Lecture Notes on 28 April 2014

* Given the following recurrence relation
  f(n) = 2 * f(n - 1) + 1 for n > 0
  f(n) = 0 for n = 0

  * Write a recursive function that returns the nth term in the series

  def func (n):
    if (n == 0):
      return 0
    else:
      return 2 * func(n - 1) + 1

  * What is f(8)?
  n	f(n)
  0	0
  1	1
  2	3
  3	7
  4	15
  5	31
  6	63
  7	127
  8	255

  f(8) = 255

  * Write an iterative version of the recursive function.

  def func_iter (n):
    sum = 0
    for i in range (n):
      sum += 2 * sum + 1
    return sum

  def func_alg (n):
    return 2 ** n - 1


* Recursive version of Fibonacci series
  def fib (n):
    if (n == 0) or (n == 1): 
      return n
    else:
      return fib (n - 1) + fib (n - 2)


* Iterative version of Fibonacci series
  def fib (n):
    if (n == 0) or (n == 1):
      return n
    f1 = 0
    f2 = 1
    f = f1 + f2
    for i in range (n - 2):
      f1 = f2
      f2 = f
      f = f1 + f2
    return f