Formal calculation systems

cientific computation on the computer has two different and complementary aspects.

On the one hand, there is traditional numerical calculation, which exploits the ability of computers to rapidly perform a large number of arithmetic operations on real numbers. The results of this type of calculation are approximate and depend on the choice of real number representation and the way in which arithmetic operations are performed. For numerical calculations, the accumulation of rounding errors and problems of poor conditioning are frequent obstacles to obtaining reproducible, reliable results.

On the other hand, symbolic or formal calculation consists in having the computer perform exact mathematical calculations: developments, transformations, simplifications of formulas. A typical aspect of formal calculation is that the symbols in the formulas are not necessarily replaced by specific numerical values, but are retained during the calculation process. All automatic calculations, i.e. those whose steps obey a precise and coherent set of rules, can be carried out by computer. Common engineering formulas, such as differential and integral calculus, can be processed by computer programs with far greater reliability than manual calculations, and with incomparably greater limits on the size of the calculations.

Formal calculus is a fast-growing field of research. In all sectors where mathematicians obtain constructive results, new specific algorithms are developed, transformed into programs and tested.

A computer algebra system is a software package containing either symbolic calculation programs for a specialized field, or programs that are general enough to perform all the mathematical calculations normally performed by mathematical engineers. A formal calculation system is presented to the user in the form of a set of commands and a more or less elaborate language enabling him to organize his calculations, and sometimes even to extend the system itself. The most widespread calculus systems also contain numerical programs and graphical utilities to facilitate interpretation and exploitation of results.