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

DOI:
https://doi.org/10.1090/S0025-5718-04-01657-6

Published electronically:
May 17, 2004

MathSciNet review:
2085414

Full-text PDF Free Access

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 variables by fixing a direction in . 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 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