[Haskell-beginners] Understanding recursion in Haskell.

[Adrian Neumann]

Am 18.03.2009 um 06:28 schrieb Caitlin:

>
> Hi.
>
> As a Haskell beginner, I was wondering if someoneone could explain  
> how the following programs function (pardon the pun)?
>


This function takes some type which has an ordering defined, i.e. you  
can compare its elements to one another

> maximum' :: (Ord a) => [a] -> a

it doesn't work for an empty list

> maximum' [] = error "maximum of empty list"

the maximum of a one element list is the lone element. this is the  
base case which will be eventually reached by the recursion

> maximum' [x] = x

should the list have more than one element

> maximum' (x:xs)

compare the first element to the maximum of the other elements. if  
it's greater, it's the maximum

>     | x > maxTail = x

otherwise the maximum of the other elements is the maximum of the  
whole list

>     | otherwise = maxTail

how to compute the maximum of the other elements? just use this  
function again. after a while we will only have one element left and  
reach the base case above.

>     where maxTail = maximum' xs
>
>

This function takes a number and a list of some type a

> take' :: (Num i, Ord i) => i -> [a] -> [a]

first, ignore the list and check whether n is <= 0. in this case  
return an empty list. this is the base case, that's eventually  
reached by the recursion

> take' n _
>     | n <= 0   = []

otherwise, check if the list is empty. this is another base case.

> take' _ []     = []

if neither n<=0 or the list empty, take the first element, x, and put  
it on front of the prefix of length (n-1) of the other elements. use  
take' again, to get that prefix. after a while either n is 0 or there  
are no more elements in the list and we reach the  base case

> take' n (x:xs) = x : take' (n-1) xs
>
>

Take two lists

>
> zip' :: [a] -> [b] -> [(a,b)]

if either one of them is empty, stop

> zip' _ [] = []
> zip' [] _ = []

otherwise prepend a tuple, build from the two first elements to the  
zipped list of the other elements. after a while one of the lists  
should become empty and the base case is reached.

> zip' (x:xs) (y:ys) = (x,y):zip' xs ys
>
>
>
> quicksort :: (Ord a) => [a] -> [a]

empty list -> nothing to do

> quicksort [] = []
> quicksort (x:xs) =

otherwise take the first element of the list and use it to split the  
list in two halves. one with all the elements that are smaller or  
equal than x, the other one with all those that are bigger. now sort  
them and put x in the middle. that should give us a sorted whole. how  
to sort them? just use quicksort again! after some splitting the  
lists will become empty and the recursion stops.

>     let smallerSorted = quicksort [a | a <- xs, a <= x]
>         biggerSorted = quicksort [a | a <- xs, a > x]
>     in  smallerSorted ++ [x] ++ biggerSorted

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