Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains

Abstract

Abstract. In this paper we consider the computational complexity of solving initial-value problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of R n and C n , under the Computable Analysis setting. We show that the solution can be computed in polynomial time over its maximal interval of definition, provided it satisfies a very generous bound on its growth, and that the function admits an analytic extension to the complex plane.

Computational benefit of smoothness: Parameterized bit-complexity of numerical operators on analytic functions and Gevrey’s hierarchy

“…(a) For 0 < r < 1 and 0 < q ≤ 1 and p ≥ 0. 6 (b) Let 1 2 ≤ r < 1 < A ∈ N, 0 < q ≤ 1, and N ∈ N + . Then A · r m q ≤ 2 −N holds for all m ≥ C · N 1/q , where C :=  2 log 2 A/ log 2 (1/r) …”

“…Similarly, solving ordinary first-order differential equations with C 1 right-hand side is PSPACE-complete [32,25,27] but maps polynomial-time computable analytic functions to polynomial-time computable ones [45,6]. Note that this and the upper complexity bounds in Fact 1(e) and Fact 1(f) are non-uniform, 2 too: They fix an arbitrary polynomial-time computable analytic or Gevrey input function f and consider the worst-case complexity of the output function g = Λ(f ) in terms of the precision parameter n only while disregarding the running time's dependence on (parameters of) f and the information about f employed by the algorithm.…”

Computational Complexity of Smooth Differential Equations

“…could yield functions h, such as h(t) = exp t − 1, that grow too fast to be polynomial-time (or even polynomialspace) computable. Bournez, Graça and Pouly [1,Theorem 2] report that the statement about the analytic case holds true if we restrict the growth of h appropriately. 4 In the last part of the proof of this fact in the book [10,Theorem 3.7], the definition of f needs to be replaced by, e.g.,…”

“…because the notion of polynomial-time computability of real functions in this paper is defined only when the domain is a bounded closed region. 3 This makes the equation (1.1) ill-defined 1 As shown by Müller [13] and Ko and Friedman [11], polynomial-time computability of an analytic function on a compact interval is equivalent to that of its Taylor sequence at a point (although the latter is a local property, polynomial-time computability on the whole interval is implied by analytic continuation; see [13,Corollary 4.5] or [3,Theorem 11]). This implies the polynomial-time computability of h, since we can efficiently compute the Taylor sequence of h from that of g. 2 Another common terminology (which we used in the abstract) is to say that g is of class C k if it is of class C (i,j) for all i, j with i + j ≤ k. 3 Although we could extend our definition to functions with unbounded domain [6, Section 4.1], the results in Whether smoothness of the input function reduces the complexity of the output has been studied for operators other than solving differential equations, and the following negative results are known.…”

Computability and Computational Complexity of the Evolution of Nonlinear Dynamical Systems

“…It is not only polynomial ODEs which can be solved in polynomial time over their maximal interval of definition, many analytic ODEs can also be solved in polynomial time, as long as the function defining the ODE and its solution do not grow quicker than a very generous bound [37].…”