Differentiate and integrate in Haskell

Use HMath.Calculus for checked finite differences and numerical quadrature of Double callbacks. Callbacks must terminate and be side-effect free. These operations are Haskell APIs; they are not Python tensor methods.

Use an explicit step and panel count

Estimate the derivative of \(x^2\) at 3 and its integral from 0 to 1:

import qualified HMath.Calculus as Calculus

main :: IO ()
main = do
    let derivative = Calculus.derivative Calculus.Central 1e-5 (\x -> x*x) 3
        integral = Calculus.integrateComposite Calculus.Simpson 100 (\x -> x*x) 0 1
        expect expected result = case result of
            Left err -> fail (show err)
            Right actual ->
                if abs (actual - expected) <= 1e-8
                    then pure ()
                    else fail ("Unexpected estimate: " ++ show actual)
    expect 6 derivative
    expect (1/3) integral
    putStrLn "derivative and integral checks passed"

The expected output is derivative and integral checks passed. The differentiation step must be positive, finite, and representable at the evaluation point. Making it arbitrarily small can worsen cancellation. Simpson's panel count must be positive and even; ThreeEighths requires a multiple of three. A fixed-panel calculation does not return an error bound.

Request adaptive refinement

import qualified HMath.Calculus as Calculus

main :: IO ()
main = case Calculus.adaptiveIntegrate Calculus.defaultTolerance (\x -> x*x) 0 1 of
    Left err -> fail (show err)
    Right report -> do
        if abs (Calculus.estimateValue report - 1/3) <= 1e-9
            && Calculus.estimateError report >= 0
            && Calculus.evaluations report <= 100000
            then putStrLn "adaptive integration check passed"
            else fail (show report)

The default adaptive tolerance uses 1e-10 absolute, 1e-8 relative, and 100,000 evaluations. Its stopping target is \(\max(\mathrm{absoluteTolerance},\ \mathrm{relativeTolerance}\,|\mathrm{estimateValue}|)\). Absolute tolerance must be positive; relative tolerance must be nonnegative; both must be finite. A nonempty interval needs at least 15 evaluations.

If refinement cannot meet the target within the budget, the adaptive integrator returns Left (NonConvergence ...). Non-finite sampled values also fail. estimateError is an estimate from the approximation rule, not a rigorous bound for arbitrary callbacks. Inspect the full Calculus API and numerical conventions.