An optimized perturbation expansion for a global O(2) theory

Evans, T. S.; Ivin, M; Möbius, Matthias E.

doi:10.1016/s0550-3213(00)00120-6

Cited by 8 publications

(18 citation statements)

References 25 publications

(62 reference statements)

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“…Therefore, here and in what follows, we start the analysis with the second-order approximant. For the considered case, we get in the second order Both approximants f * 2 (x) and f * 3 (x) perfectly reproduce function (14). For comparison, we construct the Padé approximants based on the same number of terms in the series (2).…”

Section: Combination Of Functions From R With Exponentialsmentioning

confidence: 96%

“…First of all, to improve the convergence property of a perturbative sequence, it is necessary to introduce control functions defined by an optimization procedure [3][4][5]. This idea makes the foundation of the optimized perturbation theory that is now widely employed for various applications [3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The second pivotal idea is to consider the successive passage from one approximation to the next one as a dynamical evolution on the manifold of approximants, which is formalized by the notion of group self-similarity [17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

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Self-similar factor approximants

2003

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The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents the passage from one approximant to another as the motion realized by a dynamical system with the property of group self-similarity. The derived approximants, because of their form, are named the self-similar factor approximants. These complement the obtained earlier self-similar exponential approximants and self-similar root approximants. The specific feature of the self-similar factor approximants is that their control functions, providing convergence of the computational algorithm, are completely defined from the accuracy-through-order conditions. These approximants contain the Padé approximants as a particular case, and in some limit they can be reduced to the self-similar exponential approximants previously introduced by two of us. It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions which include a variety of transcendental functions. For other functions, not pertaining to this exactly reproducible class, the factor approximants provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Padé approximants. This is illustrated by a number of examples showing the generality and accuracy of the factor approximants even when conventional techniques meet serious difficulties.

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Section: Combination Of Functions From R With Exponentialsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Self-similar factor approximants

2003

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“…[10], there appeared a series of papers [23][24][25][26][27][28] advertizing the same idea of introducing control functions for rendering perturbation theory convergent. Nowdays the optimized perturbation theory is widely used for various problems, being employed under different guises and called by different names, such as modified perturbation theory, renormalized perturbation theory, variational perturbation theory, controlled perturbation theory, self-consistent perturbation theory, oscillatorrepresentation method, delta expansion, optimized expansion, nonperturbative expansion, and so on [23][24][25][26][27][28][29][30][31][32]. Many problems of quantum mechanics, statistical mechanics, condensed matter physics, and quantum field theory are successively treated by this optimized approach.…”

Section: Optimized Perturbation Theorymentioning

confidence: 99%

Self-similar structures and fractal transforms in approximation theory

Yukalov

Yukalova

2002

Chaos, Solitons & Fractals

112

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An overview is given of the methods for treating complicated problems without small parameters, when the standard perturbation theory based on the existence of small parameters becomes useless. Such complicated problems are typical of quantum physics, many-body physics, physics of complex systems, and various aspects of applied physics and applied mathematics. A general approach for dealing with such problems has been developed, called Self-Similar Approximation Theory. A concise survey of the main ideas of this approach is presented, with the emphasis on the basic notion of group self-similarity. The techniques are illustrated by examples from quantum field theory.

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“…The scalar quantum field theory with ϕ 4 -interaction is an ideal ground for testing new methods in perturbation theory [2,48]. Much efforts has been invested into studying critical phenomena by means of the O(n)-symmetric ϕ 4 -theory.…”

Section: Critical Indices From Wilson Expansionmentioning

confidence: 99%

Summation of power series by self-similar factor approximants

Yukalov

Gluzman

Sornette

2003

Physica A: Statistical Mechanics and its Applications

105

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A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the self-similar renormalization to the latter rather to the former. This results in self-similar factor approximants extrapolating the sought functions from the region of asymptotically small variables to their whole domains. The method of constructing crossover formulas, interpolating between small and large values of variables is also analysed. The techniques are illustrated on different series which are typical of problems in statistical mechanics, condensed-matter physics, and, generally, in many-body theory.

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An optimized perturbation expansion for a global O(2) theory

Abstract: 08/10/12 meb waitng for arxiv response. 10/10/12 OK to pub, arxiv ok and peer review versio

Cited by 8 publications

References 25 publications

Self-similar factor approximants

Self-similar factor approximants

Self-similar structures and fractal transforms in approximation theory

Summation of power series by self-similar factor approximants

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