ON THE COMPLEXITY OF NUMERICAL ANALYSIS

• Published in: SIAM journal on computing 2009-01, Vol.38 (5), p.1987-2006
• Main Authors: ALLENDER, Eric, BÜRGISSER, Peter, KJELDGAARD-PEDERSEN, Johan, MILTERSEN, Peter Bro
• Format: Article
• Language: English
• Subjects: Algebra; Algorithmics. Computability. Computer arithmetics; Applied sciences; Boolean; Computer science; Computer science; control theory; systems; Exact sciences and technology; Logic and foundations; Mathematical logic, foundations, set theory; Mathematics; Miscellaneous; Multiplication & division; Numerical analysis; Recursion theory; Sciences and techniques of general use; Theoretical computing; Traveling salesman problem
• ISSN: 0097-5397; 1095-7111
• DOI: 10.1137/070697926

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: (a) the Blum-Shub-Smale model of computation over the reals; and (b) a problem we call the "generic task of numerical computation," which captures an aspect of doing numerical computation in floating point, similar to the "long exponent model" that has been studied in the numerical computing community. We show that both of these approaches 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 $N>0$. In the Blum-Shub-Smale model, polynomial-time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. The generic task of numerical computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain 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. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of the arithmetic circuit identity testing (ACIT) problem. In particular, we show that if $n!$ is not ultimately easy, then ACIT has subexponential complexity.