On the computational complexity of ordinary differential equations
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This article studies the effective convergence of numerical solutions of initial value problems (IVPs) for ordinary differential equations (ODEs). A convergent sequence {Ym} of numerical solutions is said to be effectively convergent to the exact solution if there is an algorithm that computes an N ∈ ℕ, given an arbitrary n ∈ ℕ as input, such that the error between Ym and the exact solution is less than 2-n for all m ≥ N. It is proved that there are convergent numerical solutions generated from Euler’s method which are not effectively convergent. It is also shown that a theoretically-proved-computable solution using Picard’s iteration method might not be computable by classical numerical methods, which suggests that sometimes there is a gap between theoretical computability and practical numerical computations concerning solutions of ODEs. Moreover, it is noted that the main theorem (Theorem 4.1) provides an example of an IVP with a nonuniform Lipschitz function for which the numerical solutions generated by Euler’s method are still convergent.