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Dandelin-Gräffe Method
In mathematics, Graeffe's method or Dandelin–Lobachesky–Graeffe method is an algorithm for finding all of the roots of a polynomial. It was developed independently by Germinal Pierre Dandelin in 1826 and Lobachevsky in 1834. In 1837 Karl Heinrich Gräffe also discovered the principal idea of the method. The method separates the roots of a polynomial by squaring them repeatedly. This squaring of the roots is done implicitly, that is, only working on the coefficients of the polynomial. Finally, Viète's formulas are used in order to approximate the roots. Dandelin–Graeffe iteration Let be a polynomial of degree :p(x) = (x-x_1)\cdots(x-x_n). Then :p(-x) = (-1)^n (x+x_1)\cdots(x+x_n). Let be the polynomial which has the squares x_1^2, \cdots, x_n^2 as its roots, :q(x)= \left (x-x_1^2 \right )\cdots \left (x-x_n^2 \right ). Then we can write: :\begin q(x^2) & = \left (x^2-x_1^2 \right )\cdots \left (x^2-x_n^2 \right ) \\ & = (x-x_1)(x+x_1) \cdots (x-x_n) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root-finding Algorithm
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number such that . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeros, expressed either as floating-point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound). Solving an equation is the same as finding the roots of the function . Thus root-finding algorithms allow solving any equation defined by continuous functions. However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists. Most nume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germinal Pierre Dandelin
Germinal Pierre Dandelin (12 April 1794 – 15 February 1847) was a French mathematician, soldier, and professor of engineering. Life He was born near Paris to a French father and Belgian mother, studying first at Ghent then returning to Paris to study at the École Polytechnique. He was wounded fighting under Napoleon. He worked for the Ministry of the Interior under Lazare Carnot. Later he became a citizen of the Netherlands, a professor of mining engineering in Belgium, and then a member of the Belgian army. Work He is the eponym of the Dandelin spheres, of Dandelin's theorem in geometry (for an account of that theorem, see Dandelin spheres), and of the Dandelin–Gräffe numerical method of solution of algebraic equations. He also published on the stereographic projection, algebra, and probability theory. References * Biography in ''Dictionary of Scientific Biography'' (New York 1970–1990). * Florian Cajori, ''The Dandelin–Gräffe method'', in ''A his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lobachevsky
Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for his fundamental study on Dirichlet integrals, known as the Lobachevsky integral formula. William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. Biography Nikolai Lobachevsky was born either in or near the city of Nizhny Novgorod in the Russian Empire (now in Nizhny Novgorod Oblast, Russia) in 1792 to parents of Russian and Polish origin – Ivan Maksimovich Lobachevsky and Praskovia Alexandrovna Lobachevskaya.Victor J. Katz. ''A history of mathematics: Introduction''. Addison-Wesley. 2009. p. 842. Stephen Hawking. ''God Created the Integers: The Mathematical Br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Heinrich Gräffe
Karl Heinrich Gräffe (7 November 1799 – 2 December 1873) was a German mathematician, who was professor at University of Zurich. Life and work Gräffe's father migrated to North America, leaving the family business of jewelry in his hands. Even so, Gräffe succeeded, studying at night, entering the '' Carolineum'' of Brunswick in 1821. From 1823, he studied at the University of Göttingen with professors Gauss and Thibaut, doctorate in 1825. In 1828 he was appointed professor of the Zurich Institute of Technology and, as of 1833, associate professor at the University of Zurich from the date of its creation. Simultaneously, also he was professor of the ''Obere Industrieschule''. Gräffe is known for having been the first to enunciate a method to approximate the roots of any polynomial, a method known today as the Dandelin-Gräffe method In mathematics, Graeffe's method or Dandelin–Lobachesky–Graeffe method is an algorithm for finding all of the roots of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viète's Formulas
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta"). Basic formulas Any general polynomial of degree ''n'' :P(x) = a_nx^n + a_x^ + \cdots + a_1 x + a_0 (with the coefficients being real or complex numbers and ) has (not necessarily distinct) complex roots by the fundamental theorem of algebra. Vieta's formulas relate the polynomial's coefficients to signed sums of products of the roots as follows: :\begin r_1 + r_2 + \dots + r_ + r_n = -\dfrac \\ (r_1 r_2 + r_1 r_3+\cdots + r_1 r_n) + (r_2r_3 + r_2r_4+\cdots + r_2r_n)+\cdots + r_r_n = \dfrac \\ \quad \vdots \\ r_1 r_2 \dots r_n = (-1)^n \dfrac. \end Vieta's formulas can equivalently be written as : \sum_ \left(\prod_^k r_\right)=(-1)^k\frac for (the indices are sorted in increasing order to ensure each product of roots is used exactly once ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
