1. Programming
Recursion

2. Recursion: Example 0 What does the following program do?
#include <iostream>
using namespace std;
int fac(int n){
// Assume n >= 0
int product;
if(n <= 1)
return 1;
product = n * fac(n-1);
return product; }
void main(){ // driver function
int number;
do{
cout << "Enter integer (negative to stop): ";
cin >> number;
if(number >= 0)
cout << fac(number) << endl;
}while(number >= 0);

3. Recursion: Example 0 Assume the number typed is 3. fac(3) :
3 <= 1 ? No.
product3 = 3 * fac(2)
fac(2) :
2 <= 1 ? No.
product2 = 2 * fac(1)
fac(1) :
1 <= 1 ? Yes.
return 1
product2 = 2 * 1 = 2
return product2
product3 = 3 * 2 = 6
return product3
fac(3) has the value 6

4. Recursion
Recursion is one way to decompose a task into smaller subtasks.
At least one of the subtasks is a smaller example of the same task.
The smallest example of the same task has a non-recursive solution.
Example: The factorial function
n! = n * (n-1) * (n-2) * ... * 1 or
n! = n * (n-1)! and 1! = 1

5. Recursion
A recursive solution may be simpler to write (once you get used to the idea) than a non-recursive solution.
But a recursive solution may not be as efficient as a non-recursive solution of the same problem.

6. Iterative Factorial // Non-recursive factorial function
// Compute the factorial using a loop
int fac(int n){
// Assume n >= 0
int k, product;
if(n <=1)
return 1;
product = 1;
for(k=1; k<=n; k++)
product*= k;
return product; }

7. Other Recursive Applications
Fibonacci numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
where each number is the sum of the preceding two.
Recursive definition:
F(0) = 0
F(1) = 1
F(n) = F(n-1) + F(n-2)

8. Other Recursive Applications
Binary search:
Compare search element with middle element of the array:
If not equal, then apply binary search to half of the array (if not empty) where the search element would be.

9. Recursion General Form
How to write recursively?
int rec(1-2 parameters){
if(stopping condition)
return stopping value;
// second stopping condition if needed
return value/rec(revised parameters)
+-*/ rec(revised parameters); }

10. Recursion: Example 1 How to write exp(int x, int y) recursively?
int exp(int x, int y){
if(y==0)
return 1;
return x * exp(x, y-1); }

11. Recursion: Example 2
Write a recursive function that takes a double array and its size as input and returns the sum of the array:
double asum(int a[], int size){
if(size==0)
return 0;
return asum(a, size-1)+a[size-1]; }

12. Recursion: Example 3
Write a recursive function that takes a double array and its size as input and returns the product of the array:
double aprod(int a[], int size){
if(size==0)
return 1;
return aprod(a, size-1)*a[size-1]; }

13. Recursion: Example 4
Write a recursive function that counts the number of zero digits in a non-negative integer
zeros(10200) returns 3
int zeros(int n){
if(n==0)
return 1;
if(n < 10)
return 0;
if(n%10 == 0)
return 1 + zeros(n/10);
else
return zeros(n/10); }

14. Recursion: Example 5
Write a recursive function to determine how many factors m are part of n. For example, if n=48 and m=4, then the result is 2 (48=4*4*3).
int factors(int n, int m){
if(n%m != 0)
return 0;
return 1 + factors(n/m, m); }