Clean up imports everywhere.

[numerical-analysis.git] / src / ODE / IVP.hs

diff --git a/src/ODE/IVP.hs b/src/ODE/IVP.hs
index 5d715a6356df5fd291c325a0b831ca469facb94e..6f798f46846a44ef4006145a440f87837cf0a2cc 100644 (file)
--- a/src/ODE/IVP.hs
+++ b/src/ODE/IVP.hs
@@ -1,4 +1,5 @@
 {-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE RebindableSyntax #-}
 
 -- | Numerical solution of the initial value problem,
 --
@@ -8,11 +9,18 @@
 --   for x in [x0, xN].
 --
 
-module ODE.IVP
+module ODE.IVP (
+  eulers_method,
+  eulers_method1,
+  eulers_methodH )
 where
 
-import Misc (partition)
-
+import Misc ( partition )
+import NumericPrelude hiding ( abs )
+import qualified Algebra.Field as Field ( C )
+import qualified Algebra.ToInteger as ToInteger ( C )
+import qualified Algebra.ToRational as ToRational ( C )
+import qualified Algebra.RealField as RealField ( C )
 
 -- | A single iteration of Euler's method over the interval
 --   [$x0$, $x0$+$h$].
@@ -26,7 +34,7 @@ import Misc (partition)
 --   >>> eulers_method1 x0 y0 f h
 --   2.0
 --
-eulers_method1 :: (RealFrac a, RealFrac b)
+eulers_method1 :: (Field.C a, ToRational.C a, Field.C b)
                => a -- ^ x0, the initial point
                -> b -- ^ y0, the initial value at x0
                -> (a -> b -> b) -- ^ The function f(x,y)
@@ -35,7 +43,7 @@ eulers_method1 :: (RealFrac a, RealFrac b)
 eulers_method1 x0 y0 f h =
   y0 +  h'*y'
   where
-    h' = fromRational $ toRational h
+    h' = fromRational'$ toRational h
     y' = (f x0 y0)
 
 
@@ -48,6 +56,7 @@ eulers_method1 x0 y0 f h =
 --
 --   Examples:
 --
+--   >>> import Algebra.Absolute (abs)
 --   >>> let x0 = 0.0
 --   >>> let xN = 1.0
 --   >>> let y0 = 1.0
@@ -56,8 +65,14 @@ eulers_method1 x0 y0 f h =
 --   >>> let yN = head $ reverse ys
 --   >>> abs ((exp 1) - yN) < 1/10^3
 --   True
+--   >>> head ys == y0
+--   True
 --
-eulers_method :: forall a b c. (RealFrac a, RealFrac b, Integral c)
+eulers_method :: forall a b c. (Field.C a,
+                                ToRational.C a,
+                                Field.C b,
+                                ToInteger.C c,
+                                Enum c)
                => a -- ^ x0, the initial point
                -> b -- ^ y0, the initial value at x0
                -> a -- ^ xN, the terminal point
@@ -65,19 +80,20 @@ eulers_method :: forall a b c. (RealFrac a, RealFrac b, Integral c)
                -> c -- ^ n, the number of intervals to use.
                -> [b]
 eulers_method x0 y0 xN f n =
-  go xs y0 f
+  y0 : go xs y0 f
   where
     xs = partition n x0 xN
 
     -- The 'go' function actually does all the work. It takes a list
-    -- of intervals [(x0,x1), (x1, x2)...] and peels off the first
+    -- of intervals [(v0,v1), (v1, v2)...] and peels off the first
     -- one. It then runs the single-step Euler's method on that
     -- interval, and afterwards recurses down the rest of the list.
+    --
     go :: [(a,a)] -> b -> (a -> b -> b) -> [b]
     go [] _ _ = []
-    go ((x0,x1):rest) y0 f = y1 : (go rest y1 f)
+    go ((v0,v1):rest) w0 g = w1 : (go rest w1 g)
       where
-        y1 = eulers_method1 x0 y0 f (x1 - x0)
+        w1 = eulers_method1 v0 w0 g (v1 - v0)
 
 
 -- | Perform as many iterations of Euler's method over the interval
@@ -99,7 +115,7 @@ eulers_method x0 y0 xN f n =
 --   >>> ys == ys'
 --   True
 --
-eulers_methodH :: (RealFrac a, RealFrac b)
+eulers_methodH :: (RealField.C a, ToRational.C a, Field.C b)
                => a -- ^ x0, the initial point
                -> b -- ^ y0, the initial value at x0
                -> a -- ^ xN, the terminal point
@@ -109,4 +125,5 @@ eulers_methodH :: (RealFrac a, RealFrac b)
 eulers_methodH x0 y0 xN f h =
   eulers_method x0 y0 xN f n
   where
+    n :: Integer
     n = floor $ (xN - x0) / h