Lecture Notes on 3 and 5 April 2017

Recursive Definition of Factorial
n! = 1 * 2 * 3 ... * (n - 1) * n

n! = n * (n - 1)!
0! = 1

Recursive Function for Factorial
def fact (n):
  if (n == 0):
    return 1
  else:
    return n * fact (n - 1)

Computation of 4! goes all the way to 0
fact (4)

4 * fact (3) = 4 * 6 = 24

3 * fact (2) = 3 * 2 = 6

2 * fact (1) = 2 * 1 = 2

1 * fact (0) = 1 * 1 = 1

fact(0) return 1


def main():
  var = fact (4)
  print (var)

main()

Fibonacci Series
n      0 1 2 3 4 5 6  7  8  9 10
fib(n) 0 1 1 2 3 5 8 13 21 34 55

Recursive Function to Compute Fibonacci
def fib (n):
  if (n == 0) or (n == 1):
    return n
  else:
    return fib (n - 1) + fib (n - 2)

fib (6) = fib(5) + fib(4) = 5 + fib (4)
fib (5) = fib (4) + fib(3) = 3 + 2 = 5
fib(4) = fib (3) + fib (2) = 2 + 1 = 3
fib(3) = fib (2) + fib (1) = 1 + 1 = 2
fib (2) = fib (1) + fib (0) = 1 + 0 = 1
fib (1) = 1
fib (0) = 0

Iterative Version of summing n numbers
def sum_iter (n):
  sum_n = 0
  for i in range (1, n + 1):
    sum_n += i
  return sum_n

Recursive Version of summing n numbers
def sum_rec (n):
  if (n == 1):
    return n
  else:
    return n + sum_rec (n - 1)

Iterative version of summing the digits of a number
def sum_digits (n):
  sum_d = 0
  while (n > 0):
    sum_d += (n % 10)
    n = n // 10
  return sum_d

Recursive version of summing the digits of a number
def sum_digits_rec (n):
  if (n == 0):
    return n
  else:
    return (n % 10) + sum_digits_rec (n // 10)

Euclid's Greatest Common Divisor Algorithm
def gcd (p, q):
  if (p == q):
    return p
  else:
    if (p > q):
      return gcd (p - q, q)
    else:
      return gcd (p, q - p)

Recurrence Relation

f(n) = 0 when n = 0
f(n) = 1 + 2 * f(n - 1) when n > 0

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

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

	n	f(n)
	0	0
	1	1
	2	3
	3	7
	4	15
	5	31
	6	63	
	7	127