BibTeX
@ARTICLE{
         Volin1985Aco,
       author = "{Yu.} M. Volin and G. M. Ostrovskii",
       title = "Automatic computation of derivatives with the use of the multilevel differentiating
         technique --- {I}: {A}lgorithmic basis",
       journal = "Computers and Mathematics with Applications",
       volume = "11",
       year = "1985",
       pages = "1099--1114",
       doi = "10.1016/0898-1221(85)90188-9",
       referred = "[Christianson1996SSU], [Irim91a], [Kubo91a], [Tesf91a], [Wexl87a].",
       keywords = "point algorithm; differentiation arithmetic; forward method; back method.",
       abstract = "Consider the vector-valued function $$ y = f(x), \quad f : R^m
         \rightarrow R. $$ The authors consider calculating the partial derivatives $$
         \frac{\partial y}{\partial x_i} = \frac{\partial f}{\partial x_i} $$
         for each $ 1 \le i \le m $. The first method of calculation, called the {\sl forward
         method \/}, uses the variational form $$ \partial y = \frac{\partial
         f}{\partial x_i} \partial x_i $$. The second method of calculation, called the {\sl
         back method \/}, uses the adjoint variational form $$ \partial \overline{x}_i =
         \left(\frac{\partial f} {\partial x_i}ight)^T \partial \overline{y} $$
         where $\overline{.}$ means adjoint variable and sign T means transposition. The forward method
         has been discussed elsewhere in the literature, see [Rall81a]. The authors discuss the relative
         merits of both methods. They provide results which show that the forward method is the more space
         efficient and the back method is the more time efficient. The authors combine the two methods to
         produce a {\sl multilevel method \/} which is a compromise of the two methods. This has
         the property that the memory requirements are better than the back method and the time requirements
         are better than the forward method.",
       ad_theotech = "General"
}