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Sequence Transformation
In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration, that is, for improving the rate of convergence of a slowly convergent sequence or series. Sequence transformations are also commonly used to compute the antilimit of a divergent series numerically, and are used in conjunction with extrapolation methods. Overview Classical examples for sequence transformations include the binomial transform, Möbius transform, Stirling transform and others. Definitions For a given sequence :S=\_,\, the transformed sequence is :\mathbf(S)=S'=\_,\, where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e. :s_n' = T(s_n,s_,\dots,s_) for some k which often depends on 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] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the ''n''th perfect number. Examples Integer sequences that have their own name include: *Abundant numbers *Baum–Sweet sequence *Bell numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shanks Transformation
In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941.Weniger (2003). Formulation For a sequence \left\_ the series :A = \sum_^\infty a_m\, is to be determined. First, the partial sum A_n is defined as: :A_n = \sum_^n a_m\, and forms a new sequence \left\_. Provided the series converges, A_n will also approach the limit A as n\to\infty. The Shanks transformation S(A_n) of the sequence A_n is the new sequence defined byBender & Orszag (1999), pp. 368–375.Van Dyke (1975), pp. 202–205. :S(A_n) = \frac = A_ - \frac where this sequence S(A_n) often converges more rapidly than the sequence A_n. Further speed-up may be obtained by repeated use of the Shanks transformation, by computing S^2(A_n)=S(S(A_n)), S^3(A_n)=S(S(S(A_n))), etc. No ... [...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] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Operator
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector—without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is the multiplication of a vector by a scalar (where the product is a vector), and is to be distinguished from inner product of two vectors (where the product is a scalar). Definition In general, if ''K'' is a field and ''V'' is a vector space over ''K'', then scalar multiplication is a function from ''K'' × ''V'' to ''V''. The result of applying this function to ''k'' in ''K'' and v in ''V'' is denoted ''k''v. Properties Scalar multiplication obeys the following rules ''(vector in boldface)'': * Additivity in the scalar: (''c' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extrapolation Method
In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results. Extrapolation may also mean extension of a method, assuming similar methods will be applicable. Extrapolation may also apply to human experience to project, extend, or expand known experience into an area not known or previously experienced so as to arrive at a (usually conjectural) knowledge of the unknownExtrapolation entry at Merriam–Webster (e.g. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergent Sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence 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 transf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
