Computational Complexity of Real Powering and Improved Solving Linear Differential Equations

We re-consider the problem of solving systems of differential equations approximately up to guaranteed absolute error \(1/2^n\) from the rigorous perspective of sequential and parallel time (i.e. Boolean circuit depth, equivalently: Turing machine space) complexity. While solutions to general smooth ODEs are known “PSPACE-complete” [Kawamura’10], we show that (i) The Cauchy problem for linear ODEs can be solved in NC\(^2\), that is, within polylogarithmic parallel time \(\mathcal {O}(\log ^2 n)\) by Boolean circuits of polynomial size. (ii) The Cauchy problem for linear analytic PDEs, having a unique solution by the Cauchy-Kovalevskaya theorem, can be also solved in polylogarithmic parallel time, thus generalizing the case of analytic ODEs [Bournez/Graça/Pouly’11]. (iii) Well-posed Cauchy and boundary-value problems for linear PDEs in classes of continuously differentiable functions are solvable in the counting complexity class \({\#\mathsf P} ^{\#\mathsf P} \): improving over common numerical approaches yielding exponential sequential time or parallel polynomial time. Our results build on efficient algorithms and their analyses for real polynomial, matrix and operator powering which do not occur in the discrete case and may be of independent interest.