ELLIPTIC_INTEGRAL
Elliptic Integrals

ELLIPTIC_INTEGRAL, a FORTRAN90 library which evaluates elliptic integral functions using Carlson's elliptic functions.

The complete and incomplete elliptic integrals of the first, second and third kind can be evaluated, with parameters A (angle in degrees), K (sine of A) or M (the modulus, K^2).

Routines are also supplied to evaluate Jacobi's elliptic functions CN, DN and SN.

Languages:

ELLIPTIC_INTEGRAL is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

SPECIAL_FUNCTIONS, a FORTRAN90 library which evaluates special functions, including Airy, Associated Legendre Bessel, Beta, Complete Elliptic Integral, Confluent Hypergeometric, Cosine Integral, Elliptic Integral, Error, Exponential Integral, Fresnel Integral, Gamma, Hankel, Hypergeometric, Incomplete Beta, Incomplete Gamma, Jacobian Elliptic, Kelvin, Lambda, Legendre, Mathieu, Modified Spherical Bessel, Parabolic Cylinder, Psi, Riccati-Bessel, Sine Integral, Spheroidal Wave, Struve, Whittaker, as well as Bernoulli Numbers, Euler Numbers, Hermite Polynomials, Laguerre Polynomials, Legendre Polynomials, by Shanjie Zhang, Jianming Jin;

TEST_VALUES, a FORTRAN90 library which supplies test values of various mathematical functions.

TOMS577, a FORTRAN90 library which evaluates Carlson's elliptic integral functions RC, RD, RF and RJ. This is a version of ACM TOMS algorithm 577;

Reference:

1. Roland Bulirsch,,
   Numerical calculation of elliptic integrals and elliptic functions,,
   Numerische Mathematik,,
   Volume 7, Number 1, 1965, pages 78-90.
2. Bille Carlson,
   Computing Elliptic Integrals by Duplication,
   Numerische Mathematik,
   Volume 33, 1979, pages 1-16.
3. Bille Carlson, Elaine Notis,
   Algorithm 577, Algorithms for Incomplete Elliptic Integrals,
   ACM Transactions on Mathematical Software,
   Volume 7, Number 3, pages 398-403, September 1981.

Source Code:

• elliptic_integral.f90, the source code.

Examples and Tests:

• elliptic_integral_test.f90, a sample calling program.
• elliptic_integral_test.txt, the output file.

List of Routines:

• ELLIPTIC_EA evaluates the complete elliptic integral E(A).
• ELLIPTIC_EA_VALUES returns values of the complete elliptic integral E(A).
• ELLIPTIC_EK evaluates the complete elliptic integral E(K).
• ELLIPTIC_EK_VALUES returns values of the complete elliptic integral E(K).
• ELLIPTIC_EM evaluates the complete elliptic integral E(M).
• ELLIPTIC_EM_VALUES returns values of the complete elliptic integral E(M).
• ELLIPTIC_FA evaluates the complete elliptic integral F(A).
• ELLIPTIC_FA_VALUES returns values of the complete elliptic integral F(A).
• ELLIPTIC_FK evaluates the complete elliptic integral F(K).
• ELLIPTIC_FK_VALUES returns values of the complete elliptic integral F(K).
• ELLIPTIC_FM evaluates the complete elliptic integral F(M).
• ELLIPTIC_FM_VALUES returns values of the complete elliptic integral F(M).
• ELLIPTIC_INC_EA evaluates the incomplete elliptic integral E(PHI,A).
• ELLIPTIC_INC_EA_VALUES: values of the incomplete elliptic integral E(PHI,A).
• ELLIPTIC_INC_EK evaluates the incomplete elliptic integral E(PHI,K).
• ELLIPTIC_INC_EK_VALUES: values of the incomplete elliptic integral E(PHI,K).
• ELLIPTIC_INC_EM evaluates the incomplete elliptic integral E(PHI,M).
• ELLIPTIC_INC_EM_VALUES: values of the incomplete elliptic integral E(PHI,M).
• ELLIPTIC_INC_FA evaluates the incomplete elliptic integral F(PHI,A).
• ELLIPTIC_INC_FA_VALUES: values of the incomplete elliptic integral F(PHI,A).
• ELLIPTIC_INC_FK evaluates the incomplete elliptic integral F(PHI,K).
• ELLIPTIC_INC_FK_VALUES: values of the incomplete elliptic integral F(PHI,K).
• ELLIPTIC_INC_FM evaluates the incomplete elliptic integral F(PHI,M).
• ELLIPTIC_INC_FM_VALUES: values of the incomplete elliptic integral F(PHI,M).
• ELLIPTIC_INC_PIA evaluates the incomplete elliptic integral Pi(PHI,N,A).
• ELLIPTIC_INC_PIA_VALUES: values of incomplete elliptic integral Pi(PHI,N,A).
• ELLIPTIC_INC_PIK evaluates the incomplete elliptic integral Pi(PHI,N,K).
• ELLIPTIC_INC_PIK_VALUES: values of incomplete elliptic integral Pi(PHI,N,K).
• ELLIPTIC_INC_PIM evaluates the incomplete elliptic integral Pi(PHI,N,M).
• ELLIPTIC_INC_PIM_VALUES: values of incomplete elliptic integral Pi(PHI,N,M).
• ELLIPTIC_PIA evaluates the complete elliptic integral Pi(N,A).
• ELLIPTIC_PIA_VALUES returns values of the complete elliptic integral Pi(N,A).
• ELLIPTIC_PIK evaluates the complete elliptic integral Pi(N,K).
• ELLIPTIC_PIK_VALUES returns values of the complete elliptic integral Pi(N,K).
• ELLIPTIC_PIM evaluates the complete elliptic integral Pi(N,M).
• ELLIPTIC_PIM_VALUES returns values of the complete elliptic integral Pi(N,M).
• JACOBI_CN evaluates the Jacobi elliptic function CN(U,M).
• JACOBI_CN_VALUES returns some values of the Jacobi elliptic function CN(U,M).
• JACOBI_DN evaluates the Jacobi elliptic function DN(U,M).
• JACOBI_DN_VALUES returns some values of the Jacobi elliptic function DN(U,M).
• JACOBI_SN evaluates the Jacobi elliptic function SN(U,M).
• JACOBI_SN_VALUES returns some values of the Jacobi elliptic function SN(U,M).
• RC computes the elementary integral RC(X,Y).
• RD computes an incomplete elliptic integral of the second kind, RD(X,Y,Z).
• RF computes an incomplete elliptic integral of the first kind, RF(X,Y,Z).
• RJ computes an incomplete elliptic integral of the third kind, RJ(X,Y,Z,P).
• SNCNDN evaluates Jacobi elliptic functions.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.