On the complexity of numerical analysis

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 PosSLP 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.