Summary
We study zero finding for functions fromC r([0, 1]) withf(0) ⋅f(1) < 0 and for monotone functions fromC([0, 1]). We show that a lower boundγ n with a constantγ holds for the average error of any method usingn function or derivative evaluations. Here the average is defined with respect to conditionalr-fold Wiemer measures or Ulam measures, and the error is understood in the root or residual sense. As in the worst case, we therefore cannot obtain superlinear convergence even for classes of smooth functions which have only simple zeros.
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Ritter, K. Average errors for zero finding: Lower bounds for smooth or monotone functions. Aeq. Math. 48, 194–219 (1994). https://doi.org/10.1007/BF01832985
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DOI: https://doi.org/10.1007/BF01832985