Numerical computation of real or complex elliptic integrals

@article{Carlson2005NumericalCO,
  title={Numerical computation of real or complex elliptic integrals},
  author={Bille C. Carlson},
  journal={Numerical Algorithms},
  year={2005},
  volume={10},
  pages={13-26}
}
  • B. C. Carlson
  • Published 6 September 1994
  • Mathematics
  • Numerical Algorithms
Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind). Numerical check values, consistency checks, and relations to Legendre's integrals and Bulirsch's integrals are included. 
Bounds for symmetric elliptic integrals
Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms
  • Fredrik Johansson
  • Computer Science, Mathematics
    Texts & Monographs in Symbolic Computation
  • 2019
We describe algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary
CONTINUOUS BRANCHES OF INVERSES OF THE 12 JACOBI ELLIPTIC FUNCTIONS FOR REAL ARGUMENT
Continuous complex branches of the inverses of each of the 12 Jacobi elliptic functions, for real argument, are constructed in terms of real Incomplete Elliptic Integrals of the First Kind. These
GENERALIZATIONS OF INCOMPLETE ELLIPTIC INTEGRALS OF FIRST AND SECOND KINDS
In this paper, we obtain analytical solutions of incomplete elliptic integrals of first and second kinds. Further, we generalize these incomplete elliptic integrals in the forms of multiple series
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
REDUCTION FROM AN INTEGRAL OF A RATIONAL FUNCTION WITH RESPECT TO A COSINE FUNCTION / A SQUARE ROOT OF A QUADRATIC OF THAT TO ELLIPTIC INTEGRALS INVOLVED
An integral of some rational functions involving a square root of a quadratic of a single cosine function is reduced to elliptic integrals and elementary functions if any. For simplicity interval of
Short note on a relation between the inverse of the cosine and Carlson's elliptic integral $R_D$
We prove a simple relation for a special case of Carlson's elliptic integral $R_D$. The findings are applied to derive explicit formulae for the asymptotics of certain moments of the angular central
On efficient evaluation of integrals entering boundary equations of 3D potential and elasticity theory
The paper presents recurrent formulae for efficient evaluation of all the integrals needed for solving static 3D potential and elasticity problems by the boundary elements method. The power-type
...
...

References

SHOWING 1-10 OF 20 REFERENCES
Numerical calculation of elliptic integrals and elliptic functions
TLDR
This paper contains anALGOL program for the incomplete elliptic integral of the third kind based on a theory described in [4] and replaces the inadequate one based on the Gauβ-transformation which was published in [2].
A table of elliptic integrals of the second kind
By evaluating elliptic integrals in terms of standard R-functions instead of Legendre's integrals, many (in one case 144) formulas in previous tables are unified. The present table includes only
A table of elliptic integrals: one quadratic factor
Integration in terms of real quantities is accomplished for 33 integrands that are rational except for the square root of a cubic or quartic polynomial with exactly one pair of conjugate complex
Elliptic Integrals of the First Kind
The reciprocal square root of any real polynomial with known zeros and degree not exceeding four is integrated in terms of a standard integral by a new quadratic transformation which preserves
A table of elliptic integrals: cubic cases
Forty-one integrands that are rational except for the square root of a cubic polynomial with known real zeros are integrated in terms of {ital R}-functions for which Fortran codes are available. In
A table of elliptic integrals of the third kind
As many as 72 elliptic integrals of the third kind in previous tables are unified by evaluation in terms of R-functions instead of Legendre's integrals. The present table includes only integrals
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
Symmetric Elliptic Integrals of the Third Kind
Legendre's incomplete elliptic integral of the third kind can be replaced by an integral which possesses permutation symmetry instead of a set of linear transformations. Two such symmetric integrals
Handbook of elliptic integrals for engineers and scientists
TLDR
The Handbook of Elliptic Integrals for Engineers and Scientists introduces an integral operator on the set of means and investigates its properties.
On Computing Elliptic Integrals and Functions
TLDR
Elliptic integral and function direct computation method using successive quadratic Landen and Gauss transformations to solve the inequality of the LaSalle inequality.
...
...