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Descartes' rule of signs – Counting polynomial real roots based on coefficients; Marden's theorem – On zeros of derivatives of cubic polynomials; Newton's identities – Relations between power sums and elementary symmetric functions; Quadratic function#Upper bound on the magnitude of the roots
Descartes' rule of signs asserts that the difference between the number of sign variations in the sequence of the coefficients of a polynomial and the number of its positive real roots is a nonnegative even integer. It results that if this number of sign variations is zero, then the polynomial does not have any positive real roots, and, if this ...
1) has exactly 0 or 1 sign variations . The second case is possible if and only if p (x) has a single root within (a , b). The Alesina–Galuzzi "a_b roots test" From equation (1) the following criterion is obtained for determining whether a polynomial has any roots in the interval (a , b): Perform on p (x) the substitution x ← a + b x 1 + x {\displaystyle x\leftarrow {\frac {a+bx}{1+x ...
All results described in this article are based on Descartes' rule of signs. If p(x) is a univariate polynomial with real coefficients, let us denote by # + (p) the number of its positive real roots, counted with their multiplicity, [1] and by v(p) the number of sign variations in the sequence of its coefficients.
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 ...
Taken together with Descartes' Rule of Signs, this leads to an upper bound on the number of the real roots a polynomial has inside an open interval. Although Budan's Theorem , as this result was known, was taken up by, among others, Pierre Louis Marie Bourdon (1779-1854), in his celebrated algebra textbook, it tended to be eclipsed by an ...
Root-finding algorithm. In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f, from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number x such that f(x) = 0. As, generally, the zeros of a function cannot ...
First an upper bound on the number of solutions is computed. This bound has to be as sharp as possible. ... [15] and based on Descartes' rule of signs. This ...