import NumericPrelude hiding (abs)
import qualified Algebra.Absolute as Absolute
-import qualified Algebra.Field as Field
+import qualified Algebra.Additive as Additive
+import qualified Algebra.Algebraic as Algebraic
import qualified Algebra.RealRing as RealRing
import qualified Algebra.RealField as RealField
+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.
-- We also return the number of iterations required.
--
fixed_point_with_iterations :: (Normed a,
- Field.C b,
- Absolute.C b,
- Ord b)
+ 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@.