Hello,
I would like to compute \int_0^{x_i}log(2^{P_i(t)exp(-t^2/2)} + 1)dt for a large number of values x_i where P_i is a different polynomial (degree <6) for each x_i value.
The naive approach is to use the function quadgk provided in QuadGK.jl for each value x_i and polynomial P_i. Is there a way to speed-up these multiple integrations by reusing computation from one case to the other?
If I would like to use Gaussian quadrature (FastGaussianQuadrature.jl), what type of nodes would you recommend?
See my answer here: Julia integral calculation - community module or own module? - #15 by stevengj
…though if your polynomials change matters are more difficult.
Could you explain more of the context here? Where do your integrals and polynomials come from?
How many is the large number and how broad is their distribution? If you get them all ahead of time, a decent approach would be to sort them, compute the integrals from x[i-1] to x[i] and add them back to get the full intervals.
Sorry, I should have mentioned that the samples x_i as well as the polynomials P_i are independent of each other. Large is about 300-400 different values of x_i.
I guess you never hit the singularity, so if you want to use Gauss quadrature then gausslegendre would probably make the most sense, though it will be a different grid for each x_i