Differentiation in PASCAL-SC: type GRADIENT
ACM Transactions on Mathematical Software (TOMS)
Journal of Optimization Theory and Applications
Numerical derivatives and nonlinear analysis
Numerical derivatives and nonlinear analysis
Automatic evaluation of derivatives
Applied Mathematics and Computation
An algorithm for exact evaluation of multivariate functions and their derivatives to any order
Computational Statistics & Data Analysis
APL '88 Proceedings of the international conference on APL
Automatic Differentiation of Computer Programs
ACM Transactions on Mathematical Software (TOMS)
A simple automatic derivative evaluation program
Communications of the ACM
Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics) (Siam Studies in Applied Mathematics, 2.)
APL '93 Proceedings of the international conference on APL
ACM Transactions on Mathematical Software (TOMS)
Computing multivariable Taylor series to arbitrary order
APL '95 Proceedings of the international conference on Applied programming languages
Implicit extension of Taylor series method for initial value problems
ICCMSE '03 Proceedings of the international conference on Computational methods in sciences and engineering
Taylor series method with numerical derivatives for initial value problems
Journal of Computational Methods in Sciences and Engineering - Computational and Mathematical Methods for Science and Engineering Conference 2002 - CMMSE-2002
Implicit extension of taylor series method for initial value problems
Journal of Computational Methods in Sciences and Engineering
Implicit extension ofTaylor series method with numerical derivatives for initial value problems
Computers & Mathematics with Applications
Algorithm 924: TIDES, a Taylor Series Integrator for Differential EquationS
ACM Transactions on Mathematical Software (TOMS)
Computers & Mathematics with Applications
Journal of Computational and Applied Mathematics
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For any typical multivariable expression f, point a in the domain of f, and positive integer maxorder, this method produces the numerical values of all partial derivatives at a up through order maxorder. By the technique known as automatic differentiation, theoretically exact results are obtained using numerical (as opposed to symbolic) manipulation. The key ideas are a hyperpyramid data structure and a generalized Leibniz's rule. Any expression in n variables corresponds to a hyperpyramid array, in n-dimensional space, containing the numerical values of all unique partial derivatives (not wasting space on different permutations of derivatives). The arrays for simple expressions are combined by hyperpyramid operators to form the arrays for more complicated expressions. These operators are facilitated by a generalized Leibniz's rule which, given a product of multivariable functions, produces any partial derivative by forming the minimum number of products (between two lower partials) together with a product of binomial coefficients. The algorithms are described in abstract pseudo-code. A section on implementation shows how these ideas can be converted into practical and efficient programs in a typical computing environment. For any specific problem, only the expression itself would require recoding.