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However, the most efficient optimizations can be obtained if the gradient information can be employed. In that case, we can use for example the family of conjugate-gradient algorithms, or the Broyden-Fletcher-Goldfarb-Shanno quasi-Newton scheme -- we use the implementation of these algorithms included in the GSL mathematical library.  Finally, note that for the real time parametrization mentioned above, some ad-hoc algorithms, developed already explicitly  forthe first applications of  QOCT, exist. These may in some circumstances be faster than the general purpose ones. Some of those have also been implemented in octopus. octopus (see, e.g. Refs~\cite{Zhu1998,Zhu1998a,Ohtsuki999}).  \item In order to compute the gradient, one needs to implement of a \emph{backwards-propagation} scheme for the costate, which obviously does not differ from the ones that are already implemented in octopus for the normal forwards propagation. Note, however that in some cases the backwards propagation does not have the exact same simple linear form than the forwards propagation, and may include inhomogeneous or non-linear terms.