src/Roots/Fast.hs

   1 {-# LANGUAGE RebindableSyntax #-}
   2 {-# LANGUAGE ScopedTypeVariables #-}
   3
   4 -- | The Roots.Fast module contains faster implementations of the
   5 --   'Roots.Simple' algorithms. Generally, we will pass precomputed
   6 --   values to the next iteration of a function rather than passing
   7 --   the function and the points at which to (re)evaluate it.
   8
   9 module Roots.Fast (
  10   bisect,
  11   fixed_point_iterations,
  12   fixed_point_with_iterations,
  13   has_root,
  14   trisect )
  15 where
  16
  17 import Data.List ( find )
  18 import Data.Maybe ( fromMaybe )
  19
  20 import Normed ( Normed(..) )
  21
  22 import NumericPrelude hiding ( abs )
  23 import qualified Algebra.Absolute as Absolute ( C )
  24 import qualified Algebra.Additive as Additive ( C )
  25 import qualified Algebra.Algebraic as Algebraic ( C )
  26 import qualified Algebra.RealRing as RealRing ( C )
  27 import qualified Algebra.RealField as RealField ( C )
  28
  29
  30 has_root :: (RealField.C a,
  31              RealRing.C b,
  32              Absolute.C b)
  33          => (a -> b) -- ^ The function @f@
  34          -> a       -- ^ The \"left\" endpoint, @a@
  35          -> a       -- ^ The \"right\" endpoint, @b@
  36          -> Maybe a -- ^ The size of the smallest subinterval
  37                    --   we'll examine, @epsilon@
  38          -> Maybe b -- ^ Precoumpted f(a)
  39          -> Maybe b -- ^ Precoumpted f(b)
  40          -> Bool
  41 has_root f a b epsilon f_of_a f_of_b
  42   | (signum (f_of_a')) * (signum (f_of_b')) /= 1 = True
  43   | (b - a) <= epsilon' = False
  44   | otherwise =
  45       (has_root f a c (Just epsilon') (Just f_of_a') Nothing) ||
  46         (has_root f c b (Just epsilon') Nothing (Just f_of_b'))
  47   where
  48     -- If the size of the smallest subinterval is not specified,
  49     -- assume we just want to check once on all of [a,b].
  50     epsilon' = fromMaybe (b-a) epsilon
  51
  52     -- Compute f(a) and f(b) only if needed.
  53     f_of_a'  = fromMaybe (f a) f_of_a
  54     f_of_b'  = fromMaybe (f b) f_of_b
  55
  56     c = (a + b)/2
  57
  58
  59 bisect :: (RealField.C a,
  60            RealRing.C b,
  61            Absolute.C b)
  62        => (a -> b) -- ^ The function @f@ whose root we seek
  63        -> a       -- ^ The \"left\" endpoint of the interval, @a@
  64        -> a       -- ^ The \"right\" endpoint of the interval, @b@
  65        -> a       -- ^ The tolerance, @epsilon@
  66        -> Maybe b -- ^ Precomputed f(a)
  67        -> Maybe b -- ^ Precomputed f(b)
  68        -> Maybe a
  69 bisect f a b epsilon f_of_a f_of_b
  70   -- We pass @epsilon@ to the 'has_root' function because if we want a
  71   -- result within epsilon of the true root, we need to know that
  72   -- there *is* a root within an interval of length epsilon.
  73   | not (has_root f a b (Just epsilon) (Just f_of_a') (Just f_of_b')) = Nothing
  74   | f_of_a' == 0 = Just a
  75   | f_of_b' == 0 = Just b
  76   | (b - c) < epsilon = Just c
  77   | otherwise =
  78       -- Use a 'prime' just for consistency.
  79       let f_of_c' = f c in
  80         if (has_root f a c (Just epsilon) (Just f_of_a') (Just f_of_c'))
  81         then bisect f a c epsilon (Just f_of_a') (Just f_of_c')
  82         else bisect f c b epsilon (Just f_of_c') (Just f_of_b')
  83   where
  84     -- Compute f(a) and f(b) only if needed.
  85     f_of_a'  = fromMaybe (f a) f_of_a
  86     f_of_b'  = fromMaybe (f b) f_of_b
  87
  88     c = (a + b) / 2
  89
  90
  91
  92 trisect :: (RealField.C a,
  93             RealRing.C b,
  94             Absolute.C b)
  95         => (a -> b) -- ^ The function @f@ whose root we seek
  96         -> a       -- ^ The \"left\" endpoint of the interval, @a@
  97         -> a       -- ^ The \"right\" endpoint of the interval, @b@
  98         -> a       -- ^ The tolerance, @epsilon@
  99         -> Maybe b -- ^ Precomputed f(a)
 100         -> Maybe b -- ^ Precomputed f(b)
 101         -> Maybe a
 102 trisect f a b epsilon f_of_a f_of_b
 103   -- We pass @epsilon@ to the 'has_root' function because if we want a
 104   -- result within epsilon of the true root, we need to know that
 105   -- there *is* a root within an interval of length epsilon.
 106   | not (has_root f a b (Just epsilon) (Just f_of_a') (Just f_of_b')) = Nothing
 107   | f_of_a' == 0 = Just a
 108   | f_of_b' == 0 = Just b
 109   | otherwise =
 110       -- Use a 'prime' just for consistency.
 111     let (a', b', fa', fb')
 112           | has_root f d b (Just epsilon) (Just f_of_d') (Just f_of_b') =
 113               (d, b, f_of_d', f_of_b')
 114           | has_root f c d (Just epsilon) (Just f_of_c') (Just f_of_d') =
 115               (c, d, f_of_c', f_of_d')
 116           | otherwise =
 117               (a, c, f_of_a', f_of_c')
 118     in
 119       if (b-a) < 2*epsilon
 120       then Just ((b+a)/2)
 121       else trisect f a' b' epsilon (Just fa') (Just fb')
 122   where
 123     -- Compute f(a) and f(b) only if needed.
 124     f_of_a'  = fromMaybe (f a) f_of_a
 125     f_of_b'  = fromMaybe (f b) f_of_b
 126
 127     c = (2*a + b) / 3
 128
 129     d = (a + 2*b) / 3
 130
 131     f_of_c' = f c
 132     f_of_d' = f d
 133
 134
 135
 136 -- | Iterate the function @f@ with the initial guess @x0@ in hopes of
 137 --   finding a fixed point.
 138 fixed_point_iterations :: (a -> a) -- ^ The function @f@ to iterate.
 139                        -> a       -- ^ The initial value @x0@.
 140                        -> [a]     -- ^ The resulting sequence of x_{n}.
 141 fixed_point_iterations =
 142   iterate
 143
 144
 145 -- | Find a fixed point of the function @f@ with the search starting
 146 --   at x0. This will find the first element in the chain f(x0),
 147 --   f(f(x0)),... such that the magnitude of the difference between it
 148 --   and the next element is less than epsilon.
 149 --
 150 --   We also return the number of iterations required.
 151 --
 152 fixed_point_with_iterations :: forall a b. (Normed a,
 153                                 Additive.C a,
 154                                 RealField.C b,
 155                                 Algebraic.C b)
 156                             => (a -> a)  -- ^ The function @f@ to iterate.
 157                             -> b        -- ^ The tolerance, @epsilon@.
 158                             -> a        -- ^ The initial value @x0@.
 159                             -> (Int, a) -- ^ The (iterations, fixed point) pair
 160 fixed_point_with_iterations f epsilon x0 =
 161   (fst winning_pair)
 162   where
 163     xn = fixed_point_iterations f x0
 164     xn_plus_one = tail xn
 165
 166     abs_diff v w = norm (v - w)
 167
 168     -- The nth entry in this list is the absolute value of x_{n} -
 169     -- x_{n+1}. They're of type "b" because we're going to compare
 170     -- them against epsilon.
 171     differences = zipWith abs_diff xn xn_plus_one :: [b]
 172
 173     -- This produces the list [(n, xn)] so that we can determine
 174     -- the number of iterations required.
 175     numbered_xn = zip [0..] xn :: [(Int,a)]
 176
 177     -- A list of pairs, (xn, |x_{n} - x_{n+1}|).
 178     pairs = zip numbered_xn differences
 179
 180     -- The pair (xn, |x_{n} - x_{n+1}|) with
 181     -- |x_{n} - x_{n+1}| < epsilon. The pattern match on 'Just' is
 182     -- "safe" since the list is infinite. We'll succeed or loop
 183     -- forever.
 184     Just winning_pair = find (\(_, diff) -> diff < epsilon) pairs