[ANN] Packages for integration

Announcements
1

Hello,

I have uploaded two new packages to Hackage:

Let me also mention another package of mine, older, for integration on a simplex: scubature (pure Haskell implementation).

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2

I wonder how it compares to the integration package. I assume it has better performance and/or precision?

3

Ah this is “tanh-sinh”. Yes, numerical-integration is probably better. I experimented it in R (package RcppNumerical) and it gives good results even for highly oscillatory functions.

4

You gave my an idea: it would be interesting to compare the two packages on a couple of functions. Maybe I’ll do it.

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5

Here are two functions whose integral is hard to compute, because the functions are highly oscillatory. The exact values of the integrals are given in the code.

import Numerical.Integration

example :: IO (Double, Double, Int) -- value, error estimate, error code
example = integration (\t -> 8 * cos(2*t/(1-t)) / (64*(1-t)**2 + t**2)) 0 1 1e-4 100000

value :: Double -- approx 1.7677e-7
value = exp (-6) * pi / (2 * exp 10)


example' :: IO (Double, Double, Int) -- value, error estimate, error code
example' = integration (\t -> 6 * cos(2*t/(1-t)) / (36*(1-t)**2 + t**2)) 0 1 1e-4 50000

value' :: Double -- approx 9.6513e-6 
value' = exp (-6) * pi / (2 * exp 6)

For the first one, using 100000 subdivisions of the interval (0, 1), the result (after a while) is approximately 1.756e-7

For the second one, using 50000 subdivisions of the interval (0, 1), the result is approximately 9.649e-6.

So the results are not bad.

I didn’t try “tanh-sinh” but I would bet it provides bad results.

A note: the error code (which should be 0) is totally crazy, a huge number, I don’t know why.