+-- | 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.
+--
+fixed_point :: (Normed a, Additive.C a, Algebraic.C b, RealField.C b)
+ => (a -> a) -- ^ The function @f@ to iterate.
+ -> b -- ^ The tolerance, @epsilon@.
+ -> a -- ^ The initial value @x0@.
+ -> a -- ^ The fixed point.
+fixed_point f epsilon x0 =
+ snd $ F.fixed_point_with_iterations f epsilon x0
+
+
+-- | Return the number of iterations required to find a fixed point of
+-- the function @f@ with the search starting at x0 and tolerance
+-- @epsilon@. We delegate to the version that returns the number of
+-- iterations and simply discard the fixed point.
+fixed_point_iteration_count :: (Normed a,
+ 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@.
+ -> Int -- ^ The fixed point.
+fixed_point_iteration_count f epsilon x0 =
+ fst $ F.fixed_point_with_iterations f epsilon x0
+
+
+-- | Returns a list of ratios,
+--
+-- ||x^{*} - x_{n+1}|| / ||x^{*} - x_{n}||^{p}
+--
+-- of fixed point iterations for the function @f@ with initial guess
+-- @x0@ and @p@ some positive power.
+--
+-- This is used to determine the rate of convergence.
+--
+fixed_point_error_ratios :: forall a b. (Normed a,
+ Additive.C a,
+ RealField.C b,
+ Algebraic.C b)
+ => (a -> a) -- ^ The function @f@ to iterate.
+ -> a -- ^ The initial value @x0@.
+ -> a -- ^ The true solution, @x_star@.
+ -> Integer -- ^ The power @p@.
+ -> [b] -- ^ The resulting sequence of x_{n}.
+fixed_point_error_ratios f x0 x_star p =
+ zipWith (/) en_plus_one en_exp
+ where
+ xn = F.fixed_point_iterations f x0
+ en = map (\x -> norm (x_star - x)) xn :: [b]
+ en_plus_one = tail en
+ en_exp = map (^p) en
+
+