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ellipke
Complete elliptic integrals of the first and second kind
K = ellipke(M)
[K,E] = ellipke(M)
[K,E] = ellipke(M,tol)
The complete elliptic integral of the first kind [1] is:
where F, the elliptic integral of the first kind, is
:
The complete elliptic integral of the second kind,
is
:
Some definitions of K and E use the modulus k instead of the parameter m. They are related by
:
K = ellipke(M)
returns the complete elliptic integral of the first kind for the elements of M.
[K,E] = ellipke(M)
returns the complete elliptic integral of the first and second kinds.
[K,E] = ellipke(M,tol)
computes the Jacobian elliptic functions to accuracy tol. The default is eps; increase this for a less accurate but more quickly computed answer.
ellipke computes the complete elliptic integral using the method of the arithmetic-geometric mean described in [1], section 17.6. It starts with the triplet of numbers:
ellipke computes successive iterations of ai, bi, and ci with
:
stopping at iteration n when cn 
0, within the tolerance specified by eps. The complete elliptic integral of the first kind is then
:
ellipke is limited to the input domain 
.
ellipj Jacobi elliptic functions
[1] Abramowitz, M. and I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, 17.6.