On the Use of Transcendentals for Program Testing

The element z is called a transcendental for the class F if functions in F can be uniquely identified by their values at z. Conditions for the existence of transcendentals are discussed for certain classes of polynomials, multinomials, and rational functions. Of particular interest are those transcendentals having an exact representation in computer arithmetic. Algorithms are presented for reconstruction of the coefficients of a polynomial from its value at a transcendental. The theory is illustrated by application to polynomials, quadratic forms, and quadrature formulas.