# American Institute of Mathematical Sciences

March  2014, 34(3): 903-914. doi: 10.3934/dcds.2014.34.903

## Computing of B-series by automatic differentiation

 1 University of Bergen, Department of Mathematics, Postbox 7800, N-5020 Bergen, Norway, Norway

Received  January 2013 Revised  April 2013 Published  August 2013

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.
Citation: Ferenc A. Bartha, Hans Z. Munthe-Kaas. Computing of B-series by automatic differentiation. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 903-914. doi: 10.3934/dcds.2014.34.903
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