src/RealFunction.hs

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