×

zbMATH — the first resource for mathematics

Evaluating higher derivative tensors by forward propagation of univariate Taylor series. (English) Zbl 0952.65028
The paper describes a new approach to compute higher order derivatives of a function of several variables. The approach is based on the calculation of one-dimensional Taylor developments of the function along conveniently selected directions, i.e. directional Taylor developments. The directions are given by the lattice points of nonnegative integers whose coordinates add up to a fixed derivative order. It is shown that those developments allow to interpolate all the partial derivatives whose degrees are smaller or equal than the order considered initially.
The paper includes a discussion and estimates of the computational complexities, such as data structures and memory access pattern, as well as some time run results, which show that the proposed approach can be useful in applications that require the calculation of higher order derivatives in several variables.

MSC:
65D25 Numerical differentiation
65F30 Other matrix algorithms (MSC2010)
68W30 Symbolic computation and algebraic computation
65Y20 Complexity and performance of numerical algorithms
65T99 Numerical methods in Fourier analysis
Software:
ADOL-C
PDF BibTeX XML Cite
Full Text: DOI
References:
[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 , Automatic differentiation of algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1991. Theory, implementation, and application. · Zbl 0747.00030
[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. · Zbl 0884.65015
[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. · Zbl 0672.65031 · doi:10.1016/0377-0427(89)90152-0 · doi.org
[6] Andreas Griewank, On automatic differentiation, Mathematical programming (Tokyo, 1988) Math. Appl. (Japanese Ser.), vol. 6, SCIPRESS, Tokyo, 1989, pp. 83 – 107. · Zbl 0696.65015
[7] R. Seydel, F. W. Schneider, T. Küpper, and H. Troger , Bifurcation and chaos: analysis, algorithms, applications, International Series of Numerical Mathematics, vol. 97, Birkhäuser Verlag, Basel, 1991. · Zbl 0718.58003
[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. · Zbl 0895.70002
[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. · Zbl 0892.65011 · doi:10.1145/146847.146924 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.