src/Roots/Fast.hs

   1 -- | The Roots.Fast module contains faster implementations of the
   2 --   'Roots.Simple' algorithms. Generally, we will pass precomputed
   3 --   values to the next iteration of a function rather than passing
   4 --   the function and the points at which to (re)evaluate it.
   5
   6 module Roots.Fast
   7 where
   8
   9 import Data.List (find)
  10
  11 import Vector
  12
  13
  14 has_root :: (Fractional a, Ord a, Ord b, Num b)
  15          => (a -> b) -- ^ The function @f@
  16          -> a       -- ^ The \"left\" endpoint, @a@
  17          -> a       -- ^ The \"right\" endpoint, @b@
  18          -> Maybe a -- ^ The size of the smallest subinterval
  19                    --   we'll examine, @epsilon@
  20          -> Maybe b -- ^ Precoumpted f(a)
  21          -> Maybe b -- ^ Precoumpted f(b)
  22          -> Bool
  23 has_root f a b epsilon f_of_a f_of_b =
  24   if not ((signum (f_of_a')) * (signum (f_of_b')) == 1) then
  25     -- We don't care about epsilon here, there's definitely a root!
  26     True
  27   else
  28     if (b - a) <= epsilon' then
  29       -- Give up, return false.
  30       False
  31     else
  32       -- If either [a,c] or [c,b] have roots, we do too.
  33       (has_root f a c (Just epsilon') (Just f_of_a') Nothing) ||
  34         (has_root f c b (Just epsilon') Nothing (Just f_of_b'))
  35   where
  36     -- If the size of the smallest subinterval is not specified,
  37     -- assume we just want to check once on all of [a,b].
  38     epsilon' = case epsilon of
  39                  Nothing -> (b-a)
  40                  Just eps -> eps
  41
  42     -- Compute f(a) and f(b) only if needed.
  43     f_of_a'  = case f_of_a of
  44                  Nothing -> f a
  45                  Just v -> v
  46
  47     f_of_b'  = case f_of_b of
  48                  Nothing -> f b
  49                  Just v -> v
  50
  51     c = (a + b)/2
  52
  53
  54
  55 bisect :: (Fractional a, Ord a, Num b, Ord b)
  56        => (a -> b) -- ^ The function @f@ whose root we seek
  57        -> a       -- ^ The \"left\" endpoint of the interval, @a@
  58        -> a       -- ^ The \"right\" endpoint of the interval, @b@
  59        -> a       -- ^ The tolerance, @epsilon@
  60        -> Maybe b -- ^ Precomputed f(a)
  61        -> Maybe b -- ^ Precomputed f(b)
  62        -> Maybe a
  63 bisect f a b epsilon f_of_a f_of_b
  64   -- We pass @epsilon@ to the 'has_root' function because if we want a
  65   -- result within epsilon of the true root, we need to know that
  66   -- there *is* a root within an interval of length epsilon.
  67   | not (has_root f a b (Just epsilon) (Just f_of_a') (Just f_of_b')) = Nothing
  68   | f_of_a' == 0 = Just a
  69   | f_of_b' == 0 = Just b
  70   | (b - c) < epsilon = Just c
  71   | otherwise =
  72       -- Use a 'prime' just for consistency.
  73       let f_of_c' = f c in
  74         if (has_root f a c (Just epsilon) (Just f_of_a') (Just f_of_c'))
  75         then bisect f a c epsilon (Just f_of_a') (Just f_of_c')
  76         else bisect f c b epsilon (Just f_of_c') (Just f_of_b')
  77   where
  78     -- Compute f(a) and f(b) only if needed.
  79     f_of_a'  = case f_of_a of
  80                  Nothing -> f a
  81                  Just v -> v
  82
  83     f_of_b'  = case f_of_b of
  84                  Nothing -> f b
  85                  Just v -> v
  86
  87     c = (a + b) / 2
  88
  89
  90
  91 -- | Iterate the function @f@ with the initial guess @x0@ in hopes of
  92 --   finding a fixed point.
  93 fixed_point_iterations :: (a -> a) -- ^ The function @f@ to iterate.
  94                        -> a       -- ^ The initial value @x0@.
  95                        -> [a]     -- ^ The resulting sequence of x_{n}.
  96 fixed_point_iterations f x0 =
  97   iterate f x0
  98
  99
 100 -- | Find a fixed point of the function @f@ with the search starting
 101 --   at x0. This will find the first element in the chain f(x0),
 102 --   f(f(x0)),... such that the magnitude of the difference between it
 103 --   and the next element is less than epsilon.
 104 --
 105 --   We also return the number of iterations required.
 106 --
 107 fixed_point_with_iterations :: (Vector a, RealFrac b)
 108                             => (a -> a)  -- ^ The function @f@ to iterate.
 109                             -> b        -- ^ The tolerance, @epsilon@.
 110                             -> a        -- ^ The initial value @x0@.
 111                             -> (Int, a) -- ^ The (iterations, fixed point) pair
 112 fixed_point_with_iterations f epsilon x0 =
 113   (fst winning_pair)
 114   where
 115     xn = fixed_point_iterations f x0
 116     xn_plus_one = tail xn
 117
 118     abs_diff v w = norm_2 (v - w)
 119
 120     -- The nth entry in this list is the absolute value of x_{n} -
 121     -- x_{n+1}.
 122     differences = zipWith abs_diff xn xn_plus_one
 123
 124     -- This produces the list [(n, xn)] so that we can determine
 125     -- the number of iterations required.
 126     numbered_xn = zip [0..] xn
 127
 128     -- A list of pairs, (xn, |x_{n} - x_{n+1}|).
 129     pairs = zip numbered_xn differences
 130
 131     -- The pair (xn, |x_{n} - x_{n+1}|) with
 132     -- |x_{n} - x_{n+1}| < epsilon. The pattern match on 'Just' is
 133     -- "safe" since the list is infinite. We'll succeed or loop
 134     -- forever.
 135     Just winning_pair = find (\(_, diff) -> diff < epsilon) pairs
 136