-- | We are given a function @f@ and an interval [a,b]. The bisection
--- method checks finds a root by splitting [a,b] in half repeatedly.
+-- method finds a root by splitting [a,b] in half repeatedly.
--
-- If one is found within some prescribed tolerance @epsilon@, it is
-- returned. Otherwise, the interval [a,b] is split into two
--
-- Examples:
--
--- >>> bisect cos 1 2 0.001
--- Just 1.5712890625
+-- >>> let actual = 1.5707963267948966
+-- >>> let Just root = bisect cos 1 2 0.001
+-- >>> root
+-- 1.5712890625
+-- >>> abs (root - actual) < 0.001
+-- True
--
-- >>> bisect sin (-1) 1 0.001
-- Just 0.0
F.bisect f a b epsilon Nothing Nothing
+-- | We are given a function @f@ and an interval [a,b]. The trisection
+-- method finds a root by splitting [a,b] into three
+-- subintervals repeatedly.
+--
+-- If one is found within some prescribed tolerance @epsilon@, it is
+-- returned. Otherwise, the interval [a,b] is split into two
+-- subintervals [a,c] and [c,b] of equal length which are then both
+-- checked via the same process.
+--
+-- Returns 'Just' the value x for which f(x) == 0 if one is found,
+-- or Nothing if one of the preconditions is violated.
+--
+-- Examples:
+--
+-- >>> let actual = 1.5707963267948966
+-- >>> let Just root = trisect cos 1 2 0.001
+-- >>> root
+-- 1.5713305898491083
+-- >>> abs (root - actual) < 0.001
+-- True
+--
+-- >>> trisect sin (-1) 1 0.001
+-- Just 0.0
+--
+trisect :: (RealField.C a, RealRing.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 a
+trisect f a b epsilon =
+ F.trisect f a b epsilon Nothing Nothing
+
+
-- | Find a fixed point of the function @f@ with the search starting
-- at x0. We delegate to the version that returns the number of
-- iterations and simply discard the number of iterations.
-> (a -> a) -- ^ The derivative of @f@
-> a -- ^ Initial guess, x-naught
-> [a]
-newton_iterations f f' x0 =
- iterate next x0
+newton_iterations f f' =
+ iterate next
where
next xn =
xn - ( (f xn) / (f' xn) )
-> a -- ^ Initial guess, x-naught
-> a -- ^ Second initial guess, x-one
-> [a]
-secant_iterations f x0 x1 =
- iterate2 g x0 x1
+secant_iterations f =
+ iterate2 g
where
g prev2 prev1 =
let x_change = prev1 - prev2