+{-# LANGUAGE RebindableSyntax #-}
+
module Integration.Trapezoid
where
import Misc (partition)
+import NumericPrelude hiding (abs)
+import qualified Algebra.Field as Field
+import qualified Algebra.RealField as RealField
+import qualified Algebra.ToInteger as ToInteger
+import qualified Algebra.ToRational as ToRational
+
-- | Use the trapezoid rule to numerically integrate @f@ over the
-- interval [@a@, @b@].
--
-- >>> trapezoid_1 f (-1) 1
-- 2.0
--
-trapezoid_1 :: (RealFrac a, Fractional b, Num b)
+trapezoid_1 :: (Field.C a, ToRational.C a, Field.C b)
=> (a -> b) -- ^ The function @f@
-> a -- ^ The \"left\" endpoint, @a@
-> a -- ^ The \"right\" endpoint, @b@
-> b
trapezoid_1 f a b =
- (((f a) + (f b)) / 2) * (realToFrac (b - a))
-
+ (((f a) + (f b)) / 2) * coerced_interval_length
+ where
+ coerced_interval_length = fromRational' $ toRational (b - a)
-- | Use the composite trapezoid rule to numerically integrate @f@
-- over @n@ subintervals of [@a@, @b@].
--
-- Examples:
--
+-- >>> import Algebra.Absolute (abs)
-- >>> let f x = x^2
-- >>> let area = trapezoid 1000 f (-1) 1
-- >>> abs (area - (2/3)) < 0.00001
-- True
--
+-- >>> import Algebra.Absolute (abs)
-- >>> let area = trapezoid 1000 sin 0 pi
-- >>> abs (area - 2) < 0.0001
-- True
--
-trapezoid :: (RealFrac a, Fractional b, Num b, Integral c)
+trapezoid :: (RealField.C a,
+ ToRational.C a,
+ RealField.C b,
+ ToInteger.C c,
+ Enum c)
=> c -- ^ The number of subintervals to use, @n@
-> (a -> b) -- ^ The function @f@
-> a -- ^ The \"left\" endpoint, @a@