src/RealFunction.hs

   1 {-# OPTIONS_GHC -Wno-orphans #-}
   2 {-# LANGUAGE FlexibleInstances #-}
   3
   4 module RealFunction (
   5   RealFunction,
   6   cmult,
   7   fexp )
   8 where
   9
  10 -- Presumably this is faster without a newtype wrapper, and that's why
  11 -- we're about to define a bunch of orphan instances below. Note the
  12 -- GHC pragma thingy at the top of this file to ignore those warnings.
  13 type RealFunction a = (a -> Double)
  14
  15
  16 -- | A 'Show' instance is required to be a 'Num' instance.
  17 instance Show (RealFunction a) where
  18     -- | There is nothing of value that we can display about a
  19     --   function, so simply print its type.
  20     show _ = "<RealFunction>"
  21
  22
  23 -- | An 'Eq' instance is required to be a 'Num' instance.
  24 instance Eq (RealFunction a) where
  25     -- | Nothing else makes sense here; always return 'False'.
  26     _ == _ = error "You can't compare functions for equality."
  27
  28
  29 -- | The 'Num' instance for RealFunction allows us to perform
  30 --   arithmetic on functions in the usual way.
  31 instance Num (RealFunction a) where
  32     (f1 + f2)  x = (f1 x) + (f2 x)
  33     (f1 - f2)  x = (f1 x) - (f2 x)
  34     (f1 * f2)  x = (f1 x) * (f2 x)
  35     (negate f) x = -1 * (f x)
  36     (abs    f) x = abs (f x)
  37     (signum f) x = signum (f x)
  38     fromInteger i _ = fromInteger i
  39
  40
  41 -- | Takes a constant, and a function as arguments. Returns a new
  42 --   function representing the original function times the constant.
  43 --
  44 --   ==== __Examples__
  45 --
  46 --   >>> let square x = x**2
  47 --   >>> square 1
  48 --   1.0
  49 --   >>> square 2
  50 --   4.0
  51 --   >>> let f = cmult 2 square
  52 --   >>> f 1
  53 --   2.0
  54 --   >>> f 2
  55 --   8.0
  56 --
  57 cmult :: Double -> (RealFunction a) -> (RealFunction a)
  58 cmult coeff f = (*coeff) . f
  59
  60
  61 -- | Takes a function @f@ and an exponent @n@. Returns a new function,
  62 --   @g@, defined by g(x) = (f(x))^n. This is /not/ @f@ composed
  63 --   with itself @n@ times.
  64 --
  65 --   ==== __Examples__
  66 --
  67 --   >>> let square x = x**2
  68 --   >>> square 2
  69 --   4.0
  70 --   >>> let f = fexp square 3
  71 --   >>> f 2
  72 --   64.0
  73 --
  74 fexp :: (RealFunction a) -> Int -> (RealFunction a)
  75 fexp f n
  76      | n == 0 = const 1
  77      | otherwise = \x -> (f x)^n