Publication: Efficient recurrence relations for univariate and multivariate Taylor series coefficients
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Efficient recurrence relations for univariate and multivariate Taylor series coefficients

- Article in a journal -
 

Author(s)
Richard D. Neidinger

Published in
Dynamical Systems and Differential Equations

Year
2013

Publisher
AIMS

Abstract
The efficient use of Taylor series depends, not on symbolic differentiation, but on a standard set of recurrence formulas for each of the elementary functions and operations. These relationships are often rediscovered and restated, usually in a piecemeal fashion. We seek to provide a fairly thorough and unified exposition of efficient recurrence relations in both univariate and multivariate settings. Explicit formulas all stem from the fact that multiplication of functions corresponds to a Cauchy product of series coefficients, which is more efficient than the Leibniz rule for nth-order derivatives. This principle is applied to function relationships of the form h'=v*u', where the prime indicates a derivative or partial derivative. Each standard (calculator button) function corresponds to an equation, or pair of equations, of this form. A geometric description of the multivariate operation helps clarify and streamline the computation for each desired multi-indexed coefficient. Several research communities use such recurrences including the Differential Transform Method to solve differential equations with initial conditions.

AD Theory and Techniques
General, Taylor Arithmetic

BibTeX
@ARTICLE{
         Neidinger2013Err,
       title = "Efficient recurrence relations for univariate and multivariate Taylor series
         coefficients",
       author = "Richard D. Neidinger",
       publisher = "AIMS",
       year = "2013",
       journal = "Dynamical Systems and Differential Equations",
       volume = "Special, Proceedings of the 9th AIMS International Conference (Orlando, USA)",
       pages = "587--596",
       abstract = "The efficient use of Taylor series depends, not on symbolic differentiation, but on
         a standard set of recurrence formulas for each of the elementary functions and operations. These
         relationships are often rediscovered and restated, usually in a piecemeal fashion. We seek to
         provide a fairly thorough and unified exposition of efficient recurrence relations in both
         univariate and multivariate settings. Explicit formulas all stem from the fact that multiplication
         of functions corresponds to a Cauchy product of series coefficients, which is more efficient than
         the Leibniz rule for nth-order derivatives. This principle is applied to function relationships of
         the form h'=v*u', where the prime indicates a derivative or partial derivative. Each
         standard (calculator button) function corresponds to an equation, or pair of equations, of this
         form. A geometric description of the multivariate operation helps clarify and streamline the
         computation for each desired multi-indexed coefficient. Several research communities use such
         recurrences including the Differential Transform Method to solve differential equations with initial
         conditions.",
       url = "http://aimsciences.org/journals/displayPaperPro.jsp?paperID=9241",
       ad_theotech = "General, Taylor Arithmetic"
}


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