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2013, 2013(special): 587-596. doi: 10.3934/proc.2013.2013.587

## Efficient recurrence relations for univariate and multivariate Taylor series coefficients

 1 Davidson College, Box 7002, Davidson, NC 28035-7002, United States

Received  July 2012 Revised  April 2013 Published  November 2013

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.
Citation: Richard D. Neidinger. Efficient recurrence relations for univariate and multivariate Taylor series coefficients. Conference Publications, 2013, 2013 (special) : 587-596. doi: 10.3934/proc.2013.2013.587
##### References:
 [1] B. Altman, "Higher-Order Automatic Differentiation of Multivariate Functions in MATLAB," Undergraduate Honors Thesis, Davidson College, 2010. [2] F. Dangello and M. Seyfried, "Introductory Real Analysis," Houghton Mifflin, 2000, Section 7.4. [3] W. Dunham, "Euler: The Master of Us All," MAA, 1999, Chapter 3. [4] A. Griewank and A. Walther, "Evaluating Derivatives," 2nd edition, SIAM, 2008, Section 13.2. [5] M-J. Jang, C-L. Chen and Y-C. Liu, Two-dimensional differential transform for partial differential equations, Appl. Math. Computation, 121 (2001), 261-270. [6] R.E. Moore, "Methods and Applications of Interval Analysis," SIAM, 1979, Section 3.4. [7] R.D. Neidinger, Computing multivariable Taylor series to arbitrary order,, APL Quote Quad, 25 (1995), 134-144. [8] R.D. Neidinger, Automatic differentiation and MATLAB object-oriented programming, SIAM Review, 52 (2010), 545-563. [9] G.E. Parker and J.S. Sochacki, Implementing the Picard iteration, Neural, Parallel and Scientific Computation, 4 (1996), 97-112. [10] L.B. Rall, Early automatic differentiation: the Ch'in-Horneralgorithm, Reliable Computing, 13 (2007), 303-308. [11] J. Waldvogel, Der Tayloralgorithmus, J. Applied Math. and Physics (ZAMP), 35 (1984), 780-789.

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##### References:
 [1] B. Altman, "Higher-Order Automatic Differentiation of Multivariate Functions in MATLAB," Undergraduate Honors Thesis, Davidson College, 2010. [2] F. Dangello and M. Seyfried, "Introductory Real Analysis," Houghton Mifflin, 2000, Section 7.4. [3] W. Dunham, "Euler: The Master of Us All," MAA, 1999, Chapter 3. [4] A. Griewank and A. Walther, "Evaluating Derivatives," 2nd edition, SIAM, 2008, Section 13.2. [5] M-J. Jang, C-L. Chen and Y-C. Liu, Two-dimensional differential transform for partial differential equations, Appl. Math. Computation, 121 (2001), 261-270. [6] R.E. Moore, "Methods and Applications of Interval Analysis," SIAM, 1979, Section 3.4. [7] R.D. Neidinger, Computing multivariable Taylor series to arbitrary order,, APL Quote Quad, 25 (1995), 134-144. [8] R.D. Neidinger, Automatic differentiation and MATLAB object-oriented programming, SIAM Review, 52 (2010), 545-563. [9] G.E. Parker and J.S. Sochacki, Implementing the Picard iteration, Neural, Parallel and Scientific Computation, 4 (1996), 97-112. [10] L.B. Rall, Early automatic differentiation: the Ch'in-Horneralgorithm, Reliable Computing, 13 (2007), 303-308. [11] J. Waldvogel, Der Tayloralgorithmus, J. Applied Math. and Physics (ZAMP), 35 (1984), 780-789.
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