Geometric Numerical Integration and Applications



 

1) The Baker-Campbell-Hausdorff (BCH) series up to grade 20

We have developed an algorithm which allows us to get the coefficients of the BCH series

Z = log(exp(X) exp(Y))

expressed in a basis of the free Lie algebra generated by X and Y. We have used the classical P. Hall basis and the Lyndon basis. It is contained in the paper



Sometimes the so-called symetric Baker-Campbell-Hausdorff series

Z = log(exp(X/2) exp(Y) exp(X/2)) = X + Y - 1/24 [X,[X,Y]] -1/12 [Y,[X,Y]] ...

is also required. We have therefore computed the coefficients of Z in both bases.


The tables of coefficients, together with additional documentation, can be found here.


2) The Baker-Campbell-Hausdorff (BCH) series up to grade 10 in terms of right-nested commutators

In the paper

we present a technique that allows us to get compact expressions of the BCH series in terms of independent right-nested commutators, both in the normal
and in the symmetric case. It turns out that the number of non-vanishing terms is smaller than in the classical P. Hall and Lyndon bases.

Explicit expressions are contained in the Mathematica file BCHExplicit.nb. For completeness, we also provide the explicit expression of all the identities existing among right-nested commutators up to grade m=10 in the file IdentitiesExplicit.nb.