A question regarding quadrature rules for the weight function w(x)=1/(1+x^2) about quadgk.jl HOT 2 CLOSED

What is your interval (a,b)?

• If the interval includes x=±1, where your w(x) blows up, then your integral does not converge (unless f vanishes at these points), because this is not an integrable singularity. For the same reason, QuadGK.gauss will not work for this weight function.
• Or maybe you want to compute a Cauchy Principal value? In that case, you can use a simple transformation to remove the singularity: #44 (comment)
• If your interval does not include the singularity, why do you need to precompute the effect of the weight function at all? Why not just pass the whole integrand F(x)=w(x)f(x) to quadgk or some other quadrature method? But yes, you could use QuadGK.gauss to obtain a custom weighted quadrature scheme in this case.

from quadgk.jl.

Hi @stevengj,

Thank you very much for your help. So, my interval can arbitrarily be chosen (a, b>=0).

If the interval includes x=±1, where your w(x) blows up, then your integral does not converge (unless f vanishes at these points), because this is not an integrable singularity. For the same reason, QuadGK.gauss will not work for this weight function.

Do you mean that at these values my w(x) will equal to 0? But I think w(x)=1/(1+x^2) = 1 when x=+-1.

Or maybe you want to compute a Cauchy Principal value? In that case, you can use a simple transformation to remove the singularity: #44 (comment)

Yes, indeed. That is what I want. Here my weight function could be (1)w(x) = e^{-x^2} (normal distribution) or (2) w(x) = 1/(1+x^2) (Half-Cauchy distribution, https://distribution-explorer.github.io/continuous/halfcauchy.html). In case (1), we can use Gauss–Laguerre quadrature (https://en.wikipedia.org/wiki/Gauss–Laguerre_quadrature). But in case (2), I could not find a proper a quadrature rule (see https://en.wikipedia.org/wiki/Gaussian_quadrature in Section "Other Forms"). Perhaps, I would need to do a transformation as you suggested to remove the singularity. Or another transformation is x = tan(u). So my integral will be I = \int_{a}^{b} w(x)f(x)dx = \int_{arctan(a)}^{arctan(b)} f(tan(u))du. Then, I can use your function QuadGK.gauss to get notes and weights. I am sorry that I do not know how to display LaTeX formulaes in GitHub.

If your interval does not include the singularity, why do you need to precompute the effect of the weight function at all? Why not just pass the whole integrand F(x)=w(x)f(x) to quadgk or some other quadrature method? But yes, you could use QuadGK.gauss to obtain a custom weighted quadrature scheme in this case.

Yes, indeed. I also thought that. But I thought that using QuadGK.gauss on F(x) might not give a good approximation compared to a special technique using different polynomials on w(x). So, I will try with your suggestions. Also, do you have any suggestions for other quadrature methods that I could look at?

Thank you very much for your help!

from quadgk.jl.