Foundations of Computational Mathematics

, Volume 17, Issue 2, pp 625–626

Erratum to: Higher-Order Averaging, Formal Series and Numerical Integration II: The Quasi-Periodic Case

Erratum

DOI: 10.1007/s10208-016-9311-2

Chartier, P., Murua, A. & Sanz-Serna, J.M. Found Comput Math (2017) 17: 625. doi:10.1007/s10208-016-9311-2

1 Erratum to: Found Comput Math (2012) 12:471–508 DOI 10.1007/s10208-012-9118-8

After formula (22), the tree $$u$$ must have the label $$\mathbf{l}$$ at the root and the label $$\mathbf{k}$$ at the leaf. Furthermore, the condition $$\mathbf{k}\ne \mathbf{0}$$ must be imposed.

The formula at the second bullet point after (56) must be changed to $$f_{\mathbf{k}_1\cdots \mathbf{k}_r}(y) := \partial _yf_{\mathbf{k}_2\cdots \mathbf{k}_r}(y)f_{\mathbf{k}_1}(y)$$.

In the unnumbered displayed formula before (61), the left-hand side must be $$H(y,\theta )$$ rather than $$\epsilon H(y,\theta )$$. In formula (61), the leftt-hand side must be $$\epsilon \bar{H}$$ rather than $$\bar{H}$$.

The unnumbered displayed formula before (64) may be simplified to read:
\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} Y= & {} \varepsilon \sum _{\mathbf{k}} \bar{\beta }_\mathbf{k}f_\mathbf{k}+ \varepsilon ^2 \sum _{\mathbf{k}> \mathbf{l}} \bar{\beta }_{\mathbf{l}\mathbf{k}} [f_\mathbf{l},f_\mathbf{k}] \\&+\,\varepsilon ^3 \left( \sum _{\mathbf{k}\ne \mathbf{l}} \bar{\beta }_{\mathbf{l}\mathbf{l}\mathbf{k}} [f_\mathbf{l},[f_\mathbf{l},f_\mathbf{k}]] + \sum _{\mathbf{l}>\mathbf{k}<\mathbf{m}, \ \mathbf{m}\ne \mathbf{l}} \bar{\beta }_{\mathbf{m}\mathbf{l}\mathbf{k}}\, [f_\mathbf{m},[f_\mathbf{l},f_\mathbf{k}]] \right) \\&+\,\mathcal {O}(\varepsilon ^4), \end{aligned}
where $$<$$ is some total ordering in the set of multi-indices $$\mathbb {Z}^d$$, such that $$\mathbf{k}> \mathbf{0}$$ for $$\mathbf{k}\ne \mathbf{0}$$. Formula (64) is wrong and must be replaced by
\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} Y = \varepsilon f_{\mathbf{0}} + \varepsilon ^2 F_2 + \varepsilon ^3 F_3 + \mathcal {O}(\varepsilon ^4), \end{aligned}
where, with $$<$$ as before,
\begin{aligned} F_2= & {} \sum _{\mathbf{k}> -\mathbf{k}} \frac{i}{\mathbf{k}\cdot \omega } ([f_{\mathbf{k}}-f_{-\mathbf{k}},f_{\mathbf{0}}] + [f_{-\mathbf{k}},f_{\mathbf{k}}]), \\ F_3= & {} \sum _{\mathbf{k}\ne \mathbf{0}}\frac{1}{(\mathbf{k}\cdot \omega )^2} \Big ([f_\mathbf{0},[f_\mathbf{0},f_\mathbf{k}]] + [f_\mathbf{k},[f_\mathbf{k},f_{-\mathbf{k}}]] \\&-\, \frac{1}{2} [f_\mathbf{k},[f_\mathbf{k},f_{-2\mathbf{k}}]] + [f_{-\mathbf{k}},[f_\mathbf{k},f_{\mathbf{0}}]] \Big ) \\&+ \sum _{\mathbf{0}\ne \mathbf{m}\ne -\mathbf{l}\ne \mathbf{0}} \frac{-1}{(\mathbf{l}\cdot \omega ) ((\mathbf{m}+ \mathbf{l})\cdot \omega )} \, [f_\mathbf{m},[f_\mathbf{l},f_\mathbf{0}]] \\&+ \sum _{-\mathbf{l}> \mathbf{k}< \mathbf{l}, \ \mathbf{k}\ne \mathbf{0}} \frac{1}{(\mathbf{k}\cdot \omega ) (\mathbf{l}\cdot \omega )} \, [f_{-\mathbf{l}},[f_\mathbf{l},f_\mathbf{k}]] \\&+ \sum _{\begin{array}{c} \mathbf{m}> \mathbf{k}< -\mathbf{k}\\ \mathbf{m}+\mathbf{k}\ne \mathbf{0} \end{array}} \frac{-1}{(\mathbf{k}\cdot \omega ) (\mathbf{m}\cdot \omega )} \, [f_{\mathbf{m}},[f_{-\mathbf{k}},f_\mathbf{k}]] \\&+ \sum _{\begin{array}{c} \mathbf{0}\ne \mathbf{m}\ne \pm \mathbf{l}\ne \mathbf{0}\\ \mathbf{m}> -\mathbf{m}-\mathbf{l}< \mathbf{l} \end{array}} \frac{-1}{(\mathbf{m}\cdot \omega ) ((\mathbf{m}+\mathbf{l})\cdot \omega )} \, [f_{\mathbf{m}},[f_{\mathbf{l}},f_{-\mathbf{m}-\mathbf{l}}]]. \end{aligned}
In Theorem 5.7, the minus sign in the formula for $$\beta _\mathbf{k}^{[j]}$$ must be deleted.

For the results in Sect. 5.3 to hold, it is necessary to assume that the Hamiltonian functions $$I_j(x)$$, $$j= 1,\dots ,d$$, are in involution.

In formula (78), the minus sign after $$=$$ must be deleted.