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Computer Algebra at RISC

Computer algebra is that part of computer science which designs, analyzes, implements, and applies algebraic algorithms. (Buchberger, Loos)

While it is arguable whether computer algebra is part of computer science or mathematics, we certainly agree with the rest of the statement. In fact, in our view computer algebra is a special form of scientific computation, and it comprises a wide range of basic goals, methods, and applications. In contrast to numerical computation the emphasis is on computing with symbols representing mathematical concepts. Of course that does not mean that computer algebra is devoid of computations with numbers. Decimal or other positional representations of integers, rational numbers and the like appear in any symbolic computation. But integers or real numbers are not the sole objects. In addition to these basic numerical entities computer algebra deals with polynomials, rational functions, trigonometric functions, algebraic numbers, etc. That does not mean that we will not need numerical algorithms any more. Both forms of scientific computation have their merits and they should be combined in a computational environment. For instance, in order to compute an approximate solution to a differential equation it might be reasonable to determine the first n terms of a power series solution by exact methods from computer algebra before handing these terms over to a numerical package for evaluating the power series.