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A classic example of recursion is the definition of the factorial function, given here in Python code: def factorial ( n ): if n > 0 : return n * factorial ( n - 1 ) else : return 1 The function calls itself recursively on a smaller version of the input (n - 1) and multiplies the result of the recursive call by n , until reaching the base case ...
The recursive program above is tail-recursive; it is equivalent to an iterative algorithm, and the computation shown above shows the steps of evaluation that would be performed by a language that eliminates tail calls. Below is a version of the same algorithm using explicit iteration, suitable for a language that does not eliminate tail calls.
A total recursive function is a partial recursive function that is defined for every input. Every primitive recursive function is total recursive, but not all total recursive functions are primitive recursive. The Ackermann function A(m,n) is a well-known example of a total recursive function (in fact, provable total), that is not primitive ...
Anonymous recursion is primarily of use in allowing recursion for anonymous functions, particularly when they form closures or are used as callbacks, to avoid having to bind the name of the function. Anonymous recursion primarily consists of calling "the current function", which results in direct recursion.
As a more general class of examples, an algorithm on a tree can be decomposed into its behavior on a value and its behavior on children, and can be split up into two mutually recursive functions, one specifying the behavior on a tree, calling the forest function for the forest of children, and one specifying the behavior on a forest, calling ...
A classic example of recursion is computing the factorial, which is defined recursively by 0! := 1 and n! := n × (n - 1)!.. To recursively compute its result on a given input, a recursive function calls (a copy of) itself with a different ("smaller" in some way) input and uses the result of this call to construct its result.
The definitions of elementary recursive functions are the same as for primitive recursive functions, except that primitive recursion is replaced by bounded summation and bounded product. All functions work over the natural numbers. The basic functions, all of them elementary recursive, are: Zero function. Returns zero: f(x) = 0.
This example is artificial since this is direct recursion, so overriding the factorial method would override this function; more natural examples are when a method in a derived class calls the same method in a base class, or in cases of mutual recursion. [4] [5]