1. 1 Gentle Introduction to Programming Session 3: Higher Order Functions, Recursion

2. 2 Admin Sorry about the technical problems in the practical session (11-13) Your feedback about it? Scala / Eclipse installation status?

3. 3 How to Find Mistakes in Your Code Compiler Errors Runtime errors What are you expected to do before contacting me

4. 4 Today Review Higher order functions Debugger Home work review Home work Recursion

5. 5 Review More Scala: var / val Strings & API Defining variables (type inference) Variable scope Importance of style Functions Why do we need them Types: methods, local (nested), literal, valu

6. 6 Review (Cont.) Odd’s talk (bioinformatics) Practical session: Account Linux Interpreter / Eclipse Understanding the compiler Command line arguments Homework – practice shell commands

7. 7 Why Functions? Why do we need functions? Code reusability Modularity Abstraction

8. 8 Function Definition in Scala

9. 9 Example

10. 10 There are many ways to define functions in Scala FuncDef.scala

11. 11 Example ((x : Int) => x * 2)(a) ((x : Int) => x * 2)(5) ((x : Int) => x * 2) 55 10

12. 12 Today Review Higher order functions Debugger Home work review Home work Recursion

13. 13 Closures (x : Int) => x + more // how much more? free variable

14. 14 Closures (Cont.) What happens if we change “more”?

15. 15 Can procedures get and return procedures? In Scala a function can: Return a function as its return value, Receive functions as arguments. Why is this useful?

16. 16 Consider the Following Three Sums 1 + 2 + … + 100 = (100 * 101)/2 1 + 4 + 9 + … + 100 2 = (100 * 101 * 102)/6 1 + 1/3 2 + 1/5 2 + … + 1/101 2 ~  2 /8 In mathematics they are all captured by the notion of a sum

17. 17 In Scala Generalization:

18. 18 The “Sum” Function

19. 19 Usage (HigherOrderFunc.scala)

20. 20 How Does it Work? def f1(start : Int, end : Int) : Double = { sum((x:Int)=>x,start,(x:Int)=>x+1,end) } f1(1,100) f1(50,60)

21. 21 How Does it Work? sum += ((x:Int) => x)(i) i = ((x:Int) => x+1)(i)

22. 22 Integration as a Function Integration under a curve f is approximated by dx ( f(a) + f(a + dx) + f(a + 2dx) + … + f(b) ) a b dx f

23. 23 Integration as a Function (Cont.) a b dx f

24. 24 arctan(a) = ∫(1/(1+y 2 ))dy 0 a Integration.scala

25. 25 Function as a Contract If the operands have the specified types, the procedure will result in a value of the specified type otherwise, its behavior is undefined (maybe an error, maybe random behavior) A contract between the caller and the procedure. Caller responsible for argument number and types function responsible to deliver correct result

26. 26 Example

27. 27 The Functions “Stack” g() ->h() f() ->g() main -> f()

28. 28 Today Review Higher order functions Debugger Home work review Home work Recursion

29. 29 The Debugger Some programs may compile correctly, yet not produce the desirable results These programs are valid and correct Scala programs, yet not the programs we meant to write! The debugger can be used to follow the program step by step and may help detecting bugs in an already compiled program

30. 30 Debugger – Add Breakpoint Right click on the desired line “Toggle Breakpoint”

31. 31 Debugger – Start Debugging breakpoint debug

32. 32 Debugger – Debug Perspective

33. 33 Debugger – Debugging Current state Current location Back to Scala perspective

34. 34 Today Review Higher order functions Debugger Home work review Home work Recursion

35. 35 Exercise – Integer Division Input: Two integers – A and B Output: How many times A contains B (it is the result of the integer division A/B) Do not use the operators ‘/’, ‘*’ Solution:

36. 36 Integer Division Solution

37. 37 Exercise – Power of Two Input: integer A Output: is there an integer N such that A == 2^N? Solution:

38. 38 Power of Two Solution

39. 39 Write a program that prints an upside-down half triangle of *. The height of the pyramid is the input. ***** *** ** **** * Exercise – Triangle Printout

40. 40 Triangle Printout Solution

41. 41 Today Review Higher order functions Debugger Home work review Home work Recursion

42. 42 Approximating Square Root Given integer x > 0, how to find an approximation of x? Newton was the first to notice that for any positive n, and when x 0 =1, the following series converges to sqrt(n):

43. 43 Algorithm x = 2g = 1 x/g = 2g = ½ (1+ 2) = 1.5 x/g = 4/3g = ½ (3/2 + 4/3) = 17/12 = 1.416666 x/g = 24/17g = ½ (17/12 + 24/17) = 577/408 = 1.4142156 Algorithm: Make a guess g Improve the guess by averaging g, x/g Keep improving the guess until it is good enough Example: calculate the square root of 2

44. 44 Exercise 1 Write a program that receives from the user: An integer x > 0 A double representing the approximation’s accuracy Calculates an approximation of x within the required accuracy Use the function Math.abs(x) which returns the absolute value of x

45. 45 Exercise 2 Write a function that uses the formula :  2 /6=1/1+1/4+1/9+1/16+…+1/n 2 (where n goes to infinity) in order to approximate . The function should accept an argument n which determines the number of terms in the formula. It should return the approximation of  Write a program that gets an integer n from the user, and approximate  using n terms of the above formula Use a nested function sum, with signature:

46. 46 Exercise 3 Write a function deriv : Input: a function f (Double=>Double), accuracy dx Output: function df (Double=>Double), the derivative of f. df (x) = (f(x+dx) – f(x))/dx Define a few simple functions (linear, square) and calculate their derivative at various points received by command line arguments

47. 47 For next Year – toy example of a function that returns a function (to help with the solution of ex#3)

48. 48 Today Review Higher order functions Debugger Home work review Home work Recursion

49. 49 The Functions “Stack” g() ->h() f() ->g() main -> f()

50. 50 Recursive Functions How to create a process of unbounded length? Needed to solve more complicated problems Tower of Hanoi Start with a simple example

51. 51 Tower of Hanoi

52. 52 Tower of Hanoi How to solve for arbitrary number of disks n ? Let’s name the pegs: source, target, spare All n disks are initially placed on source peg Solution: 1.Move n-1 upper disks from source to spare 2.More lower (bigger) disk from source to target 3.Move n-1 disks from spare to target (smaller. Easier?) Base condition: a single disk

53. 53 Factorial As we saw, n! = 1*2*3*… *(n-1)*n Thus, we can also define factorial the following way: 0! = 1 n! = n*(n-1)! for n>0 (n-1)! * n Recursive Definition

54. 54 A Recursive Definition Functions can also call themselves! However, usually not with the same parameters Some functions can be defined using smaller occurrences of themselves Every recursive function has a “termination condition”. The function stops calling itself when it is satisfied

55. 55 Example - Factorial Termination Condition Recursive Calll Factorial.scala

56. 56 Example - factorial factRec (1) factRec (2) ->2* factRec (1) factRec (3) ->3* factRec (2) main -> factRec (3)

57. 57 Recursive factorial – step by step FactRec(4) n 4 Returns…

58. 58 Recursive factorial – step by step FactRec(4) n 4 Returns… 4*…

59. 59 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns…

60. 60 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns…

61. 61 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns… 3*…

62. 62 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns… FactRec(2) n 2 Returns…

63. 63 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns… FactRec(2) n 2 Returns…

64. 64 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns… FactRec(2) n 2 Returns… 2*…

65. 65 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns… FactRec(2) n 2 Returns… FactRec(1) n 1 Returns…

66. 66 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns… FactRec(2) n 2 Returns… FactRec(1) n 1 Returns…

67. 67 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns… FactRec(2) n 2 Returns… FactRec(1) n 1 Returns… 1

68. 68 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns… FactRec(2) n 2 Returns… 2*1

69. 69 Recursive factorial – step by step FactRec(4) n 4 Returns… FactRec(3) n 3 Returns… 3*2

70. 70 Recursive factorial – step by step FactRec(4) n 4 Returns… 4*6

71. 71 General Form of Recursive Algorithms Base case: small (non-decomposable) problem Recursive case: larger (decomposable) problem At least one base case, and at least one recursive case. test + base case recursive case

72. 72 Short Summary Design a recursive algorithm by 1. Solving big instances using the solution to smaller instances 2. Solving directly the base cases Recursive algorithms have 1. test 2. recursive case(s) 3. base case(s)

73. 73 More Examples

74. 74 Example - Modulo Given the following iterative version of modulo calculation Find the recursive definition Modulo.scala

75. 75 Solution

76. 76 Fibonacci “Naturally” recursive Because the n th term is the sum of the (n-1) th and (n-2) th terms Therefore, the recursive definition is: fib(1) = 0 ; fib(2) = 1 /* base */ fib(n) = fib(n-1) + fib(n-2)

77. 77 Or, in Scala… Fibonacci.scala

78. 78 Recursion and Efficiency The recursive form, however elegant, may be much less efficient The number of redundant calls grow exponentially! Fib(6) Fib(4)

79. 79 Efficient Recursive Fibonacci Pass the values that were already calculated to the recursive function This technique is called Memoization How would we keep the values that were already calculated? Fib(6) Fib(4)

80. 80 Fibonacci with Memoization int main(void) { int fib[SIZE],n,i; fib[1] = 0; fib[2] = 1; for (i = 3; i < SIZE; ++i) fib[i] = -1; printf("Please enter a non-negative number\n"); scanf("%d",&n); printf("fib(%d) = %d\n",n,fib_rec_mem(n,fib)); return 0; }

81. 81 Fibonacci with Memoization int fib_rec_mem(int n,int fib[]) { if (fib[n] != -1) return fib[n]; fib[n] = fib_rec_mem(n-1,fib) + fib_rec_mem(n-2,fib); return fib[n]; }

82. 82 Odd-Even Given a function ‘odd(n : Int) : Boolean’ Return true on n that are odd, false on even n Write a function ‘even(n : Int) : Boolean’ that: Return true on n that are even, false on odd n This is easy

83. 83 Odd-Even EvenOdd.scala

84. 84 More Home Work (Recursion)

85. 85 Exercise 1 Write a program that receives two non- negative integers and computes their product recursively Hint: Notice that the product a*b is actually a+a+…+a (b times). How does this help us define the problem recursively?

86. 86 Exercise 2 Given the following iterative version of sum- of-digits calculation Write the recursive definition