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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, (2010).

[2]

F. Dangello and M. Seyfried, "Introductory Real Analysis,", Houghton Mifflin, (2000).

[3]

W. Dunham, "Euler: The Master of Us All,", MAA, (1999).

[4]

A. Griewank and A. Walther, "Evaluating Derivatives,", 2nd edition, (2008).

[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.

[6]

R.E. Moore, "Methods and Applications of Interval Analysis,", SIAM, (1979).

[7]

R.D. Neidinger, Computing multivariable Taylor series to arbitrary order,,, APL Quote Quad, 25 (1995), 134.

[8]

R.D. Neidinger, Automatic differentiation and MATLAB object-oriented programming,, SIAM Review, 52 (2010), 545.

[9]

G.E. Parker and J.S. Sochacki, Implementing the Picard iteration,, Neural, 4 (1996), 97.

[10]

L.B. Rall, Early automatic differentiation: the Ch'in-Horneralgorithm,, Reliable Computing, 13 (2007), 303.

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J. Waldvogel, Der Tayloralgorithmus,, J. Applied Math. and Physics (ZAMP), 35 (1984), 780.

show all references

References:
[1]

B. Altman, "Higher-Order Automatic Differentiation of Multivariate Functions in MATLAB,", Undergraduate Honors Thesis, (2010).

[2]

F. Dangello and M. Seyfried, "Introductory Real Analysis,", Houghton Mifflin, (2000).

[3]

W. Dunham, "Euler: The Master of Us All,", MAA, (1999).

[4]

A. Griewank and A. Walther, "Evaluating Derivatives,", 2nd edition, (2008).

[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.

[6]

R.E. Moore, "Methods and Applications of Interval Analysis,", SIAM, (1979).

[7]

R.D. Neidinger, Computing multivariable Taylor series to arbitrary order,,, APL Quote Quad, 25 (1995), 134.

[8]

R.D. Neidinger, Automatic differentiation and MATLAB object-oriented programming,, SIAM Review, 52 (2010), 545.

[9]

G.E. Parker and J.S. Sochacki, Implementing the Picard iteration,, Neural, 4 (1996), 97.

[10]

L.B. Rall, Early automatic differentiation: the Ch'in-Horneralgorithm,, Reliable Computing, 13 (2007), 303.

[11]

J. Waldvogel, Der Tayloralgorithmus,, J. Applied Math. and Physics (ZAMP), 35 (1984), 780.

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