On the Complexity of Numerical Analysis

Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, Peter Bro Miltersen
2009 SIAM journal on computing (Print)  
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 divisionfree straight-line program producing an integer N , decide whether N > 0. We show that PosSLP lies in the counting hierarchy, and we show that if A is any language in the Boolean part of P R accepted by a machine whose machine constants are
more » ... braic real numbers, then A ∈ P PosSLP . 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. Introduction The motivation for this paper comes from a desire to understand the complexity of computation over the reals in the Blum-Shub-Smale model, and more generally by a desire to understand the complexity of problems in numerical analysis. The Blum-Shub-Smale model of computation over the reals provides a very well-studied complexitytheoretic setting in which to study the computational
doi:10.1137/070697926 fatcat:fd6guotj3rehnpehbllcquaii4