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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] |
Nikola Obreshkov
Nikola Dimitrov Obreshkov ( bg, Никола Димитров Обрешков) (March 6, 1896 in VarnaAugust 11, 1963 in Sofia) was a prominent Bulgarian mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ..., working in complex analysis. See also * Obreschkoff–Ostrowski theorem References European Mathematics Society Newsletter No. 51(PDF), page 28. * 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] |
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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yun's Algorithm
In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univariate polynomial is square free if and only if it has no multiple root in an algebraically closed field containing its coefficients. This motivates that, in applications in physics and engineering, a square-free polynomial is commonly called a polynomial with no repeated roots. In the case of univariate polynomials, the product rule implies that, if divides , then divides the formal derivative of . The converse is also true and hence, f is square-free if and only if 1 is a greatest common divisor of the polynomial and its derivative. A square-free decomposition or square-free factorization of a polynomial is a factorization into powers of square-free polynomials : f = a_1 a_2^2 a_3^3 \cdots a_n^n =\prod_^n a_k^k\, where those of the that are non-constant are pairwise coprime sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Worst-case Complexity
In computer science (specifically computational complexity theory), the worst-case complexity measures the System resource, resources (e.g. running time, Computer memory, memory) that an algorithm requires given an input of arbitrary size (commonly denoted as in Big O notation, 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 growth, logarithmic) of the worst-case complexity is commonly used to compare the Algorithmic efficiency, 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 Give ... [...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), ''Sketch'' (2007 film), a Malayalam film * Sketch (2018 film), ''Sketch'' (2018 film), a Tamil film * Sketch (TV series), ''Sketch'' (TV series), a 2018 South Korean series * "Sketch", a 2008 List of Skins episodes#Series 2 (2008), episode of ''Skins'' ** Sketch (Skins character), 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 * Sketc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Ostrowski
Alexander Markowich Ostrowski ( uk, Олександр Маркович Островський; russian: Алекса́ндр Ма́ркович Остро́вский; 25 September 1893, in Kiev, Russian Empire – 20 November 1986, in Montagnola, Lugano, Switzerland) was a mathematician. 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 Landau and 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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continued Fraction
In mathematics, a continued fraction is an expression 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 a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated 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. It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is nece ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergent (continued Fraction)
In mathematics, a continued fraction is an expression 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 a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated 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. It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Properties Of Polynomial Roots
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an object * Material properties, properties by which the benefits of one material versus another can be assessed * Chemical property, a material's properties that becomes evident during a chemical reaction *Physical property, any property that is measurable whose value describes a state of a physical system *Semantic property * Thermodynamic properties, in thermodynamics and materials science, intensive and extensive physical properties of substances * Mental property, a property of the mind studied by many sciences and parasciences Computer science * Property (programming), a type of class member in object-oriented programming * .properties, a Java Properties File to store program settings as name-v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-open Interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical ... [...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 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 studyin ... [...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 comput ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
