more accurate weighted gauss via chebyshev

[stevengj]

As explained in #41 (comment)

cc @dlfivefifty

[stevengj]

One annoyance with using Chebyshev polynomials here is that they only really work well for finite intervals.

For infinite intervals in x, I do a coordinate remapping to finite intervals in t, but then I'm finding Gaussian quadrature rules for polynomials in t (the remapped coordinate) rather than x, which is not typically what people want.

So, for example, this breaks the example in the README.

[dlfivefifty]

Typically on infinite intervals you want exponential decay in the weight (otherwise polynomial * weight wouldn't converge) but this means you are fighting underflow anyways. A natural way around that is to give V(x) so that w(x) = exp(-V(x)), in other words, specify the logarithm of the weight.

I don't have a clue though what one would do next.

[stevengj]

The difficulty lies in coming up with a good starting polynomial basis to do Gram–Schmidt with for generic W(x) in the infinite case. e.g. you would use a very different bases for [exp(-x²) + exp(-(x-10)²)]/(1+x^2), exp(-|x|), and exp(-0.0001√|x|).