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Topic: Gauss Legendre algorithm

	Adrien Marie Legendre - LoveToKnow 1911
		The best known of these, which is called Legendre's theorem, is usually given in treatises on spherical trigonometry; by means of it a small spherical triangle may be treated as a plane triangle, certain corrections being applied to the angles.
		Legendre was also the author of a memoir upon triangles drawn upon a spheroid.
		Legendre's theorem is a fundamental one in geodesy, and his contributions to the subject are of the greatest importance.
	www.1911encyclopedia.org /Adrien_Marie_Legendre (1843 words)

	Adrien-Marie Legendre
		Legendre's researches connected with the gamma function are of importance, and are well known; the subject was also treated by Carl Friedrich Gauss in his memoir Disquisitiones Generales Circa Series Infinitas (1816), but in a very different manner.
		Legendre's name is most widely known on account of his Eléments de Géométrie, the most successful of the numerous attempts that have been made to supersede Euclid as a text-book on geometry.
		It will thus be seen that Legendre's works have placed him in the very foremost rank in the widely distinct subjects of elliptic functions, theory of numbers, attractions, and geodesy, and have given him a conspicuous position in connection with the integral calculus and other branches of mathematics.
	www.nndb.com /people/891/000093612 (1750 words)

	Adrien-Marie Legendre - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-04)
		Most of his work was brought to perfection by others: his work on roots of polynomials inspired Galois theory; Abel's work on elliptic functions was built on Legendre's; some of Gauss' work in statistics and number theory completed that of Legendre.
		Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely.
		In theoretical mechanics, he is known for the Legendre transform, which is used to go from the Lagrangian to the Hamiltonian formulation of mechanics.
	encyclopedia.worldsearch.com /adrien_marie_legendre.htm (233 words)

	Gauss-Legendre algorithm - Wikipedia, the free encyclopedia
		The method is based on the individual work of Carl Friedrich Gauss (1777-1855) and Adrien-Marie Legendre (1752-1833) combined with modern algorithms for multiplication and square roots.
		It was used to compute the first 206,158,430,000 decimal digits of π on September 18 to 20, 1999, and the results were checked with Borwein's algorithm.
		The algorithm has second order convergent nature, which essentially means that the number of correct digits doubles with each step of the algorithm.
	en.wikipedia.org /wiki/Salamin-Brent_algorithm (179 words)

	Legendre (Site not responding. Last check: 2007-11-04)
		Most of his work was brought to perfection by others: his work on roots of polynomials inspired Galois theory ; Abel 's work on ellipticfunctions was built on Legendre's; some of Gauss ' workin statistics and number theory completed that of Legendre.
		Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study theinverses of Jacobi 's functions and solve theproblem completely.
		In theoretical mechanics, he is known for the Legendre transform, which is used to go from the Lagrangian to theHamiltonian formulation of mechanics.
	www.therfcc.org /legendre-34847.html (214 words)

	Adrien-Marie Legendre Summary
		Legendre's interest in celestial mechanics eventually led to two further papers, one on the attraction of certain ellipsoids, and the other on the form and density of fluid planets.
		Legendre succeeded Laplace as the examiner in mathematics of students assigned to the artillery in 1799, a position he held until 1815.
		Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely.
	www.bookrags.com /Adrien-Marie_Legendre (2649 words)

	Exact Gauss Integrals
		Thus (and this is very different from classic Gauss integration methods) the program knows exactly the maximum error it makes and can check the tolerance for the integral's result.
		Gauss is calculated for an interval [ a, b ] without using the bounds.
		The enhancements to Tegral are made through the Epsilon Algorithm (followed by Square Aitken's Delta theory, 1926), which aims to find a faster parallel series that converges to the integral value.
	excalc.vestris.com /docs/ref-exact.html (434 words)

	Gauss-Legendre algorithm (Site not responding. Last check: 2007-11-04)
		Gauss, Johann Carl Friedrich (1777-1855) One of the all-time greats, Gauss began to show his mathematical brilliance at the early age of seven.
		Gauss, Carl Friedrich - Ein Genie Biografie und Werke von Gauss.
		The LLL Algorithm Papers on the LLL algorithm and its applications collected by François Koeune.
	www.serebella.com /encyclopedia/article-Gauss-Legendre_algorithm.html (545 words)

	[No title]
		In the case of the transform algorithms, only the first derivative is supported by algorithms which take into account the symmetry of the function.
		Gauss Radau is not available for the Even-Odd algorithm due to their lack of symmetry.
		Gauss and Gauss-Radau are not available for the Transform algorithm since there is no FFT code for these sets of points.
	www.labma.ufrj.br /~bcosta/pseudopack2000/algorithms.html (1097 words)

	Numerical Computation in Science and Engineering
		The topics are introduced on a need to know basis in order to concisely illustrate the practical implementation of a variety of algorithms and to demystify seemingly esoteric numerical methods.
		Algorithms that can be explained without too much elaboration and implemented within a few dozen lines of computer code are discussed in detail; those whose underlying theories require long, elaborate explanations are discussed at the level of first principles, and references for further information are given.
		Gauss elimination with row pivoting, and LU decomposition of an arbitrary matrix.
	dehesa.freeshell.org /NCSE (1042 words)

	Gauss (Site not responding. Last check: 2007-11-04)
		The gauss, abbreviated as G, is the cgs unit of magnetic flux density or magnetic induction (B), named after the German mathematician and physicist Carl Friedrich Gauss.
		One gauss is defined as one maxwell per square centimetre.
		For many years prior to 1932 the term gauss was used to designate that unit of magnetic field intensity which is now known as the oersted.
	www.kiwipedia.com /en/gauss.html (103 words)

	List of algorithms - Wikipedia, the free encyclopedia
		See also the list of data structures, list of algorithm general topics and list of terms relating to algorithms and data structures.
		Snapshot algorithm: a snapshot is the process of recording the global state of a system
		Rainflow-counting algorithm: Reduces a complex stress history to a count of elementary stress-reversals for use in fatigue analysis
	en.wikipedia.org /wiki/List_of_algorithms (1208 words)

	Adrien-Marie Legendre (Site not responding. Last check: 2007-11-04)
		Most of his work was brought to by others: his work on roots of polynomials inspired Galois theory ; Abel 's work on elliptic functions was built on Legendre's; some of Gauss ' work in statistics and number theory that of Legendre.
		In 1830 he gave a proof of Fermat's last theorem for exponent n = 5 which was given almost by Dirichlet in 1828.
		Legendre did an impressive amount of work elliptic functions including the classification of elliptic integrals but it took Abel's stroke of to study the inverses of Jacobi 's functions and solve the problem completely.
	www.freeglossary.com /Adrien_Marie_Legendre (565 words)

	Encyclopedia: Pi (Site not responding. Last check: 2007-11-04)
		Extremely long decimal expansions of Ï are typically computed with the Gauss-Legendre algorithm and Borwein's algorithm; the Salamin-Brent algorithm which was invented in 1976 has also been used in the past.
		Peter B. Borwein is a Canadian mathematician, co-developer of an algorithm for calculating Ï€ to the nth digit, co-discoverer of the billionth, four billionth, 40th billionth, and quadrillionth digits of Ï€, and professor at Simon Fraser University.
		Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate Ï.
	www.nationmaster.com /encyclopedia/Pi (8306 words)

	Gauss-Legendre algorithm (Site not responding. Last check: 2007-11-04)
		The method is based on the individual of Carl Friedrich Gauss (1777 - 1855) and Adrien-Marie Legendre (1752-1833) combined with modern algorithms for and square roots.
		It repeatedly replaces two by their arithmetic and geometric mean in order to approximate their arithmetic-geometric mean.
		The algorithm has second order convergent nature essentially means that the number of correct doubles with each step of the algorithm.
	www.freeglossary.com /Gauss-Legendre_algorithm (593 words)

	Gauss Legendre algorithm (Site not responding. Last check: 2007-11-04)
		The Gauss-Legendre algorithm is an algorithm to compute thedigits of π.
		The method is based on the individual work of Carl FriedrichGauss (1777 - 1855) and Adrien-Marie Legendre (1752-1833) combined with modern algorithms for multiplication and square roots.
		The algorithm has second order convergent nature, which essentially means that the number of correct digits doubles with eachstep of the algorithm.
	www.therfcc.org /gauss-legendre-algorithm-137855.html (167 words)

	Read about Adrien-Marie Legendre at WorldVillage Encyclopedia. Research Adrien-Marie Legendre and learn about ... (Site not responding. Last check: 2007-11-04)
		Abel's work on elliptic functions was built on Legendre's; some of Gauss' work in statistics and number theory completed that of Legendre.
		Legendre did an impressive amount of work on elliptic functions, including the classification of
		Legendre transform, which is used to go from the Lagrangian to the Hamiltonian formulation of mechanics.
	encyclopedia.worldvillage.com /s/b/Legendre (195 words)

	Encyclopedia: Borwein's algorithm (Site not responding. Last check: 2007-11-04)
		Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π.
		Borwein's algorithm (others) for an explanation of other algorithms by Jonathan and Peter Borwein to determine the digits of π.
		Categories: Pi algorithms Jonathan and Peter Borwein devised various algorithms to calculate the value of π.
	www.nationmaster.com /encyclopedia/Borwein%27s-algorithm (227 words)

	Gauss-Legendre algorithm
		The Gauss-Legendre algorithm is an algorithm to compute the digits of Pi.
		It was used to compute the first 206,158,430,000 decimal digits of Pi on September 18 to 20, 1999, and the results were checked with Borwein's algorithm.
		The text of this article is licensed under the GFDL.
	www.ebroadcast.com.au /lookup/encyclopedia/ga/Gauss-Legendre_algorithm.html (189 words)

	Encyclopedia: Gauss-Legendre algorithm (Site not responding. Last check: 2007-11-04)
		Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (GauÃŸ) (April 30, 1777 â€“ February 23, 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics.
		In mathematics, the arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a1, i.
		Categories: Pi algorithms Borweins algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π.
	www.nationmaster.com /encyclopedia/Gauss_Legendre-algorithm (513 words)

	Gauss-Legendre algorithm -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-04)
		The Gauss-Legendre algorithm is an (A precise rule (or set of rules) specifying how to solve some problem) algorithm to compute the digits of (Click link for more info and facts about π) π.
		The method is based on the individual work of (Click link for more info and facts about Carl Friedrich Gauss) Carl Friedrich Gauss (1777-1855) and (Click link for more info and facts about Adrien-Marie Legendre) Adrien-Marie Legendre (1752-1833) combined with modern algorithms for multiplication and square roots.
		It repeatedly replaces two numbers by their (The branch of pure mathematics dealing with the theory of numerical calculations) arithmetic and (The mean of n numbers expressed as the n-th root of their product) geometric mean, in order to approximate their (Click link for more info and facts about arithmetic-geometric mean) arithmetic-geometric mean.
	www.absoluteastronomy.com /encyclopedia/g/ga/gauss-legendre_algorithm1.htm (261 words)

	TVN Algorithm Testing
		The results suggest that a 24-point Gauss-Legendre integration rule could be used with TVN algorithms based on the equations (13) or (14).
		The equation (13) algorithm appears to be less sensitive to the subtractive cancellation loss of accuracy that occurs with nearly equal
		The algorithm terminates when the sum of the local error estimates is less than the requested accuracy.
	www.math.wsu.edu /faculty/genz/papers/bvnt/node10.html (839 words)

	Finance Choices - Personal Finance Wiki
		Both Legendre and Leonhard Euler speculated that π might be transcendental, a fact that was proved in 1882 by Ferdinand von Lindemann.
		A breakthrough was made in 1975, when Richard Brent and Eugene Salamin independently discovered the Brent-Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step.
		A similar algorithm that quadruples the accuracy in each step has been found by Jonathan and Peter Borwein.
	www.financechoices.co.uk /personal-finance-wiki.php?title=Pi (4014 words)

	BVT Algorithm Tests
		A quadruple precision implementation of an algorithm based on the Dunnett and Sobel (1954) paper was used for an accurate comparison.
		seconds, approximately one tenth of the time taken by the adaptive algorithm using equation (21).
		This time difference and results in Table 2 do not support the use of methods that use equation (21) with numerical integration for the efficient computation of BVT probabilities.
	www.sci.wsu.edu /math/faculty/genz/papers/bvnt/node14.html (156 words)

	SCD FY2000 ASR - Computational science research
		For the shallow water equations, this reduces the number of associated Legendre transforms per time step from nine to six, and concentrates the transforms into a projection operator that can be further optimized.
		We anticipate a significant savings in the memory used and time spent in Legendre transforms when all the components of the model have been completed, while maintaining the accuracy, stability, and parallel efficiency of the traditional spectral transform model.
		The resulting algorithm is robust as well as superior in both accuracy and speed.
	www.scd.ucar.edu /docs/asr2000/css.research.html (2128 words)

	File Name
		This algorithm can be very useful for testing if one is at a global optimum, since as in real evolution, change occurs in bursts.
		The threshold is derived using the Stein unbiased estimator with soft thresholding and a MAD estimator of the noise standard deviation.
		Gaussx 3.8.3 is compatible with Gauss 3.6; earlier versions of Gaussx, which were compatible with Gauss 3.5, crashed on 3.6 because of changes in the Gauss kernel.
	www.econotron.com /gaussx/readme2.htm (13957 words)

	Lecture 1 (Site not responding. Last check: 2007-11-04)
		One of the most elegant algorithms is called the Gauss-Legendre algorithm because it is based on old results of Gauss and Legendre, combined with modern algorithms for multiplication and square roots.
		The algorithms are related to an (incorrect) result of Ramanujan.
		Abstracts of papers on the computation of pi and elementary functions using algorithms based on complete elliptic integrals and the AGM or incomplete elliptic integrals and Landen transformations.
	web.comlab.ox.ac.uk /oucl/work/richard.brent/sp98lec1.html (241 words)

	List of algorithms - Wikipedia, the free encyclopedia
		Rainflow-counting algorithm: Reduces a complex stress history to a count of elementary stress-reversals for use in fatigue analysis
		Buddy memory allocation: Algorithm to allocate memory such that fragmentation is less.
		Powerset construction: Algorithm to convert nondeterministic automaton to deterministic automaton.
	duggmirror.com /programming/List_of_Algorithms (1881 words)

	Citations: Stability of Runge-Kutta methods for trajectory problems - Cooper (ResearchIndex) (Site not responding. Last check: 2007-11-04)
		Algorithms for the implementation of GL RK methods are given in Lambert [24] x4.10, and Ascher Petzold....
		One could therefore expect that a Gauss Legendre discretization of the wave equation (1) leads to a discrete energy momentum conservation law if the function V is zero or at most quadratic in u.
		They may appear particularly suitable for the numerical solution of highly oscillatory systems [11] although it is known that these schemes may also exhibit instabilities in various stiff situations [10, 17, 1, 2] However, whereas for the usual, highly damping stiff initial value problem....
	citeseer.ist.psu.edu /context/82525/0 (2482 words)

	Brent-Salamin Algorithm@Everything2.com
		Also known as the Gauss-Legendre algorithm, this algorithm was devised independently by Eugene Salamin and Richard Brent in 1976 as a very fast way to compute pi.
		Up until that time a laborious arctangent formula was used which had a growth rate of n-squared -- that is, doubling the number of digits you had now took four times as many iterations.
		It is entirely based upon the arithmetic-geometric mean formula which was used by Gauss to evaluate elliptic integrals.
	www.everything2.com /index.pl?node_id=998662 (345 words)

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