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Computing of B-series by automatic differentiation. (English) Zbl 1278.65020

Summary: We present an algorithm based on automatic differentiation for computing general B-series of vector fields \(f \colon \mathbb{R}^n \rightarrow \mathbb{R}^n\). The algorithm has a computational complexity depending linearly on \(n\), and provides a practical way of computing B-series up to a moderately high order \(d\). Compared to automatic differentiation for computing Taylor series solutions of differential equations, the proposed algorithm is more general, since it can compute any B-series. However, the computational cost of the proposed algorithm grows much faster in \(d\) than a Taylor series method, thus very high-order B-series are not tractable by this approach.

MSC:

65D25 Numerical differentiation
65L05 Numerical methods for initial value problems involving ordinary differential equations
65Y20 Complexity and performance of numerical algorithms
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