+-- | 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