Differentiation in Pascal-SC: Type GRADIENT

Abstract

language used is Pascal-SC, and differentiation is implemented for variables of type GRADIENT, which consists of the value of a function of n real variables and its gradient vector of first partial derivatives with respect to the independent variables. Calculations of the results of operators or functions apphed to GRADIENT variables are carried out according to the well-known rules for evaluation and differentiation of sums, differences, products, and so on. Since the differentiation is performed at compile time, the code produced is comparable in compactness and execution time to that obtained if numerical approximations are used for derivatives, and the theoretical and practical problems associated with numerical differentiation are avoided. Pascal-SC source code is given for the necessary operators and standard functions, and it is shown how to prepare code for arbitrary differentiable functions to add to the hbrary if desired. The effectiveness of the use of type GRADIENT is shown by an example of the solution of a system of nonlinear equations by Newton's method. Categories and Subject Descriptors: G.1.5 [Numerical Analysis]: Roots of Nonlinear Equations — iterattve methods, systems of equations: G.1.6 [Numerical Analysis]: Optimization--gradient methods; G.l. © 1984, ACM. All rights reserved.