Methods are presented for the automatic preparation of functions of one or more variables for economical calculation by high-speed digital computers. The cost of calculation is considered according to the factors of number of functions, complexity, requirements for precision, and the frequency with which functions are to be calculated. Contrary to classic approaches, con sideration is not given to minimizing computational error for its own sake. On the contrary, the maximum allowable error may be sought in order to minimize computational costs. In this respect, each function is represented by an error envelope that specifies the required limits of computational precision. It is the error envelope rather than the function itself which is dealt with. The approximation techniques dealt with in this paper are limited to piecewise linear ap proximation of functions of one or two independent variables. Projects requiring the maintaining and computation of large quantities of continuous functions are fre quently to be found in industry and research; for example, in the simulation of real-time processes— aircraft flight and flight trainer simulations, simula tion for control and regulation of continuous pro cesses as in chemical plants, weather calculations, radiation studies, etc. In addition, computer service centers, providing computational services to many users, may extend the range and effectiveness of their mathematical function program library by the use of the economical com putational methods of this paper.
Continuous functions of bounded nth variation
