Vincent Alexandre Popular Books

Vincent Alexandre Biography & Facts

In mathematics, Vincent's theorem—named after Alexandre Joseph Hidulphe Vincent—is a theorem that isolates the real roots of polynomials with rational coefficients. Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials, it was almost totally forgotten, having been overshadowed by Sturm's theorem; consequently, it does not appear in any of the classical books on the theory of equations (of the 20th century), except for Uspensky's book. Two variants of this theorem are presented, along with several (continued fractions and bisection) real root isolation methods derived from them. Sign variation Let c0, c1, c2, ... be a finite or infinite sequence of real numbers. Suppose l < r and the following conditions hold: If r = l+1 the numbers cl and cr have opposite signs. If r ≥ l+2 the numbers cl+1, ..., cr−1 are all zero and the numbers cl and cr have opposite signs. This is called a sign variation or sign change between the numbers cl and cr. When dealing with the polynomial p(x) in one variable, one defines the number of sign variations of p(x) as the number of sign variations in the sequence of its coefficients. Two versions of this theorem are presented: the continued fractions version due to Vincent, and the bisection version due to Alesina and Galuzzi. Vincent's theorem: Continued fractions version (1834 and 1836) If in a polynomial equation with rational coefficients and without multiple roots, one makes successive transformations of the form x = a 1 + 1 x ′ , x ′ = a 2 + 1 x ″ , x ″ = a 3 + 1 x ‴ , … {\displaystyle x=a_{1}+{\frac {1}{x'}},\quad x'=a_{2}+{\frac {1}{x''}},\quad x''=a_{3}+{\frac {1}{x'''}},\ldots } where a 1 , a 2 , a 3 , … {\displaystyle a_{1},a_{2},a_{3},\ldots } are any positive numbers greater than or equal to one, then after a number of such transformations, the resulting transformed equation either has zero sign variations or it has a single sign variation. In the first case there is no root, whereas in the second case there is a single positive real root. Furthermore, the corresponding root of the proposed equation is approximated by the finite continued fraction: a 1 + 1 a 2 + 1 a 3 + 1 ⋱ {\displaystyle a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots }}}}}}} Moreover, if infinitely many numbers a 1 , a 2 , a 3 , … {\displaystyle a_{1},a_{2},a_{3},\ldots } satisfying this property can be found, then the root is represented by the (infinite) corresponding continued fraction. The above statement is an exact translation of the theorem found in Vincent's original papers; however, the following remarks are needed for a clearer understanding: If f n ( x ) {\displaystyle f_{n}(x)} denotes the polynomial obtained after n substitutions (and after removing the denominator), then there exists N such that for all n ≥ N {\displaystyle n\geq N} either f n ( x ) {\displaystyle f_{n}(x)} has no sign variation or it has one sign variation. In the latter case f n ( x ) {\displaystyle f_{n}(x)} has a single positive real root for all n ≥ N {\displaystyle n\geq N} .... Discover the Vincent Alexandre popular books. Find the top 100 most popular Vincent Alexandre books.