C318: Jacobian Elliptic Functions sn, cn, dn

Author(s): K.S. Kölbig, H.-H. Umstätter Library: MATHLIB
Submitter: Submitted: 30.01.1980
Language: Fortran Revised: 01.12.1994

Function subprograms RELFUN and DELFUN calculate, for real argument x and real modulus k, the Jacobian elliptic functions tex2html_wrap_inline127 , tex2html_wrap_inline129 and tex2html_wrap_inline131 . The function tex2html_wrap_inline133 is the inverse of the elliptic integral of the first kind and is defined implicitly by

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The functions tex2html_wrap_inline137 and tex2html_wrap_inline139 are defined by

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This definition can be extended for tex2html_wrap_inline143 (Ref. 2) by means of

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where tex2html_wrap_inline147 . For k = 0 and tex2html_wrap_inline151 these functions are elementary:

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Note that for tex2html_wrap_inline157 the Jacobian elliptic functions are periodic (with respect to x) with period tex2html_wrap_inline161 if tex2html_wrap_inline163 and tex2html_wrap_inline165 if tex2html_wrap_inline167 , where tex2html_wrap_inline169 is the complete elliptic integral of the first kind.

On CDC and Cray computers, the double-precision version DELFUN is not available.

Structure:

SUBROUTINE subprograms
User Entry Names: RELFUN, DELFUN
Obsolete User Entry Names: ELFUN tex2html_wrap_inline171 RELFUN

Usage:

For tex2html_wrap_inline173 (type REAL), tex2html_wrap_inline175 (type DOUBLE PRECISION), 

    CALL tELFUN(X,AK2,SN,CN,DN)
X
(type according to t) The argument x.
AK2
(type according to t) The value of tex2html_wrap_inline179 (the square of the modulus).
SN
(type according to t) On exit, tex2html_wrap_inline181 .
CN
(type according to t) On exit, tex2html_wrap_inline183 .
DN
(type according to t) On exit, tex2html_wrap_inline185 .

Method:

The sequence of the Gaussian arithmetic-geometric mean is used together with the Gauss transformation and, where appropriate, the Jacobi imaginary transformation. For values tex2html_wrap_inline187 , the reciprocal modulus transformation is performed. For details see References.

Accuracy:

RELFUN (except on CDC and Cray computers) has full single-precision accuracy. For most values of the arguments, DELFUN (and RELFUN on CDC and Cray computers) has an accuracy of approximately two significant digits less than the machine precision.

Restrictions:

tex2html_wrap_inline189 tex2html_wrap_inline191 , tex2html_wrap_inline193 tex2html_wrap_inline195 , where tex2html_wrap_inline197 is the complete elliptic integral of the first kind. (See entries RELIKC and DELIKC in RELI1C (C347)).

References:

  1. M. Abramowitz and I.A. Stegun, ed., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Sections 16.12 and 17.6, 9th printing with corrections, (Dover, New York 1972).
  2. H.E. Salzer, Quick calculation of Jacobian elliptic functions, Comm. ACM 5 (1962) 399.
  3. L.V. King, On the dirct numerical calculation of elliptic functions and integrals, Cambridge Univ. Press (1924) 32.
  4. D.J. Hofsommer and R.P. van de Riet, On the numerical calculation of elliptic integrals of the first and second kind and the elliptic functions of Jacobi, Numer. Math. 5 (1963) 291-302.
  5. P.F. Byrd and M.D. Friedman, Handbook of elliptic integrals for engineers and scientists, 2nd ed., Springer-Verlag Berlin (1971).
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Michel Goossens Tue Jun 4 21:32:34 METDST 1996