Fast computation of complete elliptic integrals and Jacobian elliptic functions

@article{Fukushima2009FastCO,
  title={Fast computation of complete elliptic integrals and Jacobian elliptic functions},
  author={Toshio Fukushima},
  journal={Celestial Mechanics and Dynamical Astronomy},
  year={2009},
  volume={105},
  pages={305-328}
}
  • T. Fukushima
  • Published 25 October 2009
  • Mathematics
  • Celestial Mechanics and Dynamical Astronomy
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… 

Precise and Fast Computation of Elliptic Integrals and Elliptic Functions

Summarized is the recent progress of the new methods to compute Legendre’s complete and incomplete elliptic integrals of all three kinds and Jacobian elliptic functions. Also reviewed are the

Precise and Fast Computation of Elliptic Integrals and Functions

  • T. Fukushima
  • Mathematics
    2015 IEEE 22nd Symposium on Computer Arithmetic
  • 2015
Summarized is the recent progress of the new methods to compute Legendre's complete and incomplete elliptic integrals of all three kinds and Jacobian elliptic functions. Also reviewed are the

Numerical inversion of a general incomplete elliptic integral

Numerical computation of inverse complete elliptic integrals of first and second kinds

Fast computation of incomplete elliptic integral of first kind by half argument transformation

A new method to calculate the incomplete elliptic integral of the first kind, F(\varphi|m), by using the half argument formulas of Jacobian elliptic functions, which is significantly faster than the existing procedures.

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

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,

References

SHOWING 1-10 OF 74 REFERENCES

An extension of the Bartky-transformation to incomplete elliptic integrals of the third kind

O. Introduction Elliptic integrals and elliptic functions have for a long time attracted mathematicians, not only because of the beauty of the theory but also its usefulness for practical

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

In order to accelerate the numerical evaluation of torque-free rotation of triaxial rigid bodies, we present a fast method to compute various kinds of elliptic functions for a series of the elliptic

Chebyshev polynomial expansions of complete elliptic integrals

1. Introduction. Numerical subroutines for machine computation of the complete elliptic integrals are usually based upon the well-known Gauss arithmetic-geometric mean process [1] or Hastings-type

Expansions of elliptic motion based on elliptic function theory

New expansions of elliptic motion based on considering the eccentricitye as the modulusk of elliptic functions and introducing the new anomalyw (a sort of elliptic anomaly) defined

Computing elliptic integrals by duplication

SummaryLogarithms, arctangents, and elliptic integrals of all three kinds (including complete integrals) are evaluated numerically by successive applications of the duplication theorem. When the

Solutions of the axi-symmetric Poisson equation from elliptic integrals. I. Numerical splitting methods

In a series of two papers, we present numerical, integral-based methods to compute accurately the self-gravitating field and potential induced by tri-dimensional, axially symmetric fluids, with a

Solutions of the axi-symmetric Poisson equation from elliptic integrals - II. Semi-analytical approach

In a series of two papers, we present numerical integral-based methods to compute accurately the self-gravitating field and potential induced by a tri-dimensional, axially symmetric fluid, with

A Modified Prony Algorithm for Fitting Functions Defined by Difference Equations

This paper reformulates, generalizes, and investigates the stability of the modified Prony algorithm introduced by Osborne, and shows that the relative difference between B and the Hessian is shown to converge to zero almost surely in the case of rational fitting.

Expanding spherically symmetric models without shear

The integrability properties of the field equationLxx=F(x)L2 of a spherically symmetric shear-free fluid are investigated. A first integral, subject to an integrability condition onF(x), is found,

Analytical expansions of torque-free motions for short and long axis modes

Torque-free motion of a rigid body is integrable and its solution is expressed in terms of elliptic functions and elliptic integrals. The conventional analytical expression of the solution, however,
...