+{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE RebindableSyntax #-}
-module Integration.Simpson
+module Integration.Simpson (
+ simpson,
+ simpson_1 )
where
-import Misc (partition)
+import Misc ( partition )
-import NumericPrelude hiding (abs)
-import Algebra.Absolute (abs)
-import qualified Algebra.Field as Field
-import qualified Algebra.RealField as RealField
-import qualified Algebra.RealRing as RealRing
-import qualified Algebra.ToInteger as ToInteger
-import qualified Algebra.ToRational as ToRational
+import NumericPrelude hiding ( abs )
+import qualified Algebra.RealField as RealField ( C )
+import qualified Algebra.ToInteger as ToInteger ( C )
+import qualified Algebra.ToRational as ToRational ( C )
-- | Use the Simpson's rule to numerically integrate @f@ over the
-- interval [@a@, @b@].
-- >>> simpson_1 f (-1) 1
-- 0.0
--
+-- >>> import Algebra.Absolute (abs)
-- >>> let f x = x^2
-- >>> let area = simpson_1 f (-1) 1
-- >>> abs (area - (2/3)) < 1/10^12
-- >>> simpson_1 f 0 1
-- 0.25
--
-simpson_1 :: (RealField.C a, ToRational.C a, RealField.C b)
+simpson_1 :: forall a b. (RealField.C a, ToRational.C a, RealField.C b)
=> (a -> b) -- ^ The function @f@
-> a -- ^ The \"left\" endpoint, @a@
-> a -- ^ The \"right\" endpoint, @b@
-> b
-simpson_1 f a b =
- coefficient * ((f a) + 4*(f midpoint) + (f b))
+simpson_1 f x y =
+ coefficient * ((f x) + 4*(f midpoint) + (f y))
where
- coefficient = (fromRational' $ toRational (b - a)) / 6
- midpoint = (a + b) / 2
+ coefficient = fromRational' $ (toRational (y - x)) / 6 :: b
+ midpoint = (x + y) / 2
-- | Use the composite Simpson's rule to numerically integrate @f@
--
-- Examples:
--
+-- >>> import Algebra.Absolute (abs)
-- >>> let f x = x^4
-- >>> let area = simpson 10 f (-1) 1
-- >>> abs (area - (2/5)) < 0.0001
-- Note that the convergence here is much faster than the Trapezoid
-- rule!
--
+-- >>> import Algebra.Absolute (abs)
-- >>> let area = simpson 10 sin 0 pi
-- >>> abs (area - 2) < 0.00001
-- True