Sturm's Theorem


Sturm's theorem

[′stərmz ‚thir·əm] (mathematics) This gives a method to determine the number of real roots of a polynomial p (x) which lie between two given values of x ; the Sturm sequence of p (x) provides the necessary information.

Sturm’s Theorem

 

a theorem that provides a basis for finding nonoverlapping intervals such that each contains one real root of a given algebraic polynomial with real coefficients. The theorem was given in 1829 by J. C. F. Sturm.

For any polynomial f(x) without multiple roots, there exists a system of polynomials

f(x) = f0(x), f1(x),...,fs(x)

for which the following conditions are fulfilled: (1) fk(x) and fk+1(x), k = 0, 1,..., s – 1, do not have common roots; (2) the polynomial fs (x) has no real roots; (3) it follows from fk (α) = 0, 1 ≤ k ≤ s –1 that fk–1 (α)fk+1 (α) < 0; and (4) it follows from f(α) = 0 that the product f(x)f1(x) is increasing at the point α. Let w(c) be the number of changes of sign in the system

f(c), f1(c),...,fs(c)

If the real numbers a and b (a < b) are not roots of the polynomial f(x), then the difference w(a) – w(b) is nonnegative and equal to the number of real roots of the polynomial f(x) that lie between a and b. Thus, the number line may be divided into intervals each of which contains one real root of the polynomial f(x).