Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments

Gil, Amparo; Segura, Javier; Temme, Nico

doi:10.1145/992200.992203

Cited by 25 publications

(20 citation statements)

References 19 publications

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“…where ρ = 1 + σ 2 n r/ , L i (•) and K i (•) are real valued solutions of modified Bessel equation of purely imarginal order with Wronskian W{K i (ζ), L(ζ)} = 1/ζ. 73,74 Constant B can be found from the boundary conditions (C4d) at r = r in :…”

Section: Discussionmentioning

confidence: 99%

Vortex polarity switching in magnets with surface anisotropy

Pylypovskyi

Sheka

Kravchuk

et al. 2015

Low Temperature Physics

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Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Under the action of a perpendicular static magnetic field the vortex core undergoes a shape deformationof pillow-or barrel-shaped type, depending on the type of the surface anisotropy. This deformation plays a key point in the switching mechanism: We predict that the vortex polarity switching is accompanied (i) by a linear singularity in case of Heisenberg magnet with bulk anisotropy only and (ii) by a point singularities in case of surface anisotropy or exchange anisotropy. We study in details the switching process using spin-lattice simulations and propose a simple analytical description using a wired core model, which provides an adequate description of the Bloch point statics, its dynamics and the Bloch point mediated switching process. Our analytical predictions are confirmed by spin-lattice simulations for Heisenberg magnet and micromagnetic simulations for nanomagnet with account of a dipolar interaction.

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Section: Discussionmentioning

confidence: 99%

Vortex polarity switching in magnets with surface anisotropy

Pylypovskyi

Sheka

Kravchuk

et al. 2015

Low Temperature Physics

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“…Now the eigenfunctions e k (v) are appropriately normalized Bessel functions K ik . They were calculated using methods of Gil, Segura and Temme [24]. As in the initial data construction discussed in the last para, it is sufficient to restrict the domain of integration to a compact set [k ⋆ − 10σ; k ⋆ + 10σ].…”

Section: B Evaluation Of Semi-classical State and Observablesmentioning

confidence: 99%

Loop quantum cosmology ofk=1FRW models

et al. 2007

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The closed, k=1, FRW model coupled to a massless scalar field is investigated in the framework of loop quantum cosmology using analytical and numerical methods. As in the k = 0 case, the scalar field can be again used as emergent time to construct the physical Hilbert space and introduce Dirac observables. The resulting framework is then used to address a major challenge of quantum cosmology: resolving the bigbang singularity while retaining agreement with general relativity at large scales. It is shown that the framework fulfills this task. In particular, for states which are semi-classical at some late time, the big-bang is replaced by a quantum bounce and a recollapse occurs at the value of the scale factor predicted by classical general relativity. Thus, the 'difficulties' pointed out by Green and Unruh in the k=1 case do not arise in a more systematic treatment. As in k = 0 models, quantum dynamics is deterministic across the deep Planck regime. However, because it also retains the classical recollapse, in contrast to the k = 0 case one is now led to a cyclic model. Finally, we clarify some issues raised by Laguna's recent work addressed to computational physicists.

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“…The tau method [Rappoport 2001] numerical quadrature [Gil et al 2002a;Temme 1994], the numerical integration of Bessel's differential equation [Fabijonas 2004], the recurrence methods 47:4 M. Kodama [Gil et al 2002b[Gil et al , 2004Goldstein and Thaler 1959;Kodama 1995;Temme 1975], and the asymptotic series of Watson [1944], [Section 8.6, 8.61, and Olver [1975], Section 5, 11, are all applicable to Region R 3 . However, these methods are not always applicable to the entire region of R 3 .…”

Section: Regionmentioning

confidence: 99%

Algorithm 912

Kodama¹

2011

ACM Trans. Math. Softw.

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The present algorithm provides a module for calculating the cylindrical functions J ν (z), Y ν (z), H (1) ν (z), and H (2) ν (z), where the order ν is complex and the complex argument z satisfies −π < arg z ≤ π . The algorithm is written in Fortran 90 and calculates the functions using real and complex numbers of any intrinsic data type whose kind type parameter the user's Fortran system accepts. The methods of calculating the functions are based on two kinds of series expansions and numerical integration. Wronskian tests examine the functional values computed by this algorithm with double precision at 4,100,625 pseudorandom test points in the region |Re ν| ≤ 60, |Im ν| ≤ 60, |Re z| ≤ 300, |Im z| ≤ 300. From the results of the tests, we find that the errors of two kinds of Wronskians are less than 6.42 × 10 −14 . ACM Reference Format:Kodama, M. 2011. Algorithm 912: A module for calculating cylindrical functions of complex order and complex argument.

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Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments

Cited by 25 publications

References 19 publications

Vortex polarity switching in magnets with surface anisotropy

Vortex polarity switching in magnets with surface anisotropy

Loop quantum cosmology ofk=1FRW models

Algorithm 912

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