import Data.List (find)
+import Normed
+
import qualified Roots.Fast as F
-- | Does the (continuous) function @f@ have a root on the interval
-- >>> abs (f root) < 1/100000
-- True
--
+-- >>> import Data.Number.BigFloat
+-- >>> let eps = 1/(10^20) :: BigFloat Prec50
+-- >>> let Just root = newtons_method f f' eps 2
+-- >>> root
+-- 1.13472413840151949260544605450647284028100785303643e0
+-- >>> abs (f root) < eps
+-- True
+--
newtons_method :: (Fractional a, Ord a)
=> (a -> a) -- ^ The function @f@ whose root we seek
-> (a -> a) -- ^ The derivative of @f@
-> a -- ^ The tolerance epsilon
-> a -- ^ Initial guess, x-naught
-> Maybe a
-newtons_method f f' epsilon x0
- = find (\x -> abs (f x) < epsilon) x_n
+newtons_method f f' epsilon x0 =
+ find (\x -> abs (f x) < epsilon) x_n
where
x_n = newton_iterations f f' x0
-fixed_point_iterations :: (a -> a) -- ^ The function @f@ to iterate.
- -> a -- ^ The initial value @x0@.
- -> [a] -- ^ The resulting sequence of x_{n}.
-fixed_point_iterations f x0 =
- iterate f x0
-
-
-- | Find a fixed point of the function @f@ with the search starting
--- at x0. This will find the first element in the chain f(x0),
--- f(f(x0)),... such that the magnitude of the difference between it
--- and the next element is less than epsilon.
+-- at x0. We delegate to the version that returns the number of
+-- iterations and simply discard the number of iterations.
--
-fixed_point :: (Num a, Ord a)
+fixed_point :: (Normed a, RealFrac b)
=> (a -> a) -- ^ The function @f@ to iterate.
- -> a -- ^ The tolerance, @epsilon@.
+ -> b -- ^ The tolerance, @epsilon@.
-> a -- ^ The initial value @x0@.
-> a -- ^ The fixed point.
fixed_point f epsilon x0 =
- fst winning_pair
- where
- xn = fixed_point_iterations f x0
- xn_plus_one = tail $ fixed_point_iterations f x0
+ snd $ F.fixed_point_with_iterations f epsilon x0
- abs_diff v w =
- abs (v - w)
- -- The nth entry in this list is the absolute value of x_{n} -
- -- x_{n+1}.
- differences = zipWith abs_diff xn xn_plus_one
+-- | 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, RealFrac 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
- -- A list of pairs, (xn, |x_{n} - x_{n+1}|).
- pairs = zip xn differences
- -- The pair (xn, |x_{n} - x_{n+1}|) with
- -- |x_{n} - x_{n+1}| < epsilon. The pattern match on 'Just' is
- -- "safe" since the list is infinite. We'll succeed or loop
- -- forever.
- Just winning_pair = find (\(_, diff) -> diff < epsilon) pairs
+-- | 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 :: (Normed a, RealFrac 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
+ en_plus_one = tail en
+ en_exp = map (^p) en