X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FRoots%2FFast.hs;h=f7cde8a82a70f3f98605ab87fc94fae5a59147dc;hb=04f56a8882bb0c574b603f8c3fed9481ea934f7f;hp=d6e4d7c23f4091c7a573f01eb4d5c6572a1ca2c8;hpb=c5da1efa77844ae6159dfc781ed886fdffbbf4d1;p=numerical-analysis.git

diff --git a/src/Roots/Fast.hs b/src/Roots/Fast.hs
index d6e4d7c..f7cde8a 100644
--- a/src/Roots/Fast.hs
+++ b/src/Roots/Fast.hs
@@ -1,3 +1,5 @@
+{-# LANGUAGE RebindableSyntax #-}
+
 -- | 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
@@ -8,10 +10,18 @@ where
 
 import Data.List (find)
 
-import Vector
+import Normed
 
+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 :: (Fractional a, Ord a, Ord b, Num b)
+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@
@@ -51,8 +61,9 @@ has_root f a b epsilon f_of_a f_of_b =
     c = (a + b)/2
 
 
-
-bisect :: (Fractional a, Ord a, Num b, Ord 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@
@@ -88,6 +99,55 @@ bisect f a b epsilon f_of_a f_of_b
 
 
 
+trisect :: (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@
+        -> a       -- ^ The tolerance, @epsilon@
+        -> Maybe b -- ^ Precomputed f(a)
+        -> Maybe b -- ^ Precomputed f(b)
+        -> Maybe a
+trisect f a b epsilon f_of_a f_of_b
+  -- We pass @epsilon@ to the 'has_root' function because if we want a
+  -- result within epsilon of the true root, we need to know that
+  -- there *is* a root within an interval of length epsilon.
+  | not (has_root f a b (Just epsilon) (Just f_of_a') (Just f_of_b')) = Nothing
+  | f_of_a' == 0 = Just a
+  | 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')
+    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
+
+    c = (2*a + b) / 3
+
+    d = (a + 2*b) / 3
+
+    f_of_c' = f c
+    f_of_d' = f d
+
+
+
 -- | Iterate the function @f@ with the initial guess @x0@ in hopes of
 --   finding a fixed point.
 fixed_point_iterations :: (a -> a) -- ^ The function @f@ to iterate.
@@ -104,7 +164,10 @@ fixed_point_iterations f x0 =
 --
 --   We also return the number of iterations required.
 --
-fixed_point_with_iterations :: (Vector a, RealFrac b)
+fixed_point_with_iterations :: (Normed a,
+                                Additive.C a,
+                                RealField.C b,
+                                Algebraic.C b)
                             => (a -> a)  -- ^ The function @f@ to iterate.
                             -> b        -- ^ The tolerance, @epsilon@.
                             -> a        -- ^ The initial value @x0@.
@@ -133,4 +196,3 @@ fixed_point_with_iterations f epsilon x0 =
     -- "safe" since the list is infinite. We'll succeed or loop
     -- forever.
     Just winning_pair = find (\(_, diff) -> diff < epsilon) pairs
-