A differential equation is stable if the roots of the characteristic polynomial are in the interior of the left half plane. Likewise a difference equation is stable if the roots of the characteristic polynomial are in the interior of the unit circle. This paper concerns algorithms which test polynomials for these properties. Also of concern is the relationship between the two problems. In particular, special numerical integration formulas are developed which transform a differential equation into a difference equation. These formulas are such that the differential equation and the corresponding difference equation are both stable or else they are both unstable.