Higher order function
A higher-order function is a function that takes other functions as arguments or returns a function as result.
The major use is to abstract common behaviour into one place.
In the libraries
Many functions in the libraries are higher-order. The (probably) most commonly given examples are
Two other common ones are
curry, uncurry. A possible implementation of these is:
curry :: ((a,b)->c) -> a->b->c curry f a b = f (a,b) uncurry :: (a->b->c) -> ((a,b)->c) uncurry f (a,b)= f a b
curry's first argument must be a function which accepts a pair. It applies that function to its next two arguments.
uncurry is the inverse of
curry. Its first argument must be a function taking two values.
uncurry then applies that function to the components of the pair which is the second argument.
Simple code examples
Rather than writing
doubleList  =  doubleList (x:xs) = 2*x : doubleList xs
tripleList  =  tripleList (x:xs) = 3*x : tripleList xs
we can parameterize out the difference
multList n  =  multList n (x:xs) = n*x : multList n xs
tripleList = multList 3 doubleList = multList 2
leading to a less error prone definition of each.
But now, if we had the function
addToList n  =  addToList n (x:xs) = n+x : addToList n xs
we could parameterize the difference again
operlist n bop  =  operlist n bop (x:xs) = bop n x : operlist n bop xs
and define doubleList as
doubleList = operList 2 (*)
but this ties us into a constant parameters
and we could redefine things as
mapList f  =  mapList f (x:xs) = f x : mapList f xs
and define doubleList as
doubleList = mapList (2*)
This higher-order function "mapList" can be used in a wide range of areas to simplify code.
It is called
map in Haskell's Prelude.
In mathematics the counterpart to higher-order functions are functionals (mapping functions to scalars) and function operators (mapping functions to functions). Typical functionals are the limit of a sequence, or the integral of an interval of a function.
limit :: [Double] -> Double definiteIntegral :: (Double, Double) -> (Double -> Double) -> Double
Typical operators are the indefinite integral, the derivative, the function inverse.
indefiniteIntegral :: Double -> (Double -> Double) -> (Double -> Double) derive :: (Double -> Double) -> (Double -> Double) inverse :: (Double -> Double) -> (Double -> Double)
Here is a numerical approximation:
derive :: Double -> (Double -> Double) -> (Double -> Double) derive eps f x = (f(x+eps) - f(x-eps)) / (2*eps)