I am currently working in MATLAB (this is not relevant to the question, though), and MATLAB doesn't have built-in Mathieu functions (such as the Mathieu cosine and sine).
In principle this is just fine, because I can generate them with a RK or predictor corrector method. However, this calculates all the points in the function up to a certain $x$, and that is not very efficient if I'm only interested in one specific $f(x_i)$, for a given $x_i$.
Could you provide references with explicit algorithms to calculate the Mathieu sine and cosine functions? Any other comments or suggestions are of course also welcome.
Edit: to clarify, I'm looking to generate the functions as defined here or here (where they are referred as basic solutions). I'm not looking for the periodic solutions, but the (normalized) odd and even fundamental solutions of the equation $$ \ddot{x} + (a - 2q\cos 2 \tau)x = 0 $$ and also the characteristic exponent $\nu(a,q)$