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In mathematics, Descartes' rule of signs, described by René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting zero coefficients), and the difference ...
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 ...
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
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. Descartes's rule of signs ...
Therefore, they require starting with an interval such that the function takes opposite signs at the end points of the interval. However, in the case of polynomials there are other methods such as Descartes' rule of signs , Budan's theorem and Sturm's theorem for bounding or determining the number of roots in an interval.
Sturm's theorem provides a way for isolating real roots that is less efficient (for polynomials with integer coefficients) than other methods involving Descartes' rule of signs. However, it remains useful in some circumstances, mainly for theoretical purposes, for example for algorithms of real algebraic geometry that involve infinitesimals .
If there are two or more sign variations Descartes' rule of signs implies that there may be zero, one or more real roots inside the interval (0, ∞); in this case consider separately the roots of p(x) that lie inside the interval (0, 1) from those inside the interval (1, ∞). A special test must be made for 1.
Li discovered independently an equivalent version of what is known today as Descartes' rule of signs. References. Joseph Warren Dauben and Christoph J. Scriba, ...