Fast computation of Jacobian elliptic functions and incomplete elliptic integrals for constant values of elliptic parameter and elliptic characteristic

Fukushima, Toshio

doi:10.1007/s10569-008-9177-y

Cited by 18 publications

(19 citation statements)

References 22 publications

(17 reference statements)

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“…The key techniques used there are the utilization of Taylor series expansion and the combination of the defining relation of Jacobi's nome and Legendre's relation. This is a continuation of our trials to accelerate the procedures to compute the complete and incomplete elliptic integrals and the Jacobian elliptic functions [19,16,17,18]. The new method is sufficiently precise and significantly faster than the existing procedures to compute K(m) and/or E(m) including Cody's method as well as Innes' classic formulation [21].…”

Section: Introduction Of New Methodmentioning

confidence: 94%

Precise and fast computation of the general complete elliptic integral of the second kind

Fukushima

2011

Math. Comp.

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Abstract. We developed an efficient procedure to evaluate two auxiliary complete elliptic integrals of the second kind B(m) and D(m) by using their Taylor series expansions, the definition of Jacobi's nome, and Legendre's relation. The developed procedure is more precise than the existing ones in the sense that the maximum relative errors are 1-3 machine epsilons, and it runs drastically faster; around 5 times faster than Bulirsch's cel2 and 16 times faster than Carlson's R F and R D .

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Section: Introduction Of New Methodmentioning

confidence: 94%

Precise and fast computation of the general complete elliptic integral of the second kind

Fukushima

2011

Math. Comp.

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“…In order to circumvent the problems arisen in the numerical treatment of the potential expression, we use several formulas from the textbook of Byrd and Friedman (1945), and the computational approach by Bulirsch (1971), Carlson (1979) and Fukushima (2009), Fukushima (2010 to overcome difficulties in evaluating the Elliptic Integrals; details of it can be found in Tresaco et al (2011).…”

Section: The Annular Disk and Its Potential Functionmentioning

confidence: 99%

Computation of families of periodic orbits and bifurcations around a massive annulus

Tresaco

Elipe

Riaguas

2011

Astrophys Space Sci

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This paper studies the main features of the dynamics around a planar annular disk. It is addressed an appropriated closed expression of the gravitational potential of a massive disk, which overcomes the difficulties found in previous works in this matter concerning its numerical treatment. This allows us to define the differential equations of motion that describes the motion of a massless particle orbiting the annulus. We describe the computation methods proposed for the continuation of uni-parametric families of periodic orbits, these algorithms have been applied to analyze the dynamics around a massive annulus by means of a description of the main families of periodic orbits found, their bifurcations and linear stability.

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“…Also they are used in rotational dynamics as (1) the periodic solutions of a Kovalevskaya top (El-Sabaa 1992), (2) the integrable case of a rotational motion of a gyrostat with and without a rotor (Cavas and Vigueras 1994;Elipe and Lanchares 2008), (3) the descriptions of torque-free rotation of a triaxial rigid body (Barkin 1999;Fukushima 2008a), (4) the construction of symplectic integrator for rotation (Breiter and Buciora 2000;Fukushima 2009a), and (5) the application of the implicit midpoint integrator for satellite attitude dynamics (Hellstrom and Mikkola 2009).…”

Section: Jacobian Elliptic Functionsmentioning

confidence: 99%

Section: Jacobian Elliptic Functionsmentioning

confidence: 99%

Fast computation of complete elliptic integrals and Jacobian elliptic functions

Fukushima

2009

Celest Mech Dyn Astr

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As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, K (m) and E(m), for the standard domain of the elliptic parameter, 0 < m < 1. For the case 0 < m < 0.9, the method utilizes 10 pairs of approximate polynomials of the order of 9-19 obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, K (1 − m) and E(1 − m), are first computed by a pair of the approximate polynomials and then transformed to K (m) and E(m) by means of Jacobi's nome, q, and Legendre's identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody's Chebyshev polynomial approximation of Hastings type and Innes' formulation based on q-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, sn(u|m), cn(u|m), and dn(u|m), by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, u, after its domain is reduced to the standard range, 0 ≤ u < K (m)/4, with the help of the new method to compute K (m). The new procedure is 25-70% faster than the methods based on the Gauss transformation such as Bulirsch's algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of K (m) is not taken into account.

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Fast computation of Jacobian elliptic functions and incomplete elliptic integrals for constant values of elliptic parameter and elliptic characteristic

Cited by 18 publications

References 22 publications

Precise and fast computation of the general complete elliptic integral of the second kind

Precise and fast computation of the general complete elliptic integral of the second kind

Computation of families of periodic orbits and bifurcations around a massive annulus

Fast computation of complete elliptic integrals and Jacobian elliptic functions

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