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CONSTRUCTIVE APPROXIMATION 9 1993 SpringerVerlag New York Inc. Hypergeometric Analogues of the ArithmeticGeometric Mean Iteration
"... Abstract. The arithmeticgeometric mean iteration of Gauss and Legendre is the twoterm iteration a.+ 1 = (a. + bn)/2 and b.+ 1 = axfa~,b, with a0: = 1 and b 0: = x. The common limit is 2F1 ( 89 89 1; 1 x2) 1 and the convergence is quadratic. This is a rare object with very few close relatives. Th ..."
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Abstract. The arithmeticgeometric mean iteration of Gauss and Legendre is the twoterm iteration a.+ 1 = (a. + bn)/2 and b.+ 1 = axfa~,b, with a0: = 1 and b 0: = x. The common limit is 2F1 ( 89 89 1; 1 x2) 1 and the convergence is quadratic. This is a rare object with very few close relatives
ArithmeticGeometric Means Revisited
"... We use Maple's gfun library to study the limit formulae for a twoterm recurrence (iteration) AGN , which in the case N = 2 specializes to the wellknown ArithmeticGeometric Mean iteration of Gauss. Our main aim is to independently rediscover and prove the limit formulae for two classical cases ..."
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We use Maple's gfun library to study the limit formulae for a twoterm recurrence (iteration) AGN , which in the case N = 2 specializes to the wellknown ArithmeticGeometric Mean iteration of Gauss. Our main aim is to independently rediscover and prove the limit formulae for two classical
The arithmeticgeometric progression abstract domain
 Proc. 6 th VMCAI ’2005, Paris. LNCS 3385
, 2005
"... Abstract We present a new numerical abstract domain. This domain automatically detects and proves bounds on the values of program variables. For that purpose, it relates variable values to a clock counter. More precisely, it bounds these values with the ith iterate of the function [X ↦ → α×X+β] app ..."
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Cited by 13 (8 self)
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×X+β] applied on M, where i denotes the clock counter and the floatingpoint numbers α, β, and M are discovered by the analysis. Such properties are especially useful to analyze loops in which a variable is iteratively assigned with a barycentric mean of the values that were associated with the same variable
The arithmeticgeometric mean and fast computation of elementaryfunctions
 SIAM Rev
, 1984
"... Abstract. We produce a self contained account of the relationship between the Gaussian arithmeticgeometric mean iteration and the fast computation of elementary functions. A particularly pleasant algorithm for r is one of the byproducts. Introduction. It is possible to calculate 2 decimal places of ..."
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Cited by 17 (2 self)
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Abstract. We produce a self contained account of the relationship between the Gaussian arithmeticgeometric mean iteration and the fast computation of elementary functions. A particularly pleasant algorithm for r is one of the byproducts. Introduction. It is possible to calculate 2 decimal places
ON INEQUALITIES FOR HYPERGEOMETRIC ANALOGUES OF THE ARITHMETICGEOMETRIC MEAN
, 2007
"... ABSTRACT. In this note, we present sharp inequalities relating hypergeometric analogues of the arithmeticgeometric mean discussed in [5] and the power mean. The main result generalizes the corresponding sharp inequality for the arithmeticgeometric mean established in [10]. Key words and phrases: A ..."
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Cited by 2 (1 self)
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ABSTRACT. In this note, we present sharp inequalities relating hypergeometric analogues of the arithmeticgeometric mean discussed in [5] and the power mean. The main result generalizes the corresponding sharp inequality for the arithmeticgeometric mean established in [10]. Key words and phrases
ON THE ARITHMETICGEOMETRIC MEAN FOR CURVES OF GENUS 2
, 2007
"... We study the relationship between two genus 2 curves whose jacobians are isogenous with kernel equal to a maximal isotropic subspace of ptorsion points with respect to the Weil pairing. When p = 2 this relationship is a generalization of Gauss’s arithmeticgeometric mean for elliptic curves studied ..."
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We study the relationship between two genus 2 curves whose jacobians are isogenous with kernel equal to a maximal isotropic subspace of ptorsion points with respect to the Weil pairing. When p = 2 this relationship is a generalization of Gauss’s arithmeticgeometric mean for elliptic curves
Fast multipleprecision evaluation of elementary functions
 Journal of the ACM
, 1976
"... XI3STnXC'r. Let f(x) be one of the usual elementary functions (exp, log, artan, sin, cosh, etc.), and let M(n) be the number of singleprecision operations reqmred to multiply nbit integers. It is shown that f(x) can be evaluated, with relative error 0(2'), m O(M(n)log (n)) operations as ..."
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Cited by 107 (7 self)
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such as f, e, and e'. The algorithms depend on the theory of elhptic integrals, using the arithmeticgeometric mean iteration and ascending Landen transformations. Itsr wol~os Ar~o en~As~s: multipleprecision arithmetic, analytic complexity, arithmeticgeometric mean, computational complexity
Iterative decoding of binary block and convolutional codes
 IEEE Trans. Inform. Theory
, 1996
"... Abstract Iterative decoding of twodimensional systematic convolutional codes has been termed “turbo ” (de)coding. Using loglikelihood algebra, we show that any decoder can he used which accepts soft inputsincluding a priori valuesand delivers soft outputs that can he split into three terms: the ..."
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Cited by 600 (43 self)
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Abstract Iterative decoding of twodimensional systematic convolutional codes has been termed “turbo ” (de)coding. Using loglikelihood algebra, we show that any decoder can he used which accepts soft inputsincluding a priori valuesand delivers soft outputs that can he split into three terms
Mean shift, mode seeking, and clustering
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1995
"... AbstractMean shift, a simple iterative procedure that shifts each data point to the average of data points in its neighborhood, is generalized and analyzed in this paper. This generalization makes some kmeans like clustering algorithms its special cases. It is shown that mean shift is a modeseeki ..."
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Cited by 620 (0 self)
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AbstractMean shift, a simple iterative procedure that shifts each data point to the average of data points in its neighborhood, is generalized and analyzed in this paper. This generalization makes some kmeans like clustering algorithms its special cases. It is shown that mean shift is a mode
"GrabCut”  interactive foreground extraction using iterated graph cuts
 ACM TRANS. GRAPH
, 2004
"... The problem of efficient, interactive foreground/background segmentation in still images is of great practical importance in image editing. Classical image segmentation tools use either texture (colour) information, e.g. Magic Wand, or edge (contrast) information, e.g. Intelligent Scissors. Recently ..."
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Cited by 1140 (36 self)
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. Recently, an approach based on optimization by graphcut has been developed which successfully combines both types of information. In this paper we extend the graphcut approach in three respects. First, we have developed a more powerful, iterative version of the optimisation. Secondly, the power
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