Clean up imports everywhere.

[numerical-analysis.git] / src / Integration / Simpson.hs

diff --git a/src/Integration/Simpson.hs b/src/Integration/Simpson.hs
index b7000224764e92095398837dedf3d56eaf4ee96c..f0f57f29245db1a7db7bd10bd05a39a87c2347df 100644 (file)
--- a/src/Integration/Simpson.hs
+++ b/src/Integration/Simpson.hs
@@ -1,8 +1,17 @@
-module Integration.Simpson
+{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE RebindableSyntax #-}
+
+module Integration.Simpson (
+  simpson,
+  simpson_1 )
 where
 
-import Misc (partition)
+import Misc ( partition )
 
+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@].
@@ -17,6 +26,7 @@ import Misc (partition)
 --   >>> 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
@@ -30,7 +40,7 @@ import Misc (partition)
 --   >>> simpson_1 f 0 1
 --   0.25
 --
-simpson_1 :: (RealFrac a, Fractional b, Num b)
+simpson_1 :: (RealField.C a, ToRational.C a, RealField.C b)
             => (a -> b) -- ^ The function @f@
             -> a       -- ^ The \"left\" endpoint, @a@
             -> a       -- ^ The \"right\" endpoint, @b@
@@ -38,7 +48,7 @@ simpson_1 :: (RealFrac a, Fractional b, Num b)
 simpson_1 f a b =
   coefficient * ((f a) + 4*(f midpoint) + (f b))
   where
-    coefficient = (fromRational $ toRational (b - a)) / 6
+    coefficient = fromRational' $ (toRational (b - a)) / 6
     midpoint = (a + b) / 2
 
 
@@ -47,6 +57,7 @@ simpson_1 f a b =
 --
 --   Examples:
 --
+--   >>> import Algebra.Absolute (abs)
 --   >>> let f x = x^4
 --   >>> let area = simpson 10 f (-1) 1
 --   >>> abs (area - (2/5)) < 0.0001
@@ -55,11 +66,16 @@ simpson_1 f a b =
 --   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
 --
-simpson :: (RealFrac a, Fractional b, Num b, Integral c)
+simpson :: (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@