From: Michael Orlitzky <michael@orlitzky.com>
Date: Wed, 20 Feb 2013 23:56:13 +0000 (-0500)
Subject: Use RealField/RealRing where possible instead of their constituents.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=59c49750fd2455574fe4e67ddd7e67a20321c8a8;p=numerical-analysis.git

Use RealField/RealRing where possible instead of their constituents.
Convert src/ODE/IVP.hs to numeric-prelude.
Add RealRing/RealField instanced to BigFloat.
---

diff --git a/src/BigFloat.hs b/src/BigFloat.hs
index e23ebe6..0bebab9 100644
--- a/src/BigFloat.hs
+++ b/src/BigFloat.hs
@@ -10,6 +10,8 @@ import NumericPrelude hiding (abs)
 import qualified Algebra.Absolute as Absolute
 import qualified Algebra.Additive as Additive
 import qualified Algebra.Field as Field
+import qualified Algebra.RealField as RealField
+import qualified Algebra.RealRing as RealRing
 import qualified Algebra.Ring as Ring
 import qualified Algebra.ToRational as ToRational
 import qualified Algebra.ZeroTestable as ZeroTestable
@@ -38,3 +40,8 @@ instance Epsilon e => ZeroTestable.C (BigFloat e) where
 
 instance Epsilon e => ToRational.C (BigFloat e) where
   toRational = fromRational . P.toRational
+
+instance Epsilon e => RealRing.C (BigFloat e) where
+  floor = fromInteger . P.floor
+
+instance Epsilon e => RealField.C (BigFloat e)
diff --git a/src/ODE/IVP.hs b/src/ODE/IVP.hs
index 6bde763..b4fd3dc 100644
--- 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,
 --
@@ -12,7 +13,12 @@ module ODE.IVP
 where
 
 import Misc (partition)
-
+import NumericPrelude hiding (abs)
+import Algebra.Absolute (abs)
+import qualified Algebra.Field as Field
+import qualified Algebra.ToInteger as ToInteger
+import qualified Algebra.ToRational as ToRational
+import qualified Algebra.RealField as RealField
 
 -- | A single iteration of Euler's method over the interval
 --   [$x0$, $x0$+$h$].
@@ -26,7 +32,7 @@ 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 +41,7 @@ eulers_method1 :: (RealFrac a, RealFrac b)
 eulers_method1 x0 y0 f h =
   y0 +  h'*y'
   where
-    h' = realToFrac h
+    h' = fromRational'$ toRational h
     y' = (f x0 y0)
 
 
@@ -59,7 +65,11 @@ eulers_method1 x0 y0 f h =
 --   >>> 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
@@ -102,7 +112,7 @@ 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
diff --git a/src/Roots/Fast.hs b/src/Roots/Fast.hs
index 47fa512..78c299a 100644
--- a/src/Roots/Fast.hs
+++ b/src/Roots/Fast.hs
@@ -13,15 +13,14 @@ import Data.List (find)
 import Normed
 
 import NumericPrelude hiding (abs)
-import Algebra.Absolute
-import Algebra.Field
-import Algebra.Ring
-
-has_root :: (Algebra.Field.C a,
-             Ord a,
-             Algebra.Ring.C b,
-             Ord b,
-             Algebra.Absolute.C b)
+import qualified Algebra.Absolute as Absolute
+import qualified Algebra.Field as Field
+import qualified Algebra.RealRing as RealRing
+import qualified Algebra.RealField as RealField
+
+has_root :: (RealField.C a,
+             RealRing.C b,
+             Absolute.C b)
          => (a -> b) -- ^ The function @f@
          -> a       -- ^ The \"left\" endpoint, @a@
          -> a       -- ^ The \"right\" endpoint, @b@
@@ -61,11 +60,9 @@ has_root f a b epsilon f_of_a f_of_b =
     c = (a + b)/2
 
 
-bisect :: (Algebra.Field.C a,
-           Ord a,
-           Algebra.Ring.C b,
-           Ord b,
-           Algebra.Absolute.C b)
+bisect :: (RealField.C a,
+           RealRing.C b,
+           Absolute.C b)
        => (a -> b) -- ^ The function @f@ whose root we seek
        -> a       -- ^ The \"left\" endpoint of the interval, @a@
        -> a       -- ^ The \"right\" endpoint of the interval, @b@
@@ -119,8 +116,8 @@ fixed_point_iterations f x0 =
 --   We also return the number of iterations required.
 --
 fixed_point_with_iterations :: (Normed a,
-                                Algebra.Field.C b,
-                                Algebra.Absolute.C b,
+                                Field.C b,
+                                Absolute.C b,
                                 Ord b)
                             => (a -> a)  -- ^ The function @f@ to iterate.
                             -> b        -- ^ The tolerance, @epsilon@.
diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index 44d3d62..0a1debf 100644
--- a/src/Roots/Simple.hs
+++ b/src/Roots/Simple.hs
@@ -18,9 +18,11 @@ import Normed
 import qualified Roots.Fast as F
 
 import NumericPrelude hiding (abs)
-import Algebra.Absolute
-import Algebra.Field
-import Algebra.Ring
+import qualified Algebra.Absolute as Absolute
+import Algebra.Absolute (abs)
+import qualified Algebra.Field as Field
+import qualified Algebra.RealField as RealField
+import qualified Algebra.RealRing as RealRing
 
 -- | Does the (continuous) function @f@ have a root on the interval
 --   [a,b]? If f(a) <] 0 and f(b) ]> 0, we know that there's a root in
@@ -41,11 +43,7 @@ import Algebra.Ring
 --   >>> has_root cos (-2) 2 (Just 0.001)
 --   True
 --
-has_root :: (Algebra.Field.C a,
-             Ord a,
-             Algebra.Ring.C b,
-             Algebra.Absolute.C b,
-             Ord b)
+has_root :: (RealField.C a, RealRing.C b)
          => (a -> b) -- ^ The function @f@
          -> a       -- ^ The \"left\" endpoint, @a@
          -> a       -- ^ The \"right\" endpoint, @b@
@@ -75,11 +73,7 @@ has_root f a b epsilon =
 --   >>> bisect sin (-1) 1 0.001
 --   Just 0.0
 --
-bisect :: (Algebra.Field.C a,
-           Ord a,
-           Algebra.Ring.C b,
-           Algebra.Absolute.C b,
-           Ord b)
+bisect :: (RealField.C a, RealRing.C b)
        => (a -> b) -- ^ The function @f@ whose root we seek
        -> a       -- ^ The \"left\" endpoint of the interval, @a@
        -> a       -- ^ The \"right\" endpoint of the interval, @b@
@@ -93,10 +87,7 @@ bisect f a b epsilon =
 --   at x0. We delegate to the version that returns the number of
 --   iterations and simply discard the number of iterations.
 --
-fixed_point :: (Normed a,
-                Algebra.Field.C b,
-                Algebra.Absolute.C b,
-                Ord b)
+fixed_point :: (Normed a, RealField.C b)
             => (a -> a) -- ^ The function @f@ to iterate.
             -> b       -- ^ The tolerance, @epsilon@.
             -> a       -- ^ The initial value @x0@.
@@ -109,10 +100,7 @@ fixed_point f epsilon x0 =
 --   the function @f@ with the search starting at x0 and tolerance
 --   @epsilon@. We delegate to the version that returns the number of
 --   iterations and simply discard the fixed point.
-fixed_point_iteration_count :: (Normed a,
-                                Algebra.Field.C b,
-                                Algebra.Absolute.C b,
-                                Ord b)
+fixed_point_iteration_count :: (Normed a, RealField.C b)
                             => (a -> a) -- ^ The function @f@ to iterate.
                             -> b       -- ^ The tolerance, @epsilon@.
                             -> a       -- ^ The initial value @x0@.
@@ -130,10 +118,7 @@ fixed_point_iteration_count f epsilon x0 =
 --
 --   This is used to determine the rate of convergence.
 --
-fixed_point_error_ratios :: (Normed a,
-                             Algebra.Field.C b,
-                             Algebra.Absolute.C b,
-                             Ord b)
+fixed_point_error_ratios :: (Normed a, RealField.C b)
                    => (a -> a) -- ^ The function @f@ to iterate.
                    -> a       -- ^ The initial value @x0@.
                    -> a       -- ^ The true solution, @x_star@.
@@ -160,7 +145,7 @@ fixed_point_error_ratios f x0 x_star p =
 --   >>> tail $ take 4 $ newton_iterations f f' 2
 --   [1.6806282722513088,1.4307389882390624,1.2549709561094362]
 --
-newton_iterations :: (Algebra.Field.C a)
+newton_iterations :: (Field.C a)
                     => (a -> a) -- ^ The function @f@ whose root we seek
                     -> (a -> a) -- ^ The derivative of @f@
                     -> a       -- ^ Initial guess, x-naught
@@ -195,7 +180,7 @@ newton_iterations f f' x0 =
 --   >>> abs (f root) < eps
 --   True
 --
-newtons_method :: (Algebra.Field.C a, Algebra.Absolute.C a, Ord a)
+newtons_method :: (RealField.C a)
                  => (a -> a) -- ^ The function @f@ whose root we seek
                  -> (a -> a) -- ^ The derivative of @f@
                  -> a       -- ^ The tolerance epsilon
@@ -245,7 +230,7 @@ iterate2 f x0 x1 =
 --   >>> take 4 $ secant_iterations f 2 1
 --   [2.0,1.0,1.0161290322580645,1.190577768676638]
 --
-secant_iterations :: (Algebra.Field.C a)
+secant_iterations :: (Field.C a)
                     => (a -> a) -- ^ The function @f@ whose root we seek
                     -> a       -- ^ Initial guess, x-naught
                     -> a       -- ^ Second initial guess, x-one
@@ -273,7 +258,7 @@ secant_iterations f x0 x1 =
 --   >>> abs (f root) < (1/10^9)
 --   True
 --
-secant_method :: (Algebra.Field.C a, Algebra.Absolute.C a, Ord a)
+secant_method :: (RealField.C a)
                  => (a -> a) -- ^ The function @f@ whose root we seek
                  -> a       -- ^ The tolerance epsilon
                  -> a       -- ^ Initial guess, x-naught