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 = "10991114",
doi = "10.1016/08981221(85)901889",
referred = "[Christianson1996SSU], [Irim91a], [Kubo91a], [Tesf91a], [Wexl87a].",
keywords = "point algorithm; differentiation arithmetic; forward method; back method.",
abstract = "Consider the vectorvalued 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"
}
