X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FRoots%2FFast.hs;h=a879950556623fbcc1334fc133abc49ca7c6da30;hb=d2222aa1c04ba9c0eb80a03149abdf7d86eca032;hp=f7cde8a82a70f3f98605ab87fc94fae5a59147dc;hpb=04f56a8882bb0c574b603f8c3fed9481ea934f7f;p=numerical-analysis.git

diff --git a/src/Roots/Fast.hs b/src/Roots/Fast.hs
index f7cde8a..a879950 100644
--- a/src/Roots/Fast.hs
+++ b/src/Roots/Fast.hs
@@ -1,23 +1,31 @@
 {-# LANGUAGE RebindableSyntax #-}
+{-# LANGUAGE ScopedTypeVariables #-}
 
 -- | The Roots.Fast module contains faster implementations of the
 --   'Roots.Simple' algorithms. Generally, we will pass precomputed
 --   values to the next iteration of a function rather than passing
 --   the function and the points at which to (re)evaluate it.
 
-module Roots.Fast
+module Roots.Fast (
+  bisect,
+  fixed_point_iterations,
+  fixed_point_with_iterations,
+  has_root,
+  trisect )
 where
 
-import Data.List (find)
+import Data.List ( find )
+import Data.Maybe ( fromMaybe )
 
-import Normed
+import Normed ( Normed(..) )
+
+import NumericPrelude hiding ( abs )
+import qualified Algebra.Absolute as Absolute ( C )
+import qualified Algebra.Additive as Additive ( C )
+import qualified Algebra.Algebraic as Algebraic ( C )
+import qualified Algebra.RealRing as RealRing ( C )
+import qualified Algebra.RealField as RealField ( C )
 
-import NumericPrelude hiding (abs)
-import qualified Algebra.Absolute as Absolute
-import qualified Algebra.Additive as Additive
-import qualified Algebra.Algebraic as Algebraic
-import qualified Algebra.RealRing as RealRing
-import qualified Algebra.RealField as RealField
 
 has_root :: (RealField.C a,
              RealRing.C b,
@@ -30,33 +38,20 @@ has_root :: (RealField.C a,
          -> Maybe b -- ^ Precoumpted f(a)
          -> Maybe b -- ^ Precoumpted f(b)
          -> Bool
-has_root f a b epsilon f_of_a f_of_b =
-  if not ((signum (f_of_a')) * (signum (f_of_b')) == 1) then
-    -- We don't care about epsilon here, there's definitely a root!
-    True
-  else
-    if (b - a) <= epsilon' then
-      -- Give up, return false.
-      False
-    else
-      -- If either [a,c] or [c,b] have roots, we do too.
+has_root f a b epsilon f_of_a f_of_b
+  | (signum (f_of_a')) * (signum (f_of_b')) /= 1 = True
+  | (b - a) <= epsilon' = False
+  | otherwise =
       (has_root f a c (Just epsilon') (Just f_of_a') Nothing) ||
         (has_root f c b (Just epsilon') Nothing (Just f_of_b'))
   where
     -- If the size of the smallest subinterval is not specified,
     -- assume we just want to check once on all of [a,b].
-    epsilon' = case epsilon of
-                 Nothing -> (b-a)
-                 Just eps -> eps
+    epsilon' = fromMaybe (b-a) epsilon
 
     -- Compute f(a) and f(b) only if needed.
-    f_of_a'  = case f_of_a of
-                 Nothing -> f a
-                 Just v -> v
-
-    f_of_b'  = case f_of_b of
-                 Nothing -> f b
-                 Just v -> v
+    f_of_a'  = fromMaybe (f a) f_of_a
+    f_of_b'  = fromMaybe (f b) f_of_b
 
     c = (a + b)/2
 
@@ -87,13 +82,8 @@ bisect f a b epsilon f_of_a f_of_b
         else bisect f c b epsilon (Just f_of_c') (Just f_of_b')
   where
     -- Compute f(a) and f(b) only if needed.
-    f_of_a'  = case f_of_a of
-                 Nothing -> f a
-                 Just v -> v
-
-    f_of_b'  = case f_of_b of
-                 Nothing -> f b
-                 Just v -> v
+    f_of_a'  = fromMaybe (f a) f_of_a
+    f_of_b'  = fromMaybe (f b) f_of_b
 
     c = (a + b) / 2
 
@@ -118,26 +108,21 @@ trisect f a b epsilon f_of_a f_of_b
   | f_of_b' == 0 = Just b
   | otherwise =
       -- Use a 'prime' just for consistency.
-    let (a', b', fa', fb') =
-          if (has_root f d b (Just epsilon) (Just f_of_d') (Just f_of_b'))
-          then (d, b, f_of_d', f_of_b')
-          else
-            if (has_root f c d (Just epsilon) (Just f_of_c') (Just f_of_d'))
-            then (c, d, f_of_c', f_of_d')
-            else (a, c, f_of_a', f_of_c')
+    let (a', b', fa', fb')
+          | has_root f d b (Just epsilon) (Just f_of_d') (Just f_of_b') =
+              (d, b, f_of_d', f_of_b')
+          | has_root f c d (Just epsilon) (Just f_of_c') (Just f_of_d') =
+              (c, d, f_of_c', f_of_d')
+          | otherwise =
+              (a, c, f_of_a', f_of_c')
     in
       if (b-a) < 2*epsilon
       then Just ((b+a)/2)
       else trisect f a' b' epsilon (Just fa') (Just fb')
   where
     -- Compute f(a) and f(b) only if needed.
-    f_of_a'  = case f_of_a of
-                 Nothing -> f a
-                 Just v -> v
-
-    f_of_b'  = case f_of_b of
-                 Nothing -> f b
-                 Just v -> v
+    f_of_a'  = fromMaybe (f a) f_of_a
+    f_of_b'  = fromMaybe (f b) f_of_b
 
     c = (2*a + b) / 3
 
@@ -153,8 +138,8 @@ trisect f a b epsilon f_of_a f_of_b
 fixed_point_iterations :: (a -> a) -- ^ The function @f@ to iterate.
                        -> a       -- ^ The initial value @x0@.
                        -> [a]     -- ^ The resulting sequence of x_{n}.
-fixed_point_iterations f x0 =
-  iterate f x0
+fixed_point_iterations =
+  iterate
 
 
 -- | Find a fixed point of the function @f@ with the search starting
@@ -164,7 +149,7 @@ fixed_point_iterations f x0 =
 --
 --   We also return the number of iterations required.
 --
-fixed_point_with_iterations :: (Normed a,
+fixed_point_with_iterations :: forall a b. (Normed a,
                                 Additive.C a,
                                 RealField.C b,
                                 Algebraic.C b)
@@ -181,12 +166,13 @@ fixed_point_with_iterations f epsilon x0 =
     abs_diff v w = norm (v - w)
 
     -- The nth entry in this list is the absolute value of x_{n} -
-    -- x_{n+1}.
-    differences = zipWith abs_diff xn xn_plus_one
+    -- x_{n+1}. They're of type "b" because we're going to compare
+    -- them against epsilon.
+    differences = zipWith abs_diff xn xn_plus_one :: [b]
 
     -- This produces the list [(n, xn)] so that we can determine
     -- the number of iterations required.
-    numbered_xn = zip [0..] xn
+    numbered_xn = zip [0..] xn :: [(Int,a)]
 
     -- A list of pairs, (xn, |x_{n} - x_{n+1}|).
     pairs = zip numbered_xn differences