BibTeX
@ARTICLE{
         Neidinger1992AEM,
       author = "Neidinger, Richard D.",
       title = "An Efficient Method for the Numerical Evaluation of Partial Derivatives of Arbitrary
         Order",
       journal = "ACM Trans. Math. Software",
       volume = "18",
       number = "2",
       year = "1992",
       month = "June",
       pages = "159--173",
       url = "http://doi.acm.org/10.1145/146847.146924",
       abstract = "For any typical multivariable expression $f$, point $a$ in the domain of $f$, and
         positive integer maxorder, this method produces the numerical values of all partial derivatives at
         $a$ up through order maxorder. By the technique known as automatic differentiation, theoretically
         exact results are obtained using numerical (as opposed to symbolic) manipulation. The key ideas are
         a hyperpyramid data structure and a generalized Leibniz's rule. Any expression in $n$ variables
         corresponds to a hyperpyramid array, in $n$-dimensional space, containing the numerical values of
         all unique partial derivatives (not wasting space on different permutations of derivatives). The
         arrays for simple expressions are combined by hyperpyramid operators to form the arrays for more
         complicated expressions. These operators are facilitated by a generalized Leibniz's rule which,
         given a product of multivariable functions, produces any partial derivative by forming the minimum
         number of products (between two lower partials) together with a product of binomial coefficients.
         The algorithms are described in abstract pseudo-code. A section on implementation shows how these
         ideas can be converted into practical and efficient programs in a typical computing environment. For
         any specific problem, only the expression itself would require recoding.",
       notes = "Also appeared as Preprint, Davidson College, Davidson, N.C., 1990.",
       ad_theotech = "General"
}