Incorrect error estimate depending on a very specific value of a parameter about quadgk.jl HOT 6 CLOSED

It's not at all an "easy" integral, because the integrand has a slope discontinuity in the middle of the integration region. QuadGK, like essentially all integration algorithms, is best for smooth integrands. The discontinuity not only messes up the convergence rate, it also messes up the error estimate.

(It similarly screws up the BigFloat integral, so your error estimate is not accurate.)

from quadgk.jl.

An accurate way to do this integral is to break the integration interval up into the regions before and after the discontinuity:

julia> x1,e1 = QuadGK.quadgk(x -> exp(abs(x - 0.499)), 0, 1) # inaccurate
(1.297443190121581, 2.220446049250313e-16)

julia> x2,e2 = QuadGK.quadgk(x -> exp(abs(x - 0.499)), 0, 0.499, 1) # accurate
(1.2974441901216642, 1.1102230246251565e-16)

julia> x1-x2 # error
-1.0000000831844602e-6

The same procedure shows the error with 0.4999 is not 1e-17:

julia> x1,e1 = QuadGK.quadgk(x -> exp(abs(x - 0.4999)), 0, 1)
(1.297442547887469, 1.1102230246251565e-16)

julia> x2,e2 = QuadGK.quadgk(x -> exp(abs(x - 0.4999)), 0, 0.4999, 1)
(1.2974425578874689, 1.1102230246251565e-16)

julia> x1-x2
-9.99999993922529e-9

from quadgk.jl.

Yikes, I see, I messed that up, I thought the BigFloat version was okay.

from quadgk.jl.

Would you consider the incorrect error estimate a bug on its own? (I was reading https://www.research-collection.ethz.ch/bitstream/handle/20.500.11850/13300/eth-65-02.pdf and just wanted to see what QuadGK would do on some of the test functions.)

from quadgk.jl.

I wouldn't consider that a bug, I would consider that an inherent limitation/tradeoff of Gauss–Kronrod adaptive quadrature. It would be great to have additional quadrature schemes, of course.

from quadgk.jl.

Note that if you have a non-smooth integrand, you are better off with a lower-order quadrature — for this integrand, I get about the same accuracy, a better error estimate, and about half as many function evaluations if I pass order=3 to quadgk (the default is order=7).

from quadgk.jl.