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Ultimate complexity for numerical algorithms

Most numerical algorithms are designed for single or double precision floating point arithmetic, and their complexity is measured in terms of the total number of floating point operations. The resolution of problems with high condition numbers (e.g. when approaching a singularity or degeneracy) may require higher working precisions, in which case it is important to take the precision into account when doing complexity analyses. In this paper, we propose a new "ultimate complexity" model, which focuses on analyzing the cost of numerical algorithms for "sufficiently large" precisions. As an example application we will present an ultimately softly linear time algorithm for modular composition of univariate polynomials.