Superlinear convergence of conjugate gradients
The main goal of this paper is to illustrate that some recent results obtained in the logarithmic potential theory can be used for better understanding the phenomenon in the numerical analysis known as superlinear convergence. NEWLINENEWLINENEWLINEThe authors give a theoretical explanation for superlinear convergence behaviour observed while solving large systems of linear equations with symmetric coefficient matrices using the conjugate gradient method. A new bound on the relative error, which is valid in an asymptotic sense and depends on the asymptotic eigenvalue distribution and the ratio between the number of the iterations and the size of the system, is presented. Under some appropriate conditions the authors show that this bound is asymptotically sharp. The new results are illustrated by discussing the solution of the system of linear equations with Toeplitz coefficient matrices and a model problem arising after discretization of the Poisson equation.
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