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Real-root Isolation
In mathematics, and, more specifically in numerical analysis and computer algebra, real-root isolation of a polynomial consist of producing disjoint intervals of the real line, which contain each one (and only one) real root of the polynomial, and, together, contain all the real roots of the polynomial. Real-root isolation is useful because usual root-finding algorithms for computing the real roots of a polynomial may produce some real roots, but, cannot generally certify having found all real roots. In particular, if such an algorithm does not find any root, one does not know whether it is because there is no real root. Some algorithms compute all complex roots, but, as there are generally much fewer real roots than complex roots, most of their computation time is generally spent for computing non-real roots (in the average, a polynomial of degree has complex roots, and only real roots; see ). Moreover, it may be difficult to distinguish the real roots from the non-real roots w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Worst-case Complexity
In computer science (specifically computational complexity theory), the worst-case complexity measures the resources (e.g. running time, memory) that an algorithm requires given an input of arbitrary size (commonly denoted as in asymptotic notation). It gives an upper bound on the resources required by the algorithm. In the case of running time, the worst-case time complexity indicates the longest running time performed by an algorithm given ''any'' input of size , and thus guarantees that the algorithm will finish in the indicated period of time. The order of growth (e.g. linear, logarithmic) of the worst-case complexity is commonly used to compare the efficiency of two algorithms. The worst-case complexity of an algorithm should be contrasted with its average-case complexity, which is an average measure of the amount of resources the algorithm uses on a random input. Definition Given a model of computation and an algorithm \mathsf that halts on each input s, the mapping ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sketch Of Proof
Sketch or Sketches may refer to: * Sketch (drawing), a rapidly executed freehand drawing that is not usually intended as a finished work Arts, entertainment and media * Sketch comedy, a series of short scenes or vignettes called sketches Film and television * ''Sketch'' (2007 film), a Malayalam film * ''Sketch'' (2018 film), a Tamil film * ''Sketch'' (2024 film), an American comedy horror film * ''Sketch'' (TV series), a 2018 South Korean series * "Sketch", a 2008 episode of ''Skins'' ** Sketch (''Skins'' character) * Sketch with Kevin McDonald, a 2006 CBC television special Literature * Sketch story, or sketch, a very short piece of writing * ''Daily Sketch'', a British newspaper 1909–1971 * ''The Sketch'', a British illustrated weekly journal 1893–1959 Music * Sketch (music), an informal document prepared by a composer to assist in composition * The Sketches, a Pakistani Sufi folk rock band * ''Sketch'' (Ex Norwegian album), 2011 * ''Sketch'' (Lilas Ikuta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Ostrowski
Alexander Markowich Ostrowski (; ; 25 September 1893 – 20 November 1986) was a mathematician. Biography His father Mark having been a merchant, Alexander Ostrowski attended the Kiev College of Commerce, not a high school, and thus had an insufficient qualification to be admitted to university. However, his talent did not remain undetected: Ostrowski's mentor, Dmitry Grave, wrote to Edmund Landau and Kurt Hensel for help. Subsequently, Ostrowski began to study mathematics at Marburg University under Hensel's supervision in 1912. During World War I he was interned, but thanks to the intervention of Hensel, the restrictions on his movements were eased somewhat, and he was allowed to use the university library. After the war ended, Ostrowski moved to Göttingen where he wrote his doctoral dissertation and was influenced by David Hilbert, Felix Klein, and Landau. In 1920, after having obtained his doctorate from the University of Göttingen, Ostrowski moved to Hamburg where he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikola Obreshkov
Nikola Dimitrov Obreshkov (; March 6, 1896 in VarnaAugust 11, 1963 in Sofia) was a prominent Bulgarian mathematician, working in complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic .... See also * Obreschkoff–Ostrowski theorem References European Mathematics Society Newsletter No. 51(PDF), page 28. * 20th-century Bulgarian mathematicians 1896 births 1963 deaths Members of the Bulgarian Academy of Sciences People from Varna, Bulgaria {{Bulgaria-scientist-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continued Fraction
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite. Different fields of mathematics have different terminology and notation for continued fraction. In number theory the standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article simple continued fraction. The present article treats the case where numerators and denominators are sequences \,\ of constants or functions. From the perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis, however, they are just standard, and in the present article they will simply be called "continued fraction". Formulation A continued fraction is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergent (continued Fraction)
A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like :a_0 + \cfrac or an infinite continued fraction like :a_0 + \cfrac Typically, such a continued fraction is obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In the ''finite'' case, the iteration/recursion is stopped after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an ''infinite'' continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers a_i are called the coefficients or terms of the continued fraction. Simple co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Properties Of Polynomial Roots
In mathematics, a univariate polynomial of degree with real or complex coefficients has complex ''roots'' (if counted with their multiplicities). They form a multiset of points in the complex plane, whose geometry can be deduced from the degree and the coefficients of the polynomial. Some of these geometrical properties are related to a single polynomial, such as upper bounds on the absolute values of the roots, which define a disk containing all roots, or lower bounds on the distance between two roots. Such bounds are widely used for root-finding algorithms for polynomials, either for tuning them, or for computing their computational complexity. Some other properties are probabilistic, such as the expected number of real roots of a random polynomial of degree with real coefficients, which is less than 1+\frac 2\pi \ln (n) for sufficiently large. Notation In this article, a polynomial is always denoted : p(x)=a_0 + a_1 x + \cdots + a_n x^n, where a_0, \dots, a_n are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-open Interval
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. For example, the set of real numbers consisting of , , and all numbers in between is an interval, denoted and called the unit interval; the set of all positive real numbers is an interval, denoted ; the set of all real numbers is an interval, denoted ; and any single real number is an interval, denoted . Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. Interval arithmet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Budan's Theorem
In mathematics, Budan's theorem is a theorem for bounding the number of real roots of a polynomial in an interval, and computing the parity of this number. It was published in 1807 by François Budan de Boislaurent. A similar theorem was published independently by Joseph Fourier in 1820. Each of these theorems is a corollary of the other. Fourier's statement appears more often in the literature of the 19th century and has been referred to as Fourier's, Budan–Fourier, Fourier–Budan, and even Budan's theorem. Budan's original formulation is used in fast modern algorithms for real-root isolation of polynomials. Sign variation Let c_0, c_1, c_2, \ldots c_k be a finite sequence of real numbers. A ''sign variation'' or ''sign change'' in the sequence is a pair of indices such that c_ic_j < 0, and either or for all such that . In other words, a sign variation occurs in the sequence at each place where the signs change, when ignoring zeros. For stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Algorithm For Polynomials
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field the polynomial GCD may be computed, like for the integer GCD, by the Euclidean algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant. The similarity between the integer GCD and the polynomial GCD allows extending to univariate polynomials all the properties that may be deduced from the Euclidean algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the roots of the GCD of two polynomials are the common roots of the two polynomials, and this provides information on the roots without compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
