Further accuracy by switching basis from `s_n` to `M_n`?

rodluger / Limbdark.jl

Analytical transit light curves for limb darkened stars

MIT License

14 stars 4 forks source link

Further accuracy by switching basis from `s_n` to `M_n`? #86

Closed ericagol closed 5 years ago

ericagol commented 5 years ago

[ ] Would M_n be a better basis than s_n? Could we use recursion relations to get coefficients in terms of M_0...M_3 (or M_{n-3}...M_n)? Then we would simply need to compute the integral as a sum over these coefficients. Also, I'm wondering if these coefficients could be computed at higher precision, which might lead to higher precision for large values of n_u (which is discussed in issue #43)?

ericagol commented 5 years ago

I think that the inaccuracy at large values of n_u occurs due to odd and even terms nearly cancelling (to a part in 10^{13} in the case of n_u=50 and u_i = 1/n_u). So, if we added up the coefficients of these terms at higher precision, perhaps the result would be more numerically stable. We would still get a cancellation between the odd and even terms, but the sums would only cancel once at lower precision (assuming we compute M_n at double precision).

ericagol commented 5 years ago

I'm not sure if this is going to work; will close this issue for now.