src/Roots/Fast.hs

   1 -- | The Roots.Fast module contains faster implementations of the
   2 --   'Roots.Simple' algorithms. Generally, we will pass precomputed
   3 --   values to the next iteration of a function rather than passing
   4 --   the function and the points at which to (re)evaluate it.
   5
   6 module Roots.Fast
   7 where
   8
   9 has_root :: (Fractional a, Ord a, Ord b, Num b)
  10          => (a -> b) -- ^ The function @f@
  11          -> a       -- ^ The \"left\" endpoint, @a@
  12          -> a       -- ^ The \"right\" endpoint, @b@
  13          -> Maybe a -- ^ The size of the smallest subinterval
  14                    --   we'll examine, @epsilon@
  15          -> Maybe b -- ^ Precoumpted f(a)
  16          -> Maybe b -- ^ Precoumpted f(b)
  17          -> Bool
  18 has_root f a b epsilon f_of_a f_of_b =
  19   if not ((signum (f_of_a')) * (signum (f_of_b')) == 1) then
  20     -- We don't care about epsilon here, there's definitely a root!
  21     True
  22   else
  23     if (b - a) <= epsilon' then
  24       -- Give up, return false.
  25       False
  26     else
  27       -- If either [a,c] or [c,b] have roots, we do too.
  28       (has_root f a c (Just epsilon') (Just f_of_a') Nothing) ||
  29         (has_root f c b (Just epsilon') Nothing (Just f_of_b'))
  30   where
  31     -- If the size of the smallest subinterval is not specified,
  32     -- assume we just want to check once on all of [a,b].
  33     epsilon' = case epsilon of
  34                  Nothing -> (b-a)
  35                  Just eps -> eps
  36
  37     -- Compute f(a) and f(b) only if needed.
  38     f_of_a'  = case f_of_a of
  39                  Nothing -> f a
  40                  Just v -> v
  41
  42     f_of_b'  = case f_of_b of
  43                  Nothing -> f b
  44                  Just v -> v
  45
  46     c = (a + b)/2
  47
  48
  49
  50 bisect :: (Fractional a, Ord a, Num b, Ord b)
  51        => (a -> b) -- ^ The function @f@ whose root we seek
  52        -> a       -- ^ The \"left\" endpoint of the interval, @a@
  53        -> a       -- ^ The \"right\" endpoint of the interval, @b@
  54        -> a       -- ^ The tolerance, @epsilon@
  55        -> Maybe b -- ^ Precomputed f(a)
  56        -> Maybe b -- ^ Precomputed f(b)
  57        -> Maybe a
  58 bisect f a b epsilon f_of_a f_of_b
  59   -- We pass @epsilon@ to the 'has_root' function because if we want a
  60   -- result within epsilon of the true root, we need to know that
  61   -- there *is* a root within an interval of length epsilon.
  62   | not (has_root f a b (Just epsilon) (Just f_of_a') (Just f_of_b')) = Nothing
  63   | f_of_a' == 0 = Just a
  64   | f_of_b' == 0 = Just b
  65   | (b - c) < epsilon = Just c
  66   | otherwise =
  67       -- Use a 'prime' just for consistency.
  68       let f_of_c' = f c in
  69         if (has_root f a c (Just epsilon) (Just f_of_a') (Just f_of_c'))
  70         then bisect f a c epsilon (Just f_of_a') (Just f_of_c')
  71         else bisect f c b epsilon (Just f_of_c') (Just f_of_b')
  72   where
  73     -- Compute f(a) and f(b) only if needed.
  74     f_of_a'  = case f_of_a of
  75                  Nothing -> f a
  76                  Just v -> v
  77
  78     f_of_b'  = case f_of_b of
  79                  Nothing -> f b
  80                  Just v -> v
  81
  82     c = (a + b) / 2