Richard D. Neidinger
Abstract
Automatic differentiation, manipulating numerical vectors of coefficients, is the efficient way to compute multivariable Taylor series. This does not require symbolic differentiation or numerical approximation but uses exact formulas applied to numerical arrays. Arrays of Taylor series coefficients of any elementary function can be built-up, as the array for each component (combination or function) is a combination of its argument arrays. The functions TIMES and EXP display the algorithmic ideas that enable all of the other standard functions. We study the interesting recursive formulas for these combinations, the resulting algorithms, and the implementation in APL. To handle all coefficients in n variables up to order m, the arrays are hyper-pyramid data structures, considered conceptually as n-dimensional but implemented as one-dimensional arrays. Unlike previous work, this implementation does not require huge arrays for binomial coefficients and indirect referencing. This APL*PLUS III implementation loops through one nested reference array and takes sub-arrays from another for a practical solution to this problem that can make tremendous demands on time and space.
- 1 Berz, M. Differential algebraic description of beam dynamics to very high orders. Particle Accelerators, 24 (2), 1989, 109-124.Google Scholar
- 2 Bischof, C., Corliss, G., and Griewank, A. Structured second- and higher-order derivatives through univariate Taylor series. Optimization Methods and Software, 2, 1993, 211-232.Google ScholarCross Ref
- 3 Christianson, B. Reverse accumulation and accurate rounding error estimates for Taylor series coefficients. Optimization Methods and Software, 1 (1), 1992, 81-94.Google ScholarCross Ref
- 4 Juedes, D.W. A taxonomy of automatic differentiation tools. Automatic Differentiation of A Igorithms, Edited by Griewank and Corliss, SIAM, Philadelphia, 1991.Google Scholar
- 5 Kalman, D. and Lindell, R. A recursive approach to multivariate automatic differentiation. Preprint.Google Scholar
- 6 Moore, R.E. Methods and applications of interval analysis. SIAM Studies in Applied Mathematics, 2, SIAM, Philadelphia, Pa., 1979, 24-31. Google ScholarDigital Library
- 7 Neidinger, R.D. Automatic differentiation and APL. College Mathematics J., 20 (3), May 1989, 238-251.Google ScholarCross Ref
- 8 Neidinger, R.D. An APL approach to differential calculus yields a powerful tool. A PL89 Proceedings, A PL Quote Quad, 19, 1989, 285-288. Google ScholarDigital Library
- 9 Neidinger, R.D. Differential equations are recurrence relations in APL. A PL92 Proceedings, A PL Quote Quad, 23 (1), July 1992, 165-174. Google ScholarDigital Library
- 10 Neidinger, R.D. An efficient method for the numerical evaluation of partial derivatives of arbitrary order. A CM Transactions on Mathematical Software, 18 (2), June 1992, 159- 173. Google ScholarDigital Library
Index Terms
- Computing multivariable Taylor series to arbitrary order
Recommendations
Computing multivariable Taylor series to arbitrary order
APL '95: Proceedings of the international conference on Applied programming languagesAutomatic differentiation, manipulating numerical vectors of coefficients, is the efficient way to compute multivariable Taylor series. This does not require symbolic differentiation or numerical approximation but uses exact formulas applied to ...
Application of Taylor series in obtaining the orthogonal operational matrix
In this research first we explicitly obtain the relation between the coefficients of the Taylor series and Jacobi polynomial expansions. Then we present a new method for computing classical operational matrices (derivative, integral and product) for ...
On truncated Taylor series and the position of their spurious zeros
A truncated Taylor series, or a Taylor polynomial, which may appear when treating the motion of gravity water waves, is obtained by truncating an infinite Taylor series for a complex, analytical function. For such a polynomial the position of the ...
Comments