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On the Complexity of Numerical Analysis Allender, Eric Bürgisser, Peter Kjeldgaard-Pedersen, Johan Miltersen, Peter Bro Blum-Shub-Smale Model Euclidean Traveling Salesman Problem Counting Hierarchy We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both hinge on the question of understanding the complexity of the following problem, which we call PosSlp: Given a division-free straight-line program producing an integer N, decide whether N>0. We show that OrdSlp lies in the counting hierarchy, and combining our results with work of Tiwari, we show that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Eric Allender and Peter Bürgisser and Johan Kjeldgaard-Pedersen and Peter Bro Miltersen 2006 Is Part Of Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.06111.12 urn:nbn:de:0030-drops-6130 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.12 eng https://creativecommons.org/licenses/by/4.0/legalcode