X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FRoots%2FSimple.hs;h=44d3d62112d2c4f339fa16b65c143e65ed5bb83a;hb=fe73028041fe3becce6ce1ff268181d55d54a011;hp=6e3ff5153b20a7e6ff7589c6b2c77a1914406e02;hpb=807d976941a8dd426ecf43b18b876413a58384f2;p=numerical-analysis.git

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index 6e3ff51..44d3d62 100644
--- a/src/Roots/Simple.hs
+++ b/src/Roots/Simple.hs
@@ -1,3 +1,5 @@
+{-# LANGUAGE RebindableSyntax #-}
+
 -- | The Roots.Simple module contains root-finding algorithms. That
 --   is, procedures to (numerically) find solutions to the equation,
 --
@@ -15,6 +17,11 @@ import Normed
 
 import qualified Roots.Fast as F
 
+import NumericPrelude hiding (abs)
+import Algebra.Absolute
+import Algebra.Field
+import Algebra.Ring
+
 -- | 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
 --   [a,b] by the intermediate value theorem. Likewise when f(a) >= 0
@@ -34,7 +41,11 @@ import qualified Roots.Fast as F
 --   >>> has_root cos (-2) 2 (Just 0.001)
 --   True
 --
-has_root :: (Fractional a, Ord a, Ord b, Num b)
+has_root :: (Algebra.Field.C a,
+             Ord a,
+             Algebra.Ring.C b,
+             Algebra.Absolute.C b,
+             Ord b)
          => (a -> b) -- ^ The function @f@
          -> a       -- ^ The \"left\" endpoint, @a@
          -> a       -- ^ The \"right\" endpoint, @b@
@@ -45,8 +56,6 @@ has_root f a b epsilon =
   F.has_root f a b epsilon Nothing Nothing
 
 
-
-
 -- | We are given a function @f@ and an interval [a,b]. The bisection
 --   method checks finds a root by splitting [a,b] in half repeatedly.
 --
@@ -66,7 +75,11 @@ has_root f a b epsilon =
 --   >>> bisect sin (-1) 1 0.001
 --   Just 0.0
 --
-bisect :: (Fractional a, Ord a, Num b, Ord b)
+bisect :: (Algebra.Field.C a,
+           Ord a,
+           Algebra.Ring.C b,
+           Algebra.Absolute.C b,
+           Ord 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@
@@ -76,6 +89,65 @@ bisect f a b epsilon =
   F.bisect f a b epsilon Nothing Nothing
 
 
+-- | Find a fixed point of the function @f@ with the search starting
+--   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)
+            => (a -> a) -- ^ The function @f@ to iterate.
+            -> b       -- ^ The tolerance, @epsilon@.
+            -> a       -- ^ The initial value @x0@.
+            -> a       -- ^ The fixed point.
+fixed_point f epsilon x0 =
+  snd $ F.fixed_point_with_iterations f epsilon x0
+
+
+-- | Return the number of iterations required to find a fixed point of
+--   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)
+                            => (a -> a) -- ^ The function @f@ to iterate.
+                            -> b       -- ^ The tolerance, @epsilon@.
+                            -> a       -- ^ The initial value @x0@.
+                            -> Int       -- ^ The fixed point.
+fixed_point_iteration_count f epsilon x0 =
+  fst $ F.fixed_point_with_iterations f epsilon x0
+
+
+-- | Returns a list of ratios,
+--
+--     ||x^{*} - x_{n+1}|| / ||x^{*} - x_{n}||^{p}
+--
+--   of fixed point iterations for the function @f@ with initial guess
+--   @x0@ and @p@ some positive power.
+--
+--   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)
+                   => (a -> a) -- ^ The function @f@ to iterate.
+                   -> a       -- ^ The initial value @x0@.
+                   -> a       -- ^ The true solution, @x_star@.
+                   -> Integer -- ^ The power @p@.
+                   -> [b]     -- ^ The resulting sequence of x_{n}.
+fixed_point_error_ratios f x0 x_star p =
+  zipWith (/) en_plus_one en_exp
+  where
+    xn = F.fixed_point_iterations f x0
+    en = map (\x -> norm (x_star - x)) xn
+    en_plus_one = tail en
+    en_exp = map (^p) en
+
+
 
 -- | The sequence x_{n} of values obtained by applying Newton's method
 --   on the function @f@ and initial guess @x0@.
@@ -88,7 +160,7 @@ bisect f a b epsilon =
 --   >>> tail $ take 4 $ newton_iterations f f' 2
 --   [1.6806282722513088,1.4307389882390624,1.2549709561094362]
 --
-newton_iterations :: (Fractional a, Ord a)
+newton_iterations :: (Algebra.Field.C a)
                     => (a -> a) -- ^ The function @f@ whose root we seek
                     -> (a -> a) -- ^ The derivative of @f@
                     -> a       -- ^ Initial guess, x-naught
@@ -100,7 +172,6 @@ newton_iterations f f' x0 =
     xn - ( (f xn) / (f' xn) )
 
 
-
 -- | Use Newton's method to find a root of @f@ near the initial guess
 --   @x0@. If your guess is bad, this will recurse forever!
 --
@@ -124,7 +195,7 @@ newton_iterations f f' x0 =
 --   >>> abs (f root) < eps
 --   True
 --
-newtons_method :: (Fractional a, Ord a)
+newtons_method :: (Algebra.Field.C a, Algebra.Absolute.C a, Ord a)
                  => (a -> a) -- ^ The function @f@ whose root we seek
                  -> (a -> a) -- ^ The derivative of @f@
                  -> a       -- ^ The tolerance epsilon
@@ -136,7 +207,6 @@ newtons_method f f' epsilon x0 =
     x_n = newton_iterations f f' x0
 
 
-
 -- | Takes a function @f@ of two arguments and repeatedly applies @f@
 --   to the previous two values. Returns a list containing all
 --   generated values, f(x0, x1), f(x1, x2), f(x2, x3)...
@@ -158,6 +228,7 @@ iterate2 f x0 x1 =
       let next = f prev2 prev1 in
         next : go prev1 next
 
+
 -- | The sequence x_{n} of values obtained by applying the secant
 --   method on the function @f@ and initial guesses @x0@, @x1@.
 --
@@ -174,7 +245,7 @@ iterate2 f x0 x1 =
 --   >>> take 4 $ secant_iterations f 2 1
 --   [2.0,1.0,1.0161290322580645,1.190577768676638]
 --
-secant_iterations :: (Fractional a, Ord a)
+secant_iterations :: (Algebra.Field.C a)
                     => (a -> a) -- ^ The function @f@ whose root we seek
                     -> a       -- ^ Initial guess, x-naught
                     -> a       -- ^ Second initial guess, x-one
@@ -202,7 +273,7 @@ secant_iterations f x0 x1 =
 --   >>> abs (f root) < (1/10^9)
 --   True
 --
-secant_method :: (Fractional a, Ord a)
+secant_method :: (Algebra.Field.C a, Algebra.Absolute.C a, Ord a)
                  => (a -> a) -- ^ The function @f@ whose root we seek
                  -> a       -- ^ The tolerance epsilon
                  -> a       -- ^ Initial guess, x-naught
@@ -212,54 +283,3 @@ secant_method f epsilon x0 x1
   = find (\x -> abs (f x) < epsilon) x_n
   where
     x_n = secant_iterations f x0 x1
-
-
-
--- | Find a fixed point of the function @f@ with the search starting
---   at x0. We delegate to the version that returns the number of
---   iterations and simply discard the number of iterations.
---
-fixed_point :: (Normed a, RealFrac b)
-            => (a -> a) -- ^ The function @f@ to iterate.
-            -> b       -- ^ The tolerance, @epsilon@.
-            -> a       -- ^ The initial value @x0@.
-            -> a       -- ^ The fixed point.
-fixed_point f epsilon x0 =
-  snd $ F.fixed_point_with_iterations f epsilon x0
-
-
--- | Return the number of iterations required to find a fixed point of
---   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, RealFrac b)
-                            => (a -> a) -- ^ The function @f@ to iterate.
-                            -> b       -- ^ The tolerance, @epsilon@.
-                            -> a       -- ^ The initial value @x0@.
-                            -> Int       -- ^ The fixed point.
-fixed_point_iteration_count f epsilon x0 =
-  fst $ F.fixed_point_with_iterations f epsilon x0
-
-
--- | Returns a list of ratios,
---
---     ||x^{*} - x_{n+1}|| / ||x^{*} - x_{n}||^{p}
---
---   of fixed point iterations for the function @f@ with initial guess
---   @x0@ and @p@ some positive power.
---
---   This is used to determine the rate of convergence.
---
-fixed_point_error_ratios :: (Normed a, RealFrac b)
-                   => (a -> a) -- ^ The function @f@ to iterate.
-                   -> a       -- ^ The initial value @x0@.
-                   -> a       -- ^ The true solution, @x_star@.
-                   -> Integer -- ^ The power @p@.
-                   -> [b]     -- ^ The resulting sequence of x_{n}.
-fixed_point_error_ratios f x0 x_star p =
-  zipWith (/) en_plus_one en_exp
-  where
-    xn = F.fixed_point_iterations f x0
-    en = map (\x -> norm (x_star - x)) xn
-    en_plus_one = tail en
-    en_exp = map (^p) en