Taylor expansion of the inverse function with application to the Langevin function

Itskov, Mikhail; Dargazany, Roozbeh; Hörnes, Karl

doi:10.1177/1081286511429886

Cited by 53 publications

(59 citation statements)

References 21 publications

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“…The path in α is derived, for u ≥ 0, according to (34) where s = p α (u), and for u < 0 it is extended by symmetry…”

Section: Applying the Methods Of Steepest Descentmentioning

confidence: 99%

“…The fact that the coefficients f m are imaginary valued is a consequence of (27). The inversion map in 3) is derived by exploiting the method used in [34]. The fact that the even coefficients p 2m are real and the odd coefficients p 2m+1 are imaginary is a consequence of the fact that f m are imaginary valued, and can be proved by induction.…”

Section: Proof Of Proceduresmentioning

confidence: 99%

See 1 more Smart Citation

Coding in the Finite-Blocklength Regime: Bounds Based on Laplace Integrals and Their Asymptotic Approximations

Erseghe

2016

IEEE Trans. Inform. Theory

134

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Abstract-In this paper we provide new compact integral expressions and associated simple asymptotic approximations for converse and achievability bounds in the finite blocklength regime. The chosen converse and random coding union bounds were taken from the recent work of Polyanskyi-Poor-Verdù, and are investigated under parallel AWGN channels, the AWGN channels, the BI-AWGN channel, and the BSC. The technique we use, which is a generalization of some recent results available from the literature, is to map the probabilities of interest into a Laplace integral, and then solve (or approximate) the integral by use of a steepest descent technique. The proposed results are particularly useful for short packet lengths, where the normal approximation may provide unreliable results.Index Terms-Channel capacity, Coding for noisy channels, Converse, Finite blocklength regime, Shannon theory.

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“…The path in α is derived, for u ≥ 0, according to (34) where s = p α (u), and for u < 0 it is extended by symmetry…”

Section: Applying the Methods Of Steepest Descentmentioning

confidence: 99%

Section: Proof Of Proceduresmentioning

confidence: 99%

Coding in the Finite-Blocklength Regime: Bounds Based on Laplace Integrals and Their Asymptotic Approximations

Erseghe

2016

IEEE Trans. Inform. Theory

134

View full text Add to dashboard Cite

show abstract

“…This series has nonzero coefficients only for odd powers of y. Recently, Itskov and Dargazany (2011) proposed a simple recurrence procedure for calculating Taylor series coefficients of the inverse function. This formula was further applied to the inverse Langevin function.…”

Section: Previous Approximationsmentioning

confidence: 99%

“…The efficiency of the presented script has been thoroughly examined by comparing it with the results of Itskov et al (2011). Figure 1 shows the magnitude of the first 100 odd coefficients c k (all even are equal 0) calculated using the Mathematica script.…”

Section: Previous Approximationsmentioning

confidence: 99%

Approximation of the inverse Langevin function revisited

Jedynak

2014

Rheol Acta

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The main purpose of this paper is to provide an easy-to-use approximation formula for the inverse Langevin function. The mathematical complexity of this function makes it unfeasible for an analytical manipulation and inconvenient for computer simulation. This situation has motivated a series of papers directed on its approximation. The best known solution is given by Cohen. It is used in a lot of statistically based models of rubber-like materials. The formula is derived from rounded Padé approximation [3/2]. The main idea of the presented approach in this paper relies on improvement of the precision of approximation formula for the inverse Langevin function by using multipoint Padé approximation method. We focused our study strongly on obtaining a simple and accurate approximation. It is assumed that the proposed approximation formula may be considered a useful tool for verification of the results obtained in other ways. Our results are supported by investigating several numerical examples. The paper also presents a few applications of computer software named Mathematica which can be used to calculate symbolically one point Padé approximants and numerically multipoint Padé approximants. Using this software, we showed also how to compute higher order derivatives of the inverse function in a simple and elegant way. This issue was discussed by Itskov et al.

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“…In cases of moderate and large deformations, Taylor expansion is more favorable (see [Itskov et al 2012]). However, if chain breakage occurs very close to the fully stretched state of the chains ν < 1.04, then Padé approximants [Puso 2003] show better agreement with the exact values.…”

mentioning

confidence: 99%

Network evolution model of anisotropic stress softening in filled rubber-like materials: Parameter identification and finite element implementation

Dargazany

Khiêm

Navrath

et al. 2012

J. Mech. Mater. Struct.

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A purely micromechanical network evolution theory granting new insight into the damage mechanism was proposed previously by the authors (Dargazany and Itskov, 2009). In this follow-up paper, we further formulate the network evolution model for implementation into finite element simulations. To this end, a general internal variable formulation is developed which determines the inelastic response of the microstructure on the basis of the free energy function. The thermodynamical consistency of the network evolution model is then verified analytically. Next, the predictive capabilities of the model are demonstrated by means of several experiments especially designed to capture stress softening, permanent set, and induced anisotropy. Finally, the influence of the filler concentration on material parameters is studied.

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Taylor expansion of the inverse function with application to the Langevin function

Cited by 53 publications

References 21 publications

Coding in the Finite-Blocklength Regime: Bounds Based on Laplace Integrals and Their Asymptotic Approximations

Coding in the Finite-Blocklength Regime: Bounds Based on Laplace Integrals and Their Asymptotic Approximations

Approximation of the inverse Langevin function revisited

Network evolution model of anisotropic stress softening in filled rubber-like materials: Parameter identification and finite element implementation

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