Programming & Data Structure

Transcription

1 Functions Programming & Data Structure CS Partha Bhowmick pb CSE Department IIT Kharagpur Spring

2 Functions Callee : Caller = Bowler : Captain

3 Functions int f(){... } (1) main(){... f();... } Here f is the callee function. main is the caller function, which calls f().

4 Computation of sinx (1) sin x = x x3 3! + x5 5! x7 7! + x9 9! = i 0 = i 0 ( 1) i x 2i+1 (2i + 1)! t i, where t i = ( 1) i x 2i+1 (2i + 1)! A finite number of terms of the above power series can be used to compute an approximate value of sin x, where x is in radian.

5 Computation of sinx (2) Inductive Definitions t i = S i = { x if i = 0, t x 2 i 1 if i > 0. 2i(2i + 1) { t0 if i = 0, S i 1 + t i if i > 0. They are also called recurrence relations or recursive definitions.

6 Computation of sinx (3) Inductive Definition Iterative Process An iterative process of computation can be obtained from the inductive definition. Thus, an approximation of sin x is computed as S i, the sum from t 0 to t i, as follows. 1 Start with initial values: t 0 and S 0. 2 Repeat the following two steps for i > 0: 1 Compute t i from t i 1. 2 Compute the next approximate value of sin x by computing S i from S i 1 and t i.

7 Computation of sinx (4) Termination of Iteration The process is to be terminated after a finite number of iterations. The termination may be 1 after a fixed number of iterations, or 2 after achieving a pre-specified accuracy.

8 Computation of sinx (5) #include <stdio.h> #include <math.h> int main() { float x, preerr, apperr, rad, term, sinx; int i = 1; printf("enter the angle in degree: "); scanf("%f", &x); printf("enter the Percentage Error: "); scanf("%f", &preerr); // Initialization rad = atan(1.0)*x/45.0; term = rad; sinx = term;

9 Computation of sinx (6) do { // Iteration float factor; factor = 2.0 * i++; factor *= (factor + 1.0); factor = - rad * rad / factor; sinx += (term = factor * term); apperr = 100.0*fabs(term/sinx); } while (apperr >= preerr); printf("\nsin(%f) = %55.50f\n#Iterations = %d\n", x, sinx, i - 1); return 0; } // sin.c

10 Computation of sinx (7) $ cc -lm -Wall sin.c $./a.out Enter the angle in degree: 90 Enter the Percentage Error:.1 sin( ) = #Iterations = 4 $./a.out Enter the angle in degree: 90 Enter the Percentage Error:.01 sin( ) = #Iterations = 5

11 Computation of sinx (8) $./a.out Enter the angle in degree: 90 Enter the Percentage Error: sin( ) = #Iterations = 6 $./a.out Enter the angle in degree: 90 Enter the Percentage Error: sin( ) = #Iterations = 10

12 Computation of sinx (9) Caution! Notice the last two runs. The accuracy of the program decreases for large values of angle due to error propagation.

13 Recursion To understand recursion, you must understand recursion.

14 Recursion (1) Example: Factorial Function { 1 if n = 0, n! = n (n 1)! if n > 0. or, fact(n) = { 1 if n = 0, n fact(n 1) if n > 0.

15 Recursion (2) Example: Computation of 4! 4! = 4 3! = 4 (3 2!) = 4 (3 (2 1!)) = 4 (3 (2 (1 0!))) = 4 (3 (2 (1 1))) = 4 (3 (2 1)) = 4 (3 2) = 4 6 = 24

16 Recursion (3) Features There is no value computation in the first four steps, as the function is being unfolded. The value computation starts only after the basis of the definition is reached. Last four steps computes the values. Definition: Recursive Function A function that calls itself directly or indirectly is called a recursive function. Unfolding and delayed computation can be simulated by such a function.

17 Recursion Recursive Call (4) A() call A A() call A A() call A A() As a recursive function calls itself, the obvious question is about its termination. The basis of the inductive (recursive) definition provides the condition for termination. The function calls itself to gradually reach the termination condition or the basis!

18 Recursion (5) int fact(int n){ if (n == 0) return 1; else return n * fact(n-1);} fact(4) = 4 fact(3) = 4 (3 fact(2)) = 4 (3 (2 fact(1))) = 4 (3 (2 (1 fact(0)))) = 4 (3 (2 (1 1))) = 4 (3 (2 1)) = = 24.

19 Recursion (6) Functions during Recursive Calls For every call, there is new incarnation of all the formal parameters and the local variables (that are not static). The variable names get bound to different memory locations. Variables of one invocation are not visible from another invocation. Once a return statement is executed, all the variables of the corresponding invocation die.

20 Recursion (7) The last incarnation of a variable name dies first and the last call is returned first a principle called last in first out (LIFO).

21 Stack Frame or Activation Record (1) main() stack frame factorial(4) stack frame 4 x main() stack frame n factorial(4) return address factorial(3) 4 3 x n return address n return address Stack Region Stack Region

22 Stack Frame or Activation Record (2) main() stack frame factorial(4) factorial(3) factorial(2) factorial(1) factorial(0) return addr x main() stack frame 4 n return address factorial(4) 3 n return address factorial(3) 2 n return address factorial(2) 1 n return address factorial(1) 0 n Return 1 x 4 n return address 3 n return address 2 n return address 1 n return address Stack Region Stack Region

23 Stack Frame or Activation Record (3) main() stack frame factorial(4) factorial(3) factorial(2) factorial(1) return addr x main() stack frame 4 n return address factorial(4) 3 n return address factorial(3) 2 n return address factorial(2) 1 n Return 1 x 4 n return address 3 n return address 2 n return address Stack Region Stack Region

24 Stack Frame or Activation Record (4) main() stack frame factorial(4) return addr 4 x n main() stack frame Return 24 x Stack Region Stack Region

25 Expensive side of recursive function (1) The recursive factorial function uses more memory than its non-recursive counterpart. The non-recursive function uses fixed amount of memory for an int data, whereas the memory usage by the recursive function is proportional to the value of data. A function call and return takes some amount of extra time.

26 Recursive Function by Iteration! (1) Value computation in recursively defined factorial function starts after unfolding the recursion. But we can redesign the function so that it can start the computation from the beginning. int factiter(int n) { int acc = 1, i; for(i = n; i > 0; --i) acc *= i; return acc;}

27 gcd(m, n) (1) Inductive Definition & Recursive Function gcd(m, n) m n = { n if m = 0, gcd(n mod m, m) if m > 0. int gcd(int m, int n){ // m <= n if(m == 0) return n; return gcd(n%m, m);}

28 gcd(m, n) (2) Different Calls to gcd() gcd(0, 0) return 0 gcd(0, 5) return 5 gcd(5, 0) gcd(0, 5) return 5 gcd(18, 12) gcd(12, 18) gcd(6, 12) gcd(0, 6) return 6

29 Fibonacci Sequence (1) Fibonacci s question in the year 1202: How many pairs of rabbits will be there in a year from now? Conditions: Begin with 1 male and 1 female rabbit just born which reach sexual maturity after 1 month. The gestation period of a rabbit is 1 month. On reaching sexual maturity, a female rabbit gives birth to 1 male and 1 female rabbit in every month. Rabbits never die. Leonardo Pisano Fibonacci ( , Pisa).

30 Fibonacci Sequence (2) n fib(n)

31 Fibonacci Sequence (3) A tiling with squares whose side lengths are successive Fibonacci numbers.

32 Fibonacci Sequence (4) Fibonacci spiral from golden rectangle of size (It starts at the left-bottom corner of the first square (length 8) and passes through the farthest corners of succeeding squares.)

33 Fibonacci Sequence (5) Romanesque brocolli (Roman cauliflower) is a striking example of the Fibonacci.

34 Fibonacci Sequence (6) Spiral aloe. Numerous such cactus display the Fibonacci spiral.

35 Fibonacci Sequence (7) Nautilus shell displays Fibonacci spiral.

36 Fibonacci Sequence (8) Cancer cell division (mitosis) [Imaging by time-lapse microscopy. The DNA is shown in red, and the cell membrane is shown in cyan.]

37 Fibonacci Sequence (9) Inductive Definition { n if 0 n < 2 fib n = fib n 1 + fib n 2 if n 2. And its direct coding results to: int fibr(int n){ // efficient? if(n < 2) return n; return fibr(n-1) + fibr(n-2); }

38 Fibonacci Sequence (10) 2 1 fibr(4) fibr(5) + Recursion Tree (n = 5) 4 3 fibr(2) fibr(3) fibr(1) 1 1

39 Fibonacci Sequence (11) 3 fibr(3) 2 fibr(4) + 1 fibr(5) fibr(2) fibr(1) 6 fibr(0)

40 Fibonacci Sequence (12) 1 2 fibr(4) + 3 fibr(3) fibr(2) fibr(1) fibr(1) 6 1 fibr(0) fibr(5) +

41 Fibonacci Sequence (13) 1 fibr(5) + 2 fibr(4) fibr(3) 7 fibr(2) fibr(2) fibr(1) fibr(1) 4 fibr(0) fibr(1) fibr(0)

42 Fibonacci Sequence (14) 4 1 fibr(5) fibr(4) fibr(3) fibr(3) fibr(2) fibr(2) fibr(1) fibr(1) fibr(1) fibr(2) fibr(0)fibr(1) fibr(0) fibr(1) fibr(0)

43 Fibonacci Sequence (15) Why bad? For n = 5, #calls = 15, #additions = 17. Better option Can be done by only n 1 = 4 additions in an iterative program.

44 Fibonacci Sequence (16) Number of additions to compute fib n by the previous recursive function: n fib n add n if n = 0 or 1, add n = add n 1 + add n = fib n+1 1 if n > 1

45 Fibonacci Sequence (17) Why bad? When the recursive function is called with n as parameter, there will be n + 1 activation records (stack frames) present on the stack. On the contrary, for the iterative program, there is only a constant number of variables.

46 Fibonacci Sequence (18) Non-Recursive int fib(int n){ // non-recursive int i=1, f0=0, f1=1; if(n < 2) return n; for(i=2; i<=n; ++i) f1 += f0, f0 = f1 - f0; return f1;} i f f

47 Fibonacci Sequence (19) An Efficient Recursive Function We can write a recursive C function that will compute like the iterative program. This function has three parameters, and let it be called as fib(n,0,1), where 0 and 1 are base values corresponding to fib(0) and fib(1). int fib(int n, int f0, int f1) { if(n == 0) return f0; if(n == 1) return f1; return fib(n-1, f1, f1+f0);}

48 Fibonacci Sequence (20) #include <stdio.h> int fib(int, int, int); int main() { int n; printf("enter a non-ve integer: "); scanf("%d", &n); printf("fib(%d)=%d\n",n,fib(n,0,1)); return 0;}

49 Fibonacci Sequence (21) 1st call 2nd call 3rd call 4th call 5th call f n Return f1 5