+{-# LANGUAGE RebindableSyntax #-}
+
module Integration.Trapezoid
where
+import Misc (partition)
--- | Partition the interval [@a@, @b@] into @n@ subintervals, which we
--- then return as a list of pairs.
-partition :: (RealFrac a, Integral b)
- => b -- ^ The number of subintervals to use, @n@
- -> a -- ^ The \"left\" endpoint of the interval, @a@
- -> a -- ^ The \"right\" endpoint of the interval, @b@
- -> [(a,a)]
- -- Somebody asked for zero subintervals? Ok.
-partition 0 _ _ = []
-partition n a b
- | n < 0 = error "partition: asked for a negative number of subintervals"
- | otherwise =
- [ (xi, xj) | k <- [0..n-1],
- let k' = fromIntegral k,
- let xi = a + k'*h,
- let xj = a + (k'+1)*h ]
- where
- h = fromRational $ (toRational (b-a))/(toRational n)
-
+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) * (fromRational $ toRational (b - a))
+ (((f a) + (f b)) / 2) * (fromRational' $ toRational (b - a))
--- | Use the composite trapezoid tule to numerically integrate @f@
+-- | 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
+-- >>> abs (area - (2/3)) < 0.00001
-- True
--
--- >>> let area = trapezoid 1000 sin (-1) 1
--- >>> abs (area - 2) < 0.00001
+-- >>> 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@