Computing the topological degree with noisy information
Journal of Complexity
Iterative solution of nonlinear equations in several variables
Iterative solution of nonlinear equations in several variables
Optimal solution of nonlinear equations
Optimal solution of nonlinear equations
Complexity of nonlinear two-point boundary-value problems
Journal of Complexity
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We consider iterative methods for approximating solutions of nonlinear equations, where the iteration cannot be computed exactly, but is corrupted by additive perturbations. The cost of computing each iteration depends on the size of the perturbation. For a class of cost functions, we show that the total cost of producing an ε-approximation can be made proportional to the cost c(ε) of one single iterative step performed with the accuracy proportional to ε. We also demonstrate that for some cost functions the total cost is proportional to c(ε)2. In both cases matching lower bounds are shown. The results find natural application to establishing the complexity of nonlinear boundary-value problems, where they yield an improvement over the known upper bounds, and remove the existing gap between the upper and lower bounds.