elliptic12 sometimes gives incorrect results

[cffk]

The function elliptic12 returns incorrect results for phase equal to some odd multiples of pi/2. For example

m = 0.5;
n = 17;
elliptic12(n*pi/2, m) - n*ellipke(m)
==>
ans = -1.8541

The correct answer is ans = 0 (for any integer n).

[moiseevigor]

Fixed in 9967c61.

Root cause: tan(n*pi/2) is numerically unstable for large n (e.g. n=17). The Landen descent accumulates this error across iterations.

Fix (MATLAB elliptic12.m): Explicit period reduction before the Landen loop using the identity F(u + k*pi | m) = F(u | m) + 2*k*K(m). K(m) is derived directly from the AGM limit already computed: K = pi / (2 * a_inf).

k_per = round(u_work ./ pi);       % strip full half-periods
phin0 = u_work - k_per .* pi;      % reduced to (-pi/2, pi/2]
% ... Landen descent on phin0 ...
Ff = phin ./ (a(mn,K).*e*2) + k_per .* K_vals(K);   % add back periods

Python: Was already correct — uses k = round(u / pi) period reduction in elliptic12.py.

Verified in Python for n = 1, 3, 5, 17, 101 with m = 0.5: error < 1e-14 in all cases.