Recursion

• This is chapter 18.
• A recursive call is a function call in which the function being called is the same as the one making the call.
• A recursive function is a function that can make a call to its self.
• An example of a recursive function

  • void Countdown(size_t i) {
        cout << i <<  endl;
        if (i > 0) {
           Countdown(i-1); 
        }
        return;
    } 

  • As we can see in this code, if the input is greater than 0, the function will make a call to its self.
• Trace this on a stack.
• Recursive functions require three things
  • At least one recursive call.
  • A base case where the recursion stops.
  • A general case or recursive case where the algorithm makes progress towards a solution.
• Where are each of these in the above code?
• When I think of a recursive solution my thoughts are generally
  • To countdown I need to count the number that was input,
  • If am at 0, then stop, otherwise, countdown starting at one number less than the current state.
• In general all recursive algorithms can be written iteratively
  • But the solution may involve a stack.
  • Iterative functions are generally faster.
  • Recusive functions tend to be more simple and elegant.
  • Everything in this set of notes would be simple to change to iterative solutions.
• Write a recursive function to compute $x^n$ for integer $n>=0$
  • Algorithm

    Power(base,exp)
       if the exponent is 0, return 1
       otherwise return base * power(base, exp-1)
             

• Factorial
  $$ x! = \left\lbrace \begin{array}{lr}1& x = 0 \\x(x-1)!& x>0 \\ \end{array} \right.$$
• Fibonacci $$ fib(n) = \left\lbrace \begin{array}{lr}1& n = 0,1 \\fib(n-1)+fib(n-2)& n>1 \\ \end{array} \right.$$
• GCD

      GCD(a,b) 
         if b is 0
            return a
         else
             return gcd(b,a mod b)