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61A Lecture 6 Announcements Recursive Functions Recursive Functions 4 Recursive Functions Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly 4 Recursive Functions Definition:

  1. 61A Lecture 6

  2. Announcements

  3. Recursive Functions

  4. Recursive Functions 4

  5. Recursive Functions Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly 4

  6. Recursive Functions Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly Implication: Executing the body of a recursive function may require applying that function 4

  7. Recursive Functions Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly Implication: Executing the body of a recursive function may require applying that function 4

  8. Recursive Functions Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly Implication: Executing the body of a recursive function may require applying that function Drawing Hands, by M. C. Escher (lithograph, 1948) 4

  9. Digit Sums 2+0+1+5 = 8 5

  10. Digit Sums 2+0+1+5 = 8 • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9 5

  11. Digit Sums 2+0+1+5 = 8 • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9 • Useful for typo detection! 5

  12. Digit Sums 2+0+1+5 = 8 • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9 • Useful for typo detection! The Bank of 61A 1234 5678 9098 7658 OSKI THE BEAR 5

  13. Digit Sums 2+0+1+5 = 8 • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9 • Useful for typo detection! The Bank of 61A A checksum digit is a function of all the other 1234 5678 9098 7658 digits; It can be computed to detect typos OSKI THE BEAR 5

  14. Digit Sums 2+0+1+5 = 8 • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9 • Useful for typo detection! The Bank of 61A A checksum digit is a function of all the other 1234 5678 9098 7658 digits; It can be computed to detect typos OSKI THE BEAR • Credit cards actually use the Luhn algorithm, which we'll implement after digit_sum 5

  15. Sum Digits Without a While Statement 6

  16. Sum Digits Without a While Statement def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 6

  17. Sum Digits Without a While Statement def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" 6

  18. Sum Digits Without a While Statement def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n 6

  19. Sum Digits Without a While Statement def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) 6

  20. Sum Digits Without a While Statement def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last 6

  21. The Anatomy of a Recursive Function def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last 7

  22. The Anatomy of a Recursive Function • The def statement header is similar to other functions def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last 7

  23. The Anatomy of a Recursive Function • The def statement header is similar to other functions def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last 7

  24. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last 7

  25. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last 7

  26. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases • Base cases are evaluated without recursive calls def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last 7

  27. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases • Base cases are evaluated without recursive calls def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last 7

  28. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases • Base cases are evaluated without recursive calls • Recursive cases are evaluated with recursive calls def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last 7

  29. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases • Base cases are evaluated without recursive calls • Recursive cases are evaluated with recursive calls def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last 7

  30. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases • Base cases are evaluated without recursive calls • Recursive cases are evaluated with recursive calls def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last (Demo) 7

  31. Recursion in Environment Diagrams

  32. Recursion in Environment Diagrams Interactive Diagram 9

  33. Recursion in Environment Diagrams (Demo) Interactive Diagram 9

  34. Recursion in Environment Diagrams (Demo) Interactive Diagram 9

  35. Recursion in Environment Diagrams (Demo) • The same function fact is called multiple times Interactive Diagram 9

  36. Recursion in Environment Diagrams (Demo) • The same function fact is called multiple times Interactive Diagram 9

  37. Recursion in Environment Diagrams (Demo) • The same function fact is called multiple times • Different frames keep track of the different arguments in each call Interactive Diagram 9

  38. Recursion in Environment Diagrams (Demo) • The same function fact is called multiple times • Different frames keep track of the different arguments in each call • What n evaluates to depends upon 
 the current environment Interactive Diagram 9

  39. Recursion in Environment Diagrams (Demo) • The same function fact is called multiple times • Different frames keep track of the different arguments in each call • What n evaluates to depends upon 
 the current environment Interactive Diagram 9

  40. Recursion in Environment Diagrams (Demo) • The same function fact is called multiple times • Different frames keep track of the different arguments in each call • What n evaluates to depends upon 
 the current environment • Each call to fact solves a simpler problem than the last: smaller n Interactive Diagram 9

  41. Iteration vs Recursion 10

  42. Iteration vs Recursion Iteration is a special case of recursion 10

  43. Iteration vs Recursion Iteration is a special case of recursion 4! = 4 · 3 · 2 · 1 = 24 10

  44. Iteration vs Recursion Iteration is a special case of recursion 4! = 4 · 3 · 2 · 1 = 24 Using while: 10

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