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ellpj
Jacobian Elliptic Functions.SYNOPSIS:
double u, m, sn, cn, dn, phi;
int ellpj();
ellpj(u, m, _&sn, _&cn, _&dn, _&phi);
DESCRIPTION:
Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m), and dn(u|m) of parameter m between 0 and 1, and real argument u.
These functions are periodic, with quarter-period on the real axis equal to the complete elliptic integral ellpk(1.0-m).
Relation to incomplete elliptic integral:
If u = ellik(phi,m), then sn(u|m) = sin(phi), and cn(u|m) = cos(phi).
Phi is called the amplitude of u.
Computation is by means of the arithmetic-geometric mean algorithm, except when m is within 1e-9 of 0 or 1.
In the latter case with m close to 1, the approximation applies only for phi < pi/2.
ACCURACY:
Tested at random points with u between 0 and 10, m between 0 and 1.
Absolute error (* = relative error):
arithmetic function # trials peak rms
DEC sn 1800 4.5e-16 8.7e-17
IEEE phi 10000 9.2e-16* 1.4e-16*
IEEE sn 50000 4.1e-15 4.6e-16
IEEE cn 40000 3.6e-15 4.4e-16
IEEE dn 100000 3.9e-15 1.7e-16
Larger errors occur for m near 1.
Peak error observed in consistency check using addition theorem for sn(u+v) was 4e-16 (absolute). Also tested by the earlier relation to the incomplete elliptic integral. Accuracy deteriorates when u is large.