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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 In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . Like an .... 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_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Euler Transform
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function. Definition The binomial transform, , of a sequence, , is the sequence defined by s_n = \sum_^n (-1)^k \binom a_k. Formally, one may write s_n = (Ta)_n = \sum_^n T_ a_k for the transformation, where is an infinite-dimensional operator (mathematics), operator with matrix elements . The transform is an involution (mathematics), involution, that is, TT = 1 or, using index notation, \sum_^\infty T_ T_ = \delta_ where \delta_ is the Kronecker delta. The original series can be regained by a_n=\sum_^n (-1)^k \binom s_k. The binomial transform of a sequence is just the -th forward difference#n-th difference, forward differences of the sequence, with odd differences carrying a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Series
In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms converge to 0 monotonically, but this condition is not necessary for convergence. Examples The geometric series − + − + ⋯ sums to . The alternating harmonic series has a finite sum but the harmonic series does not. The series 1-\frac+\frac-\ldots=\sum_^\infty\frac converges to \frac, but is not absolutely convergent. The Mercator series provides an analytic power series expression of the natural logarithm, given by \sum_^\infty \frac x^n = \ln (1+x),\;\;\;, x, \le1, x\ne-1. The functions si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adriaan Van Wijngaarden
Adriaan "Aad" van Wijngaarden (2 November 1916 – 7 February 1987) was a Dutch mathematician and computer scientist. Trained as a mechanical engineer, Van Wijngaarden emphasized and promoted the mathematical aspects of computing, first in numerical analysis, then in programming languages and finally in design principles of such languages. Biography Van Wijngaarden's university education was in mechanical engineering, for which he received a degree from Delft University of Technology in 1939. He then studied for a doctorate in hydrodynamics, but abandoned the field. He joined the Nationaal Luchtvaartlaboratorium in 1945 and went with a group to England the next year to learn about new technologies that had been developed there during World War II. Van Wijngaarden was intrigued by the new idea of automatic computing. On 1 January 1947, he became the head of the Computing Department of the brand-new Centrum Wiskunde & Informatica (CWI), which was at the time known as the Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leibniz Formula For Pi
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics. Leibniz has been called the "last universal genius" due to his vast expertise across fields, which became a rarity after his lifetime with the coming of the Industrial Revolution and the spread of specialized labor. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. Leibniz contributed to the field of libra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J (programming Language)
The J programming language, developed in the early 1990s by Kenneth E. Iverson and Roger Hui, is an array programming language based primarily on APL (also by Iverson). To avoid repeating the APL special-character problem, J uses only the basic ASCII character set, resorting to the use of the dot and colon as ''inflections'' to form short words similar to '' digraphs''. Most such ''primary'' (or ''primitive'') J words serve as mathematical symbols, with the dot or colon extending the meaning of the basic characters available. Also, many characters which in other languages often must be paired (such as [] "" `` or ) are treated by J as stand-alone words or, when inflected, as single-character roots of multi-character words. J is a very terse array programming language, and is most suited to mathematical and statistical programming, especially when performing operations on matrices. It has also been used in extreme programming and network performance analysis. Like John B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Summation
In the mathematics of convergent and divergent series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series mus ..., Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σ''a''''n'', if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series. As well as being used to define values for divergent series, Euler summation can be used to speed the convergence of series. Euler summation can be generalized into a family of methods denoted (E, ''q''), where ''q'' ≥ 0. The (E, 1) sum is the ordinary Euler sum. All of these methods are strictly weaker than Borel summation; for ''q'' > 0 they are incomparable with Abel su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series (mathematics)
In mathematics, a series is, roughly speaking, an addition of Infinity, infinitely many Addition#Terms, terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance. Among the Ancient Greece, Ancient Greeks, the idea that a potential infinity, potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the Quadrature of the Parabola, quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
