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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Sequence Transformation
In mathematics, a sequence transformation is an Operator (mathematics), operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution, discrete convolution with another sequence and resummation of a sequence and nonlinear mappings, more generally. They are commonly used for series acceleration, that is, for improving the rate of convergence of a slowly convergent sequence or series (mathematics), series. Sequence transformations are also commonly used to compute the antilimit of a divergent series numerically, and are used in conjunction with extrapolation methods. Classical examples for sequence transformations include the binomial transform, Möbius transform, and Stirling transform. Definitions For a given sequence :(s_n)_,\, and a sequence transformation \mathbf, the sequence resulting from transformation by \mathbf is :\mathbf( ( s_n ) ) = ( s'_n )_, where the elements of the transformed sequence a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Convergence Acceleration
Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen * "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four Weirdoverse titles in 1997 **A 2015 crossover storyline spanning the DC Comics Multiverse * ''Convergence'' (journal), an academic journal that covers the fields of communications and media * ''Convergence'' (novel), by Charles Sheffield * ''Convergence'' (Cherryh novel), by C. J. Cherryh Music * ''Convergence'' (Front Line Assembly album), 1988 * ''Convergence'' (David Arkenstone and David Lanz album), 1996 * ''Convergence'' (Dave Douglas album), 1999 * ''Convergence'' (Warren Wolf album), 2016 Other media * ''Convergence'' (Pollock), a 1952 oil painting by Jackson Pollock * ''Convergence'' (2015 film), an American horror-thriller film * ''Convergence'' (2019 film), a British drama film *''Convergence'', a 2021 Netflix film by Orlando von ... [...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 as part of an extension to Bernoulli's method. It is most useful for accelerating the convergence of a sequence that is converging linearly. A precursor form was known to Seki Kōwa (1642 – 1708) and applied to the rectification of the circle, i.e., to the calculation of π. Definition Given a sequence X = with n = 0, 1, 2, 3, \ldots, Aitken's delta-squared process associates to this sequence the new sequence A = (a_n) = , which can also be written as A = \left( x_n-\frac \right), with \Delta x_= x_-x_ and \Delta^2 x_n=x_n -2x_ + x_=\Delta x_-\Delta x_. Both are the same sequence algebraically but the latter has improved numerical stability in computational implementation. A /math> is ill-defined if the sequence \Delta^2 = (\Delta^2 x_ ... [...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 point (mathematics), 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 a function, 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 limit (mathematics), converge to a point x_\text. If f is continuous, then one can prove that the obtained x_\text is a fixed point of f, i.e., f(x_\text)=x_\text . 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 , which consists in taking f(x) = \frac 1 2 \left(\frac a x + x\right), i.e. the mean value of and , to approach the limit x = \sqrt a (from whatever startin ... [...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 computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find 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 like economics, 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 simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |