of the eigenvalues and the singular values of thepreconditioned system has been proven. In the next chapter we construct acirculant preconditioner dealing with well-conditioned Toeplitz systems and aband-times-circulant preconditioner for ill-conditioned ones. We proveanalogous theoretical results and we give a comparison with the preconditionerproposed previously at the numerical results of the last section. In the fourthand last chapter of the thesis we study Toeplitz systems, having an unknowngenerating function. We adapt the preconditioners constructed at the previouschapters. After estimating the generating function, its roots and themultiplicities of them, we construct the corresponding preconditioners.

Krylov subspace methods for the solution of linear Toeplitz systems