DIFFERENTIATION AND GENERATION OF TAYLOR COEFFICIENTS IN PASCAL-SC

Abstract

Evaluation of derivatives and Taylor coefficients of functions defined by computer programs has many applications in scientific computation. The process of automatic differentiation of such functions has as its goal the production of machine code for the evaluation of derivatives and Taylor coefficients. This is in contrast to symbolic differentiation, where the desired output is a more or less pretty formula, and numerical differentiation, which is inaccurate and unstable. Another distinction between automatic and symbolic differentiation is that the latter usually involves considerable computational overhead, while automatic differentiation can be carried out at compile time by a compiler which permits user-defined data types and operators. This paper shows how PASCAL-SC, a compiler of this type, can be used to generate the real derivative types GRADIENT, HESSIAN, TAYLOR, and the corresponding interval types IGRADIENT, IHESSIAN, ITAYLOR. Applications of these types to solution of systems of nonlinear equations, sensitivity analysis, constrained and unconstrained optimization, and the solution of initial-value problems for systems of ordinary differential equations are indicated.