A simple algorithm for the fast calculation of higher order derivatives of the inverse function

Dargazany, Roozbeh; Hörnes, Karl; Itskov, Mikhail

doi:10.1016/j.amc.2013.06.035

Cited by 17 publications

(10 citation statements)

References 8 publications

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“…In Itskov et al (2010), this series was further continued as follows Dargazany et al 2013) proposed a simple and fast recurrence procedure calculating Taylor series coefficients of the inverse function. This formula was further applied to the inverse Langevin function.…”

Section: Introductionmentioning

confidence: 99%

A simple and accurate approximation of the inverse Langevin function

Darabi

Itskov

2015

Rheol Acta

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The inverse Langevin function cannot be represented in an explicit form and requires an approximation by a series, a non-rational or a rational function as for example by a Padé approximation. In the current paper, an analytical method based on the Padé technique and the multiple point interpolation is presented for the inverse Langevin function. Thus, a new simple and accurate approximation of the inverse Langevin function is obtained. It might be advantageous, for example, for non-Gaussian statistical theory of rubber elasticity where the inverse Langevin function plays an important role.

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Section: Introductionmentioning

confidence: 99%

A simple and accurate approximation of the inverse Langevin function

Darabi

Itskov

2015

Rheol Acta

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“…Using this software, we wrote a short script which can be used for finding a solution of higher order derivatives of the inverse function in a simple and elegant way. The mentioned problem was discussed by Jarecki and Ziabicki (2002) and Dargazany et al (2013).…”

Section: Resultsmentioning

confidence: 99%

“…They stated that the solution based on 115 series terms shows the best accuracy within the region [0, 0.95] between examined approximations. This method of calculation of higher order derivatives of the inverse function was also presented by Dargazany et al (2013). The authors compared it with other methods known in literature both with respect to the computation time and memory usage.…”

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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“…(38) cannot be solved analytically. Hence the iterative method is used to solve the equation [58]. The step size of the parameter variation is chosen based on the sensitivity of the parameter to the maximum deflection (mid-length deflection).…”

Section: Numerical Solution Methodsmentioning

confidence: 99%

Electromechanical instability of nanobridge in ionic liquid electrolyte media: influence of electrical double layer, dispersion forces and size effect

et al. 2015

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In this paper, the electromechanical response and instability of the nanobridge immersed in ionic electrolyte media is investigated. The electrochemical force field is determined using double-layer theory and linearized PoissonBoltzmann equation. The presence of dispersion forces, i.e., Casimir and van der Waals attractions are incorporated considering the correction due to the presence of liquid media between the interacting surfaces (three-layer model). The strain gradient elasticity is employed to model the size-dependent structural behavior of the nanobridge. To solve the nonlinear constitutive equation of the system, three approaches, e.g., the Rayleigh-Ritz method, Lumped parameter model and the numerical solution method are employed. Impacts of the dispersion forces and size effect on the instability characteristics as well as the effects of ion concentration and potential ratio are discussed.

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A simple algorithm for the fast calculation of higher order derivatives of the inverse function

Cited by 17 publications

References 8 publications

A simple and accurate approximation of the inverse Langevin function

A simple and accurate approximation of the inverse Langevin function

Approximation of the inverse Langevin function revisited

Electromechanical instability of nanobridge in ionic liquid electrolyte media: influence of electrical double layer, dispersion forces and size effect

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