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Vincent's Theorem
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 ''c''0, ''c''1, ''c''2, ... be a finite or infinite sequence of real numbers. Suppose ''l'' 1. To obtain an arbitrary positive root we need to assume that a_1 \ge 0. * Negative roots are obtained by replacing ''x'' by −''x'', in which case the negative roots become positive. Vincent's theorem: Bisection version (Alesin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vincent's Theorem
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 ''c''0, ''c''1, ''c''2, ... be a finite or infinite sequence of real numbers. Suppose ''l'' 1. To obtain an arbitrary positive root we need to assume that a_1 \ge 0. * Negative roots are obtained by replacing ''x'' by −''x'', in which case the negative roots become positive. Vincent's theorem: Bisection version (Alesin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vincent's Theorem
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 ''c''0, ''c''1, ''c''2, ... be a finite or infinite sequence of real numbers. Suppose ''l'' 1. To obtain an arbitrary positive root we need to assume that a_1 \ge 0. * Negative roots are obtained by replacing ''x'' by −''x'', in which case the negative roots become positive. Vincent's theorem: Bisection version (Alesin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandre Joseph Hidulphe Vincent , a Portuguese hypocoristic of the name "Alexandre"
{{Disambig ...
Alexandre may refer to: * Alexandre (given name) * Alexandre (surname) * Alexandre (film) See also * Alexander * Xano (other) Xano is the name of: * Xano, a Portuguese hypocoristic of the name "Alexandre (other) Alexandre may refer to: * Alexandre (given name) * Alexandre (surname) * Alexandre (film) See also * Alexander Alexander is a male given name. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
North Carolina State University
North Carolina State University (NC State) is a public land-grant research university in Raleigh, North Carolina. Founded in 1887 and part of the University of North Carolina system, it is the largest university in the Carolinas. The university forms one of the corners of the Research Triangle together with Duke University in Durham and the University of North Carolina at Chapel Hill. It is classified among "R1: Doctoral Universities – Very high research activity". The North Carolina General Assembly established the North Carolina College of Agriculture and Mechanic Arts, now NC State, on March 7, 1887, originally as a land-grant college. The college underwent several name changes and officially became North Carolina State University at Raleigh in 1965. However, by longstanding convention, the "at Raleigh" portion is usually omitted. Today, NC State has an enrollment of more than 35,000 students, making it among the largest in the country. NC State has historical streng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maple (software)
Maple is a symbolic and numeric computing environment as well as a multi-paradigm programming language. It covers several areas of technical computing, such as symbolic mathematics, numerical analysis, data processing, visualization, and others. A toolbox, MapleSim, adds functionality for multidomain physical modeling and code generation. Maple's capacity for symbolic computing include those of a general-purpose computer algebra system. For instance, it can manipulate mathematical expressions and find symbolic solutions to certain problems, such as those arising from ordinary and partial differential equations. Maple is developed commercially by the Canadian software company Maplesoft. The name 'Maple' is a reference to the software's Canadian heritage. Overview Core functionality Users can enter mathematics in traditional mathematical notation. Custom user interfaces can also be created. There is support for numeric computations, to arbitrary precision, as well as symbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bisection Method
In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a root. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the interval halving method, the binary search method, or the dichotomy method. For polynomials, more elaborate methods exist for testing the existence of a root in an interval (Descartes' rule of signs, Sturm's theorem, Budan's theorem). They allow extending the bisection method into efficient algorithms for finding all real roots of a polynomial; see Real-root isolation. The method The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Xcas
Xcas is a user interface to Giac, which is an open source computer algebra system (CAS) for Windows, macOS and Linux among many other platforms. Xcas is written in C++. Giac can be used directly inside software written in C++. Xcas has a compatibility modes with many popular algebra systems like WolframAlpha, Mathematica, Maple, or MuPAD. Users can use Giac/Xcas to develop formal algorithms or use it in other software. Giac is used in SageMath for calculus operations. Among other things, Xcas can solve equations (Figure 3) and differential equations (Figure 4) and draw graphs. There is a forum for questions about Xcas. CmathOOoCAS, an OpenOffice.org plugin which allows formal calculation in Calc spreadsheet and Writer word processing, uses Giac to perform calculations. Features Here is a brief overview of what Xcas is able to do:Read more commands and featurehear * Xcas has the ability of a scientific calculator that provides show input and writes pretty print * Xcas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
