BibTeX
@ARTICLE{
         Iri1984SCo,
       author = "Masao Iri",
       title = "Simultaneous Computation of Functions, Partial Derivatives, and Estimates of Rounding
         Errors --- Complexity and Practicality ---",
       journal = "Japan Journal of Applied Mathematics",
       year = "1984",
       volume = "1",
       number = "2",
       pages = "223--252",
       issn = "1868-937X",
       doi = "10.1007/BF03167059",
       referred = "[Christianson1996SSU], [Kubota1996PFP], [Rall1996AIt].",
       keywords = "differentiation arithmetic; error estimates; algebraic complexity.",
       abstract = "This paper proves theoretical results about the algebraic complexity involved in
         calculating the partial derivatives of a function. It is shown that the algebraic complexity of
         computing a function of several variables and its partial derivatives with respect to all of the
         variables is at most a constant (four, five, six or seven which is independent of the number of
         variables) times as large as that of computing the function alone. Furthermore, it is seen that a
         reasonable bound for the rounding error incurred in the final function value is ready to obtain once
         the function and its derivatives have been computed according to the procedure described."
}