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Directions for computing truncated multivariate Taylor series


Author: Richard D. Neidinger
Journal: Math. Comp. 74 (2005), 321-340
MSC (2000): Primary 65D25, 65D05, 41A05, 41A63, 65Y20
Published electronically: May 17, 2004
MathSciNet review: 2085414
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Abstract | References | Similar Articles | Additional Information

Abstract: Efficient recurrence relations for computing arbitrary-order Taylor coefficients for any univariate function can be directly applied to a function of $n$ variables by fixing a direction in $\mathbb{R} ^{n}$. After a sequence of directions, the multivariate Taylor coefficients or partial derivatives can be reconstructed or ``interpolated''. The sequence of univariate calculations is more efficient than multivariate methods, although previous work indicates a space cost for this savings and significant cost for the reconstruction. We completely eliminate this space cost and develop a much more efficient algorithm to perform the reconstruction. By appropriate choice of directions, the reconstruction reduces to a sequence of Lagrange polynomial interpolation problems in $\mathbb{R} ^{n-1}$ for which a divided difference algorithm computes the coefficients of a Newton form. Another algorithm collects like terms from the Newton form and returns the desired multivariate coefficients.


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Additional Information

Richard D. Neidinger
Affiliation: Department of Mathematics, Davidson College, Box 7002, Davidson, North Carolina 28035
Email: rineidinger@davidson.edu

DOI: https://doi.org/10.1090/S0025-5718-04-01657-6
Keywords: Automatic differentiation, multivariate, polynomial interpolation, higher-order derivatives, divided difference
Received by editor(s): May 28, 2002
Received by editor(s) in revised form: June 10, 2003
Published electronically: May 17, 2004
Article copyright: © Copyright 2004 American Mathematical Society