On the application of orthogonal polynomials to the iterative solution of linear systems of equations with indefinite or non-Hermitian matrices

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Let Ax= b be a large linear system of equations, and let the eigenvalues of the matrix A lie in one or several known simply connected regions S j , j=1(1) l, in the complex plane. We consider the iterative solution of such linear systems by methods that make use of polynomials { p n ( z)} ∞ n=0 orthogonal on the boundary of ∪ l j=1 S j . We show that for boundary curves such that the p n ( z) satisfy a three-term recurrence relation, iterative methods based on this recurrence relation yield an optimal asymptotic rate of convergence. For boundary curves for which the p n ( z) do not satisfy a three-term recurrence relation, we show that n-cyclic Richardson iteration methods with relaxation parameters chosen as the reciprocal roots of p n ( z) give nearly asymptotic optimal convergence for n sufficiently large. Our analysis suggests that in many cases the iterative methods should be based on residual polynomials p n(z) p n(0) , instead of on the corresponding kernel polynomials q z= ∑ n k=0 p k(0) p k(z) ∑ n k=0|p k(0)| 2 . Numerical examples are presented.