|
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] |
Kummer's Transformation Of Series
In mathematics, specifically in the field of numerical analysis, Kummer's transformation of series is a method used to series acceleration, accelerate the convergence of an infinite series. The method was first suggested by Ernst Kummer in 1837. Technique Let :A=\sum_^\infty a_n be an infinite sum whose value we wish to compute, and let :B=\sum_^\infty b_n be an infinite sum with comparable terms whose value is known. If the limit :\gamma:=\lim_ \frac exists, then a_n-\gamma \,b_n is always also a sequence going to zero and the series given by the difference, \sum_^\infty (a_n-\gamma\, b_n), converges. If \gamma\neq 0, this new series differs from the original \sum_^\infty a_n and, under broad conditions, converges more rapidly.Holy et al.''On Faster Convergent Infinite Series'' Mathematica Slovaca, January 2008 We may then compute A as :A=\gamma\,B + \sum_^\infty (a_n-\gamma\,b_n), where \gamma B is a constant. Where a_n\neq 0, the terms can be written as the product (1-\gamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richardson Extrapolation
In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A^\ast = \lim_ A(h). In essence, given the value of A(h) for several values of h, we can estimate A^\ast by extrapolating the estimates to h=0. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century, though the idea was already known to Christiaan Huygens in his calculation of π. In the words of Birkhoff and Rota, "its usefulness for practical computations can hardly be overestimated."Page 126 of Practical applications of Richardson extrapolation include Romberg integration, which applies Richardson extrapolation to the trapezoid rule, and the Bulirsch–Stoer algorithm for solving ordinary differential equations. Example of Richardson extrapolation Suppose that we wish to approximate A^*, and we have a method A(h) that depends on a small parameter h in such a way that A( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Wijngaarden Transformation
In mathematics and numerical analysis, the van Wijngaarden transformation is a variant on the Euler transform used to accelerate the convergence of an alternating series. One algorithm to compute Euler's transform runs as follows: Compute a row of partial sums s_ = \sum_^k(-1)^n a_n and form rows of averages between neighbors s_ = \frac2 The first column s_ then contains the partial sums of the Euler transform. Adriaan van Wijngaarden's contribution was to point out that it is better not to carry this procedure through to the very end, but to stop two-thirds of the way. A. van Wijngaarden, in: Cursus: Wetenschappelijk Rekenen B, Proces Analyse, Stichting Mathematisch Centrum, (Amsterdam, 1965) pp. 51-60 If a_0,a_1,\ldots,a_ are available, then s_ is almost always a better approximation to the sum than s_. In many cases the diagonal terms do not converge in one cycle so process of averaging is to be repeated with diagonal terms by bringing them in a row. (For example, this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forward Difference Operator
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted \Delta is the operator that maps a function to the function \Delta /math> defined by :\Delta x)= f(x+1)-f(x). A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for approximating derivatives, and the term "fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear 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 (cf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Don Zagier
Don Bernard Zagier (born 29 June 1951) is an American-German mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany. He was a professor at the ''Collège de France'' in Paris from 2006 to 2014. Since October 2014, he is also a Distinguished Staff Associate at the International Centre for Theoretical Physics (ICTP). Background Zagier was born in Heidelberg, West Germany. His mother was a psychiatrist, and his father was the dean of instruction at the American College of Switzerland. His father held five different citizenships, and he spent his youth living in many different countries. After finishing high school (at age 13) and attending Winchester College for a year, he studied for three years at MIT, completing his bachelor's and master's degrees and being named a Putnam Fellow in 1967 at the age of 16. He then wrote a doctoral dissertation on characteristic classes under Friedrich ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Cohen (number Theorist)
Henri Cohen (born 8 June 1947) is a number theorist, and a professor at the University of Bordeaux. He is best known for leading the team that created the PARI/GP computer algebra system. He introduced the Rankin–Cohen bracket and has written several textbooks in computational and algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob .... Selected publications * ; 2nd correct. print 19951st printing 1993ref> * * * * References External links Personal web page* Number theorists École Normale Supérieure alumni 20th-century French mathematicians 21st-century French mathematicians 1947 births Living people University of Bordeaux faculty {{France-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Series
In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. Examples The geometric series 1/2 − 1/4 %2B 1/8 − 1/16 %2B %E2%8B%AF sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does not. The Mercator series provides an analytic expression of the natural logarithm: \sum_^\infty \frac x^n \;=\; \ln (1+x). The functions sine and cosine used in trigonometry can be defined as alternating series in calculus even though they are introduced in elementary algebra as the ratio of sides of a right triangle. In fact, \sin x = \sum_^\infty (-1)^n \frac, and \cos x = \sum_^\infty (-1)^n \frac . When the alternating factor is removed from these series one obtains the hyperbolic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
WZ Theory in particle physics
{{disambiguation ...
WZ may refer to: * WZ sex-determination system, also known as the ZW sex-determination system * WZ theory, a technique for simplifying certain combinatorial summations in mathematics * Eswatini (FIPS 10-4 country code WZ) * ''Westdeutsche Zeitung'', a German newspaper * Wetzlar, Germany * WinZip, a computer file compression software * Wizet, a Korean online gaming developer, which uses the file extension * W and Z bosons In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Wynn (mathematician)
Peter Wynn (19312017) was an English mathematician. His main achievements concern approximation theory – in particular the theory of Padé approximants – and its application in numerical methods for improving the rate of convergence of sequences of 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 ...s. Publications # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # MathSciNet entries Reference Book * C. Brezinski and M. Redivo-Zaglia: "The genesis and early developments of Aitken’s process, Shanks' transformation, the epsilon algorithm, and related fixed point methods", Numer. Algorithms, vol.80 (2019) pp.11-133. * C. Brezinski: "Reminiscences of Peter Wynn", Num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Takakazu Seki
, Selin, Helaine. (1997). ''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures,'' p. 890 also known as ,Selin, was a Japanese mathematician and author of the Edo period. Seki laid foundations for the subsequent development of Japanese mathematics, known as ''wasan''. He has been described as "Japan's Newton". He created a new algebraic notation system and, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations. Although he was a contemporary of German polymath mathematician and philosopher Gottfried Leibniz and British polymath physicist and mathematician Isaac Newton, Seki's work was independent. His successors later developed a school dominant in Japanese mathematics until the end of the Edo period. While it is not clear how much of the achievements of ''wasan'' are Seki's, since many of them appear only in writings of his pupils, some of the results parallel or anticipate those discovered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Aitken
Alexander Craig "Alec" Aitken (1 April 1895 – 3 November 1967) was one of New Zealand's most eminent mathematicians. In a 1935 paper he introduced the concept of generalized least squares, along with now standard vector/matrix notation for the linear regression model. Another influential paper co-authored with his student Harold Silverstone established the lower bound on the variance of an estimator, now known as Cramér–Rao bound. He was elected to the Royal Society of Literature for his World War I memoir, ''Gallipoli to the Somme''. Life and work Aitken was born on 1 April 1895 in Dunedin, the eldest of the seven children of Elizabeth Towers and William Aitken. He was of Scottish descent, his grandfather having emigrated from Lanarkshire in 1868. His mother was from Wolverhampton. He was educated at Otago Boys' High School in Dunedin (1908–13) where he was school dux and won the Thomas Baker Calculus Scholarship in his last year at school. He saw active service ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
