Algebraic multilevel iteration methods and the best approximation to $1/x$ in the uniform norm

In this note, we provide simple convergence analysis for the algebraic multilevel iteration methods [37, 51]. We consider two examples of AMLI methods with different polynomial acceleration. The first one is based on shifted and scaled Chebyshev polynomial and the other on the polynomial of best approximation to x−1 on a finite interval [λmin , λmax ], 0 < λmin < λmax in the ‖ ·‖∞ norm. The construction of the latter polynomial is of interest by itself, and we have included a derivation of a recurrence relation for computing this polynomial. We have also derived several inequalities related to the error of best approximation, which we applied in the AMLI analysis.

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