Split Roots into Roots.Simple and Roots.Fast.

[numerical-analysis.git] / src / Roots / Fast.hs

diff --git a/src/Roots/Fast.hs b/src/Roots/Fast.hs
new file mode 100644 (file)
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+++ b/src/Roots/Fast.hs
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+-- | 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
+where
+
+has_root :: (Fractional a, Ord a, Ord b, Num b)
+         => (a -> b) -- ^ The function @f@
+         -> a       -- ^ The \"left\" endpoint, @a@
+         -> a       -- ^ The \"right\" endpoint, @b@
+         -> Maybe a -- ^ The size of the smallest subinterval
+                   --   we'll examine, @epsilon@
+         -> 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 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
+
+    -- 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 = (a + b)/2
+
+
+
+bisect :: (Fractional a, Ord a, Num 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@
+       -> a       -- ^ The tolerance, @epsilon@
+       -> Maybe b -- ^ Precomputed f(a)
+       -> Maybe b -- ^ Precomputed f(b)
+       -> Maybe a
+bisect 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
+  | (b - c) < epsilon = Just c
+  | otherwise =
+      -- Use a 'prime' just for consistency.
+      let f_of_c' = f c in
+        if (has_root f a c (Just epsilon) (Just f_of_a') (Just f_of_c'))
+        then bisect f a c epsilon (Just f_of_a') (Just f_of_c')
+        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
+
+    c = (a + b) / 2