Sturm's Theorem


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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 ≤ ks –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).

References in periodicals archive ?
Then, using Sturm's theorem [7], which states that all the coefficients of a modular form of weight k/2 with respect to [GAMMA] are divisible by a prime p if and only if the first
More specifically, the paper demonstrates the applicability of Descartes' Rule of Signs, Budan's Theorem, and Sturm's Theorem from the theory of equations and rules developed in the business literature by Teichroew, Robichek, and Montalbano (1965a, 1965b), Mao (1969), Jean (1968, 1969), and Pratt and Hammond (1979).
Of particular interest are Descartes' Rule of Signs, Budan's Theorem, and Sturm's Theorem from the theory of equations, and the rules developed in the business literature by Teichroew, Robichek, and Montalbano (1965a, 1965b), Mao (1969), Jean (1968, 1969), and Pratt and Hammond (1979).
Sturm's Theorem is more awkward to apply and requires greater computational skill than Budan's Theorem.
In applying Sturm's Theorem, it is necessary to realize that:
Thus, using Sturm's Theorem, the number of positive internal rates of return is determined for the pump problem as follows:
Finally, Sturm's Theorem determined that each of the two roots is in fact distinct.
The Pratt and Hammond answer includes multiple roots and establishes only an upper bound, which may be substantially larger than the actual number of positive roots, while Sturm's Theorem gives the exact number of distinct roots and does not restrict itself to only positive roots.
Sturm's Theorem provides the most practical information regarding the number of internal rates of return and their bounds.