An APL approach to differential calculus yields a powerful tool
APL '89 Conference proceedings on APL as a tool of thought
Differential equations are recurrence relations in APL
APL '92 Proceedings of the international conference on APL
An efficient method for the numerical evaluation of partial derivatives of arbitrary order
ACM Transactions on Mathematical Software (TOMS)
Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics) (Siam Studies in Applied Mathematics, 2.)
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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.