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# notes-recursion - Notes on Recursive Functions Preface Last...

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Notes on Recursive Functions Last modified: 8/5/08 Preface This page is a "how-to" on setting up recursive functions for introductory and intermediate programming courses. There are a lot of other applications and there's plenty of theory out there, but I'm not going to go into that. Here's one particularly good presentation that defines recursion; I shall not attempt to recreate what's already been done well -- I'll just link to it instead. My goal here is to familiarize you with the basics of how to create and use recursive functions. Parts of Recursive Definitions Before we write a recursive function, it's more or less essential to write a recursive definition (or do some form of pre-writing). Recursive functions work based on the values of their parameters. For the sake of simplicity to start out, we'll look at recursive functions with one parameter, n , assumed to be an integer. ALL recursive definitions are broken up into two parts: Base Values of n: usually (not always) n = 0 and/or n = 1. Idea: For these values of n , the function is defined simply. The function returns a preset value, e.g. 0, 1, the value of one of the other parameters, etc. Examples: f(0) == 0 and f(1) == 1 f(0) == parameter2 Recursive Values of n : usually higher values of n , e.g. n >= 2. Idea: For higher values of n , the function is defined in terms of itself for a different value of n (which I'll refer to as f(changed n ) below). There are some considerations. The value of n in the recursive call needs to change in such a way that it is working toward one (of the) base case(s). Usually we add or subtract something from n . We occasionally multiply or divide n by something (illustrated much further below). The value of f( n ) is obtained by doing something to f(changed n ), e.g. a combination of one of more of the following: Adding something to f(changed n ). Subtracting something from f(changed n ). Multiplying f(changed n ) by something.

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Dividing f(changed n ) by something. If we treat subtracting a as adding -a and dividing by b as multiplying by 1/ b, we can summarize with the following generalization: f( n ) = (f(changed n ) + a ) * b + c. Examples: Assuming a base case of f(0) exists and n is positive, f( n ) = f( n -1) + 1 f( n ) = f( n -1) / 5 f( n ) = f( n -1) * n Assuming base cases of f(0) and f(1) exist and n >= 2, f( n ) = f( n -2) * ( n -3) f( n ) = (f( n -2) - 1)/ n As you hopefully observed from the examples, the recursive cases make assumptions about the base cases existing. Therefore, you should define the base case first.
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