X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FRoots%2FFast.hs;h=f7cde8a82a70f3f98605ab87fc94fae5a59147dc;hb=04f56a8882bb0c574b603f8c3fed9481ea934f7f;hp=8e49750e650a12f727d2c63282686bc0bfa7dfc1;hpb=61dcb661ee35ff693a9d64d4d21a65791a480a52;p=numerical-analysis.git

diff --git a/src/Roots/Fast.hs b/src/Roots/Fast.hs
index 8e49750..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
@@ -6,7 +8,20 @@
 module Roots.Fast
 where
 
-has_root :: (Fractional a, Ord a, Ord b, Num b)
+import Data.List (find)
+
+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 :: (RealField.C a,
+             RealRing.C b,
+             Absolute.C b)
          => (a -> b) -- ^ The function @f@
          -> a       -- ^ The \"left\" endpoint, @a@
          -> a       -- ^ The \"right\" endpoint, @b@
@@ -46,8 +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@
@@ -80,3 +96,103 @@ bisect f a b epsilon f_of_a f_of_b
                  Just v -> v
 
     c = (a + b) / 2
+
+
+
+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.
+                       -> a       -- ^ The initial value @x0@.
+                       -> [a]     -- ^ The resulting sequence of x_{n}.
+fixed_point_iterations f x0 =
+  iterate f x0
+
+
+-- | Find a fixed point of the function @f@ with the search starting
+--   at x0. This will find the first element in the chain f(x0),
+--   f(f(x0)),... such that the magnitude of the difference between it
+--   and the next element is less than epsilon.
+--
+--   We also return the number of iterations required.
+--
+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@.
+                            -> (Int, a) -- ^ The (iterations, fixed point) pair
+fixed_point_with_iterations f epsilon x0 =
+  (fst winning_pair)
+  where
+    xn = fixed_point_iterations f x0
+    xn_plus_one = tail xn
+
+    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
+
+    -- This produces the list [(n, xn)] so that we can determine
+    -- the number of iterations required.
+    numbered_xn = zip [0..] xn
+
+    -- A list of pairs, (xn, |x_{n} - x_{n+1}|).
+    pairs = zip numbered_xn differences
+
+    -- The pair (xn, |x_{n} - x_{n+1}|) with
+    -- |x_{n} - x_{n+1}| < epsilon. The pattern match on 'Just' is
+    -- "safe" since the list is infinite. We'll succeed or loop
+    -- forever.
+    Just winning_pair = find (\(_, diff) -> diff < epsilon) pairs