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Minimum Polynomial Extrapolation
In mathematics, minimum polynomial extrapolation is a sequence transformation used for convergence acceleration of vector sequences, due to Cabay and Jackson. While Aitken's method is the most famous, it often fails for vector sequences. An effective method for vector sequences is the minimum polynomial extrapolation. It is usually phrased in terms of the fixed point iteration: : x_=f(x_k). Given iterates x_1, x_2, ..., x_k in \mathbb R^n, one constructs the n \times (k-1) matrix U=(x_2-x_1, x_3-x_2, ..., x_k-x_) whose columns are the k-1 differences. Then, one computes the vector c=-U^+ (x_-x_k) where U^+ denotes the Moore–Penrose pseudoinverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inv ... of U. The number 1 is then appended to the end of c, and the extrapolated limit is :s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 (c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 sequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Aitken's Method
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] |
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Fixed Point Iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of f, the fixed-point iteration is :x_=f(x_n), \, n=0, 1, 2, \dots which gives rise to the sequence x_0, x_1, x_2, \dots of iterated function applications x_0, f(x_0), f(f(x_0)), \dots which is hoped to converge to a point x_. If f is continuous, then one can prove that the obtained x_ is a fixed point of f, i.e., :f(x_)=x_ . More generally, the function f can be defined on any metric space with values in that same space. Examples * A first simple and useful example is the Babylonian method for computing the square root of ''a''>0, which consists in taking f(x)=\frac 12\left(\frac ax + x\right), i.e. the mean value of ''x'' and ''a/x'', to approach the limit x = \sqrt a (from whatever starting point x_0 \gg 0 ). This is a special case of Newton's method quoted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to 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 cells in medicine an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |