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Convergence Rate
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of convergence'' q \geq 1 and ''rate of convergence'' \mu if : \lim _ \frac=\mu. The rate of convergence \mu is also called the ''asymptotic error constant''. Note that this terminology is not standardized and some authors will use ''rate'' where this article uses ''order'' (e.g., ). In practice, the rate and order of convergence provide useful insights when using iterative methods for calculating numerical approximations. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. Strictly speaking, however, the asymptotic behavior of a sequence does not give conclusive information about any finite part of the sequence. Similar concepts are used for discretization methods. The solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endre Süli
Endre Süli (also, Endre Suli or Endre Šili) is a mathematician. He is Professor of Numerical Analysis in the Mathematical Institute, University of Oxford, Fellow and Tutor in Mathematics at Worcester College, Oxford and Adjunct Fellow of Linacre College, Oxford. He was educated at the University of Belgrade and, as a British Council Visiting Student, at the University of Reading and St Catherine's College, Oxford. His research is concerned with the mathematical analysis of numerical algorithms for nonlinear partial differential equations. Biography Süli is a Foreign Member of the Serbian Academy of Sciences and Arts (2009), Fellow of the European Academy of Sciences (FEurASc, 2010), Fellow of the Society for Industrial and Applied Mathematics (FSIAM, 2016), a Member of the Academia Europaea (MAE, 2020), and a Fellow of the Royal Society (FRS, 2021). [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michelle Schatzman
Michelle Schatzman (1949–2010) was a French mathematician, specializing in applied mathematics, who combined research as a CNRS research director and teaching as a professor at the Claude Bernard University Lyon 1. Biography Michelle Véra Schatzman was born in a secular Jewish family. Her father was French astrophysicist Évry Schatzman, who also was president of the Rationalist Union. Her mother Ruth Schatzman (née Fisher) was an associate of Russian in high schools in Lille and Paris, then a lecturer at the Paris VIII University. Michelle Schatzman married Yves Pigier in 1975. They had two children, Claude Mangoubi (née Pigier), born in 1976 in Clamart, and René Pigier, born in 1983 in Paris. They divorced in 1988. Her daughter is married to Dan Mangoubi, an Israeli mathematician, a professor at the Albert Einstein Institute of the University of Jerusalem. Education and career Michelle Schatzman entered École normale supérieure de jeunes filles in 1968. She obtained th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aitken's Delta-squared Process
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926.Alexander Aitken, "On Bernoulli's numerical solution of algebraic equations", ''Proceedings of the Royal Society of Edinburgh'' (1926) 46 pp. 289–305. Its early form was known to Seki Kōwa (end of 17th century) and was found for rectification of the circle, i.e. the calculation of π. It is most useful for accelerating the convergence of a sequence that is converging linearly. Definition Given a sequence X = _, one associates with this sequence the new sequence :A X=_, which can, with improved numerical stability, also be written as : (A X)_n = x_n-\frac, or equivalently as :(A X)_n = x_ - \frac = x_ - \frac where :\Delta x_=,\ \Delta x_=, and :\Delta^2 x_n=x_n -2x_ + x_=\Delta x_-\Delta x_,\ for n = 0, 1, 2, 3, \dots \, Obviou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Cost
In computational complexity theory, a computational resource is a resource used by some computational models in the solution of computational problems. The simplest computational resources are computation time, the number of steps necessary to solve a problem, and memory space, the amount of storage needed while solving the problem, but many more complicated resources have been defined. A computational problem is generally defined in terms of its action on any valid input. Examples of problems might be "given an integer ''n'', determine whether ''n'' is prime", or "given two numbers ''x'' and ''y'', calculate the product ''x''*''y''". As the inputs get bigger, the amount of computational resources needed to solve a problem will increase. Thus, the resources needed to solve a problem are described in terms of asymptotic analysis, by identifying the resources as a function of the length or size of the input. Resource usage is often partially quantified using Big O notation. Com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series Acceleration
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also be used, for example, to obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities. Definition Given a sequence :S=\_ having a limit :\lim_ s_n = \ell, an accelerated series is a second sequence :S'=\_ which converges faster to \ell than the original sequence, in the sense that :\lim_ \frac = 0. If the original sequence is divergent, the sequence transformation acts as an extrapolation method to the antilimit \ell. The mappings from the original to the transformed series may be linear (as defined in the article sequence transfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repulsive Fixed Point
A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the function. In physics, the term fixed point can refer to a temperature that can be used as a reproducible reference point, usually defined by a phase change or triple point. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain and the codomain of , and . For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, , has no fixed points, since is never equal to for any real number. In graphical terms, a fixed point means the point is on the line , or in other words the graph of has a point in common with that line. Fixed-point iteration In numerical analysis, ''fixed-point itera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point Theorems
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Fixed may refer to: * ''Fixed'' (EP), EP by Nine Inch Nails * ''Fixed'', an upcoming 2D adult animated film directed by Genndy Tartakovsky * Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System * Fixed, subjected to neutering * Fixed point (mathematics), a point that is mapped to itself by the function * Fixed line telephone, landline See also * * * Fix (other) * Fixer (other) * Fixing (other) * Fixture (other) A fixture can refer to: * Test fixture, used to control and automate testing * Light fixture * Plumbing fixture * Fixture (tool), a tool used in manufacturing * Fixture (property law) * A type of sporting event See also * * * Fixed (disambigua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the exponents and are nonnegative integers with , and the coefficient of each term is a specific positive integer depending on and . For example, for , (x+y)^4 = x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4. The coefficient in the term of is known as the binomial coefficient \tbinom or \tbinom (the two have the same value). These coefficients for varying and can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where \tbinom gives the number of different combinations of elements that can be chosen from an -element set. Therefore \tbinom is often pronounced as " choose ". History Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid ment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
