+{-# LANGUAGE RebindableSyntax #-}
+
-- | 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
module Roots.Fast
where
-has_root :: (Fractional a, Ord a, Ord b, Num b)
+import Data.List (find)
+
+import Normed
+
+import NumericPrelude hiding (abs)
+import qualified Algebra.Absolute as Absolute
+import qualified Algebra.Additive as Additive
+import qualified Algebra.Algebraic as Algebraic
+import qualified Algebra.RealRing as RealRing
+import qualified Algebra.RealField as RealField
+
+has_root :: (RealField.C a,
+ RealRing.C b,
+ Absolute.C b)
=> (a -> b) -- ^ The function @f@
-> a -- ^ The \"left\" endpoint, @a@
-> a -- ^ The \"right\" endpoint, @b@
c = (a + b)/2
-
-bisect :: (Fractional a, Ord a, Num b, Ord b)
+bisect :: (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@
Just v -> v
c = (a + b) / 2
+
+
+
+
+-- | Iterate the function @f@ with the initial guess @x0@ in hopes of
+-- finding a fixed point.
+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.
+--
+-- We also return the number of iterations required.
+--
+fixed_point_with_iterations :: (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, a) -- ^ The (iterations, fixed point) pair
+fixed_point_with_iterations f epsilon x0 =
+ (fst winning_pair)
+ where
+ xn = fixed_point_iterations f x0
+ xn_plus_one = tail xn
+
+ abs_diff v w = norm (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
+
+ -- This produces the list [(n, xn)] so that we can determine
+ -- the number of iterations required.
+ numbered_xn = zip [0..] xn
+
+ -- A list of pairs, (xn, |x_{n} - x_{n+1}|).
+ pairs = zip numbered_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