src/RealFunction.hs

   1 {-# LANGUAGE FlexibleInstances #-}
   2
   3 module RealFunction (
   4   RealFunction,
   5   cmult,
   6   fexp
   7   )
   8 where
   9
  10
  11 type RealFunction a = (a -> Double)
  12
  13
  14 -- | A 'Show' instance is required to be a 'Num' instance.
  15 instance Show (RealFunction a) where
  16     -- | There is nothing of value that we can display about a
  17     --   function, so simply print its type.
  18     show _ = "<RealFunction>"
  19
  20
  21 -- | An 'Eq' instance is required to be a 'Num' instance.
  22 instance Eq (RealFunction a) where
  23     -- | Nothing else makes sense here; always return 'False'.
  24     _ == _ = error "You can't compare functions for equality."
  25
  26
  27 -- | The 'Num' instance for RealFunction allows us to perform
  28 --   arithmetic on functions in the usual way.
  29 instance Num (RealFunction a) where
  30     (f1 + f2)  x = (f1 x) + (f2 x)
  31     (f1 - f2)  x = (f1 x) - (f2 x)
  32     (f1 * f2)  x = (f1 x) * (f2 x)
  33     (negate f) x = -1 * (f x)
  34     (abs    f) x = abs (f x)
  35     (signum f) x = signum (f x)
  36     fromInteger i _ = fromInteger i
  37
  38
  39 -- | Takes a constant, and a function as arguments. Returns a new
  40 --   function representing the original function times the constant.
  41 --
  42 --   ==== __Examples__
  43 --
  44 --   >>> let square x = x**2
  45 --   >>> square 1
  46 --   1.0
  47 --   >>> square 2
  48 --   4.0
  49 --   >>> let f = cmult 2 square
  50 --   >>> f 1
  51 --   2.0
  52 --   >>> f 2
  53 --   8.0
  54 --
  55 cmult :: Double -> (RealFunction a) -> (RealFunction a)
  56 cmult coeff f = (*coeff) . f
  57
  58
  59 -- | Takes a function @f@ and an exponent @n@. Returns a new function,
  60 --   @g@, defined by g(x) = (f(x))^n. This is /not/ @f@ composed
  61 --   with itself @n@ times.
  62 --
  63 --   ==== __Examples__
  64 --
  65 --   >>> let square x = x**2
  66 --   >>> square 2
  67 --   4.0
  68 --   >>> let f = fexp square 3
  69 --   >>> f 2
  70 --   64.0
  71 --
  72 fexp :: (RealFunction a) -> Int -> (RealFunction a)
  73 fexp f n
  74      | n == 0 = (\_ -> 1)
  75      | otherwise = \x -> (f x)^n