A NORMAL-FORM FOR FUNCTION RINGS OF PIECEWISE FUNCTIONS

Computer algebra systems often have to deal with piecewise continuous functions. These are, for example, the absolute value function, signum , piecewise defined functions but also functions that are the supremum or infimum of two functions. We present a new algebraic approach to t hese types of problems. This paper presents a normal form for a functi on ring containing piecewise polynomial functions of a real variable. We give a complete rule system to compute the normal form of an expres sion. The main result is that this normal form can be used to decide e xtensional equality of two piecewise functions. Also we define supremu m and infimum for piecewise functions; in fact, we show that the funct ion ring forms a lattice. Additionally, a method to solve equalities a nd inequalities in this function ring is presented. Finally, we give a ''user interface'' to the algebraic representation of the piecewise f unctions. (C) 1998 Academic Press.