The oldest method for computing the number of real roots, and the number of roots in an interval results from Sturm's theorem, but the methods based on Descartes' rule of signs and its extensions—Budan's and Vincent's theorems—are generally more efficient. For root finding, all proceed by reducing the size of the intervals in which roots are searched until getting intervals containing zero or one root. Then the intervals containing one root may be further reduced for getting a quadratic convergence of Newton's method to the isolated roots. The main computer algebra systems (Maple, Mathematica, SageMath, PARI/GP) have each a variant of this method as the default algorithm for the real roots of a polynomial.