Complexity of fixed points, I

We study the complexity (minimal cost) of computing an s-approximation to a fixed point of a contractive function with the contractive factor q < 1. This is done for the relative error criterion in Part I and for the absolute error criterion in Part II, which is in progress. The complexity depends strongly on the dimension of the domain of functions. For the one-dimensional case we develop an optimal fixed point envelope (FPE) algorithm. The cost of the FPE algorithm with use of the relative error criterion is roughly clog12elog3+q1+3q+0.9+log log11−q , where c is the cost of one function evaluation. Thus, for fixed ϵ and q close to 1 the cost of the FPE algorithm is much smaller than the cost of the simple iteration algorithm, since the latter is roughly clog1elog1q For the contractive functions of d variables, with d ≥ log(1/ϵ)/log(l/q) we show that it is impossible to essentially improve the efficiency of the simple iteration.

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