On Thursday, June 5, 2014 5:55:42 AM UTC-4, Hans W Borchers wrote: > > No, I don't like rational approximations too much. Sometimes the degrees of > the polynomials need to be quite high, and this can sometimes lead to > numerical > instabilities. >
(If the degrees are high, you can just chop up the domain into smaller pieces. You can even go to an extreme of using a lookup table of 50 or 100 different approximants, although this is harder to do in Julia because of the lack of a switch statement or equivalent.) In this case, the power series expansion is quite effective (for smaller > imaginary parts), and continued fractions is a technique I personally like > very much. And a similar approach works for E(n,x). > Good to know! > There are also relations > to reduce E(n+1,.) to E(n,.). > Although for large n it would be nice to have an algorithm that is not linear in n, and a colleague of mine came across an application where he needed the extension to complex n. > Above all it works for complex numbers. Convergence is sufficient for the > upper > half plane and extends to the whole complex plane through the symmetry > relation. > Yes, it's definitely nice to have a single implementation that works for both real and complex numbers, even if it is slightly subm
