A modification of Muller’s method

Costabile, F.; Gualtieri, Maria Italia; Luceri, R.

doi:10.1007/s10092-006-0113-9

Cited by 15 publications

(4 citation statements)

References 5 publications

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“…The recurrence relation (2.23) utilized for the above Hessenberg determinant D n (λ|x) reproduces the defining property (2.5) of the polynomials β n (λ|x). In the classical λ → 0 limit the above construction agrees to structure given in [7].…”

Section: Determinant Structuresupporting

confidence: 80%

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An extension of the Bernoulli polynomials inspired by the Tsallis statistics

Balamurugan¹,

Chakrabarti²,

Jagannathan³

2016

Preprint

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In [1,2] Carlitz introduced the degenerate Bernoulli numbers and polynomials by replacing the exponential factors in the corresponding classical generating functions with their deformed analogs: exp(t) → (1 + λt) 1/λ , and exp(tx) → (1 + λt) x/λ . The deformed exponentials reduce to their ordinary counterparts in the λ → 0 limit. In the present work we study the extension of the Bernoulli polynomials obtained via an alternate deformation exp(tx) → (1 + λtx) 1/λ that is inspired by the concepts of q-exponential function and q-logarithm used in the nonextensive Tsallis statistics.

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Section: Determinant Structuresupporting

confidence: 80%

“…Recently a determinantal approach for the Bernoulli polynomials B n (x) has been proposed [6,7]. The construction employs an upper Hessenberg matrix [8] with the entries h ,ℓ = 0 if  − ℓ ≥ 2.…”

Section: Determinant Structurementioning

confidence: 99%

An extension of the Bernoulli polynomials inspired by the Tsallis statistics

Balamurugan¹,

Chakrabarti²,

Jagannathan³

2016

Preprint

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“…Moreover, a root of a polynomial has to be computed in cases [C.1] and [C.3]. Numerically, the second order modified Mueller method [5] is used to compute it. It usually requires between 5 and 9 evaluations of the polynomial to obtain an error smaller than…”

Section: Proposition 31 (Piecewise Affine Reconstructionmentioning

confidence: 99%

Two-Size Moment Multi-Fluid Model: A Robust and High-Fidelity Description of Polydisperse Moderately Dense Evaporating Sprays

Laurent

Sibra

Doisneau

2016

Commun. Comput. Phys.

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High fidelity modeling and simulation of moderately dense sprays at relatively low cost is still a major challenge for many applications. For that purpose, we introduce a new multi-fluid model based on a two-size moment formalism in sections, which are size intervals of discretization. It is derived from a Boltzmann type equation taking into account drag, evaporation and coalescence, which are representative of the complex terms that arise in multi-physics environments. The closure of the model comes from a reconstruction of the distribution. A piecewise affine reconstruction in size is thoroughly analyzed in terms of stability and accuracy, a key point for a high-fidelity and reliable description of the spray. Robust and accurate numerical methods are then developed, ensuring the realizability of the moments. The model and method are proven to describe the spray with a high accuracy in size and size-conditioned variables, resorting to a lower number of sections compared to one size moment methods. Moreover, robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebra thanks to a tailored overall strategy. This strategy is successfully tested on a difficult 2D unsteady case, which proves the efficiency of the modeling and numerical choices.

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“…There are some traditional methods to solve the transcendental dispersion equation in the complex domain, such as Newton iteration method and Muller's method. 32 However, these methods have their own limitations in dealing with the multivariate transcendental equations. The Newton iteration method needs the firstorder derivative of the determinant that is usually difficult to obtain.…”

Section: Introductionmentioning

confidence: 99%

Properties of circumferential non-propagating waves in functionally graded piezoelectric cylindrical shells

Zhang

et al. 2019

Advances in Mechanical Engineering

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Each guided elastic wave mode is composed of propagating and non-propagating branches. Non-propagating branches associated with complex wavenumbers are substantially different from the propagating ones. Accurate calculation of the transcendental dispersion equation for complex wavenumbers and arbitrary ranges of frequencies is usually very difficult, especially for demanding cases such as those involving composite materials and curved structures. In this article, an extended Legendre orthogonal polynomial method is presented to determine non-propagating waves in a functionally graded piezoelectric cylindrical shell. The extended Legendre orthogonal polynomial method can obtain complete solutions without the tedious iterative search. Results are compared with those published earlier to validate the extended Legendre orthogonal polynomial method, and the convergence of this method is also discussed. Dispersion characteristics of the non-propagating waves in various graded piezoelectric cylindrical shells are studied and three-dimensional dispersion curves are plotted. The effects of piezoelectricity, graded fields, and the mechanical and electrical boundary conditions on dispersion curves are illustrated. The displacement amplitude, stress, and electric potential distributions are also analyzed in detail. Numerical results show that the extended Legendre orthogonal polynomial method is very efficient to retrieve the guided waves of any nature and the modes of all orders.

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A modification of Muller’s method

Cited by 15 publications

References 5 publications

An extension of the Bernoulli polynomials inspired by the Tsallis statistics

An extension of the Bernoulli polynomials inspired by the Tsallis statistics

Two-Size Moment Multi-Fluid Model: A Robust and High-Fidelity Description of Polydisperse Moderately Dense Evaporating Sprays

Properties of circumferential non-propagating waves in functionally graded piezoelectric cylindrical shells

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