Evaluating higher derivative tensors by forward propagation of univariate Taylor series

Authors:
Andreas Griewank, Jean Utke and Andrea Walther

Journal:
Math. Comp. **69** (2000), 1117-1130

MSC (1991):
Primary 65D05, 65Y20, 68Q40

Published electronically:
February 17, 2000

MathSciNet review:
1651755

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Abstract | References | Similar Articles | Additional Information

This article considers the problem of evaluating all pure and mixed partial derivatives of some vector function defined by an evaluation procedure. The natural approach to evaluating derivative tensors might appear to be their recursive calculation in the usual forward mode of computational differentiation. However, with the approach presented in this article, much simpler data access patterns and similar or lower computational counts can be achieved through propagating a family of univariate Taylor series of a suitable degree. It is applicable for arbitrary orders of derivatives. Also it is possible to calculate derivatives only in some directions instead of the full derivative tensor. Explicit formulas for all tensor entries as well as estimates for the corresponding computational complexities are given.

**1.**Christian H. Bischof, George F. Corliss, and Andreas Griewank,*Structured second- and higher-order derivatives through univariate Taylor series.*Optimization Methods and Software**2**(1993), 211-232.**2.**Martin Berz,*Differential algebraic description of beam dynamics to very high orders.*Particle Accelerators**12**(1989), 109-124.**3.**Andreas Griewank and George F. Corliss (eds.),*Automatic differentiation of algorithms*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1991. Theory, implementation, and application. MR**1143784****4.**Andreas Griewank, David Juedes, and Jean Utke,*ADOL-C: A package for the automatic differentiation of algorithms written in C/C++.*ACM Trans. Math. Software**22**(1996), 131-167.**5.**Andreas Griewank and George W. Reddien,*Computation of cusp singularities for operator equations and their discretizations*, J. Comput. Appl. Math.**26**(1989), no. 1-2, 133–153. Continuation techniques and bifurcation problems. MR**1007357**, 10.1016/0377-0427(89)90152-0**6.**Andreas Griewank,*On automatic differentiation*, Mathematical programming (Tokyo, 1988) Math. Appl. (Japanese Ser.), vol. 6, SCIPRESS, Tokyo, 1989, pp. 83–107. MR**1114312****7.**R. Seydel, F. W. Schneider, T. Küpper, and H. Troger (eds.),*Bifurcation and chaos: analysis, algorithms, applications*, International Series of Numerical Mathematics, vol. 97, Birkhäuser Verlag, Basel, 1991. MR**1109502****8.**Andreas Griewank, Computational differentiation and optimization. In J. Birge and K. Murty, editors,*Mathematical Programming: State of the Art*University of Michigan (1994), 102-131.**9.**Ulf Hutschenreiter, A new method for bevel gear tooth flank computation. In Martin Berz, Christian Bischof, George F. Corliss and Andreas Griewank, editors,*Computational Differentiation - Techniques, Applications, and Tools*SIAM (1996), 161-172.**10.**Richard D. Neidinger,*An efficient method for the numerical evaluation of partial derivatives of arbitrary order*, ACM Trans. Math. Software**18**(1992), no. 2, 159–173. MR**1167887**, 10.1145/146847.146924

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Additional Information

**Andreas Griewank**

Affiliation:
Institute of Scientific Computing, Technical University Dresden, D-01062 Dresden, Germany

Email:
griewank@math.tu-dresden.de

**Jean Utke**

Affiliation:
Framework Technologies, Inc., 10 South Riverside Plaza, Suite 1800, Chicago, Illinois 60606

Email:
utke@fti-consulting.com

**Andrea Walther**

Affiliation:
Institute of Scientific Computing, Technical University Dresden, D-01062 Dresden, Germany

Email:
awalther@math.tu-dresden.de

DOI:
http://dx.doi.org/10.1090/S0025-5718-00-01120-0

Keywords:
Higher order derivatives,
computational differentiation

Received by editor(s):
January 2, 1998

Received by editor(s) in revised form:
June 30, 1998

Published electronically:
February 17, 2000

Additional Notes:
This work was partially supported by the Deutsche Forschungsgesellschaft under grant GR 705/4-1.

Article copyright:
© Copyright 2000
American Mathematical Society