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Autumn Semester 2009 Programming and Data Structure 1
Functions
Courtsey: University of Pittsburgh-CSD-Khalifa
Autumn Semester 2009 Programming and Data Structure 2
Class Test I: Time and Venue
• Date: August 20, 2009• Time: 6:00 PM to 7:00 PM
– Students must occupy seat within 5:45 PM, and carry library card with them.
• Venue: VIKRAMSHILA COMPLEX / MAIN BUILDING
- Section 7 & 10 (AE-EC) :: AE, AG, AR, BT, CE, CH, CS , CY, EC Room V1– Section 8 & 10 (Rest) :: Room V2– Section 9 (AE-CS):: AE, AG, AR, BT, CE, CH, CS Room V3– Section 9 (Rest):: Room V4– Section 11:: F 116– Section 12:: F 142
Autumn Semester 2009 Programming and Data Structure 3
Functions: Why?
• Functions– Modularize a program– All variables declared inside functions are local
variables• Known only in function defined
– Parameters• Communicate information between functions• Local variables
• Benefits– Divide and conquer
• Manageable program development– Software reusability
• Use existing functions as building blocks for new programs
• Abstraction - hide internal details (library functions)– Avoids code repetition
Autumn Semester 2009 Programming and Data Structure 4
Functions: Why?
• Divide and conquer – Construct a program from smaller
pieces or components– Each piece more manageable than the
original program
Autumn Semester 2009 Programming and Data Structure 5
Program Modules in C
• Functions– Modules in C
– Programs written by combining user-defined functions with library functions
• C standard library has a wide variety of functions
• Makes programmer's job easier - avoid reinventing the wheel
Autumn Semester 2009 Programming and Data Structure 6
Program Modules in C
• Function calls– Invoking functions
• Provide function name and arguments (data)• Function performs operations or
manipulations• Function returns results
– Boss asks worker to complete task• Worker gets information, does task, returns
result• Information hiding: boss does not know
details
Autumn Semester 2009 Programming and Data Structure 7
Function: An Example#include <stdio.h>
int square(int x){ int y; y=x*x; return(y);}
void main() { int a,b,sum_sq;
printf(“Give a and b \n”); scanf(“%d%d”,&a,&b);
sum_sq=square(a)+square(b);
printf(“Sum of squares= %d \n”,sum_sq); }
Function definition
Name of function
parameter
Return data-type
Functions called
Parameters Passed
Autumn Semester 2009 Programming and Data Structure 8
Invoking a function call : An Example• #include <stdio.h>
• int square(int x)• {• int y;• • y=x*x;• return(y);• }
• void main()• {• int a,b,sum_sq;
• printf(“Give a and b \n”);• scanf(“%d%d”,&a,&b);
• sum_sq=square(a)+square(b);
• printf(“Sum of squares= %d \n”,sum_sq);• }
Assume value of a is 10
10a
x 10
*
y 100
returns
Autumn Semester 2009 Programming and Data Structure 9
Function Definitions
• Function definition format (continued)return-value-type function-name( parameter-list )
{ declarations and statements}
– Declarations and statements: function body (block)• Variables can be declared inside blocks (can be
nested)• Function can not be defined inside another function
– Returning control• If nothing returned
– return; – or, until reaches right brace
• If something returned – return expression;
Autumn Semester 2009 Programming and Data Structure 10
Variable Scope
• int A;• void main()
• { A = 1;• myProc();• printf ( "A =
%d\n", A);• }
• void myProc()• { int A = 2;• while( A==2 )• {• int A = 3;• printf ( "A =
%d\n", A);• break;• }• printf ( "A =
%d\n", A);• }• . . .
Printout:
--------------
A = 3
A = 2
A = 1
Autumn Semester 2009 Programming and Data Structure 11
Math Library Functions
• Math library functions – perform common mathematical calculations– #include <math.h>
• Format for calling functionsFunctionName (argument);
• If multiple arguments, use comma-separated list
– printf( "%5.2f", sqrt( 900.0 ) ); • Calls function sqrt, which returns the square root of its
argument
• All math functions return data type double
– Arguments may be constants, variables, or expressions
Autumn Semester 2009 Programming and Data Structure 12
Math Library Functions• double acos(double x) -- Compute arc cosine of x.
double asin(double x) -- Compute arc sine of x. double atan(double x) -- Compute arc tangent of x. double atan2(double y, double x) -- Compute arc tangent of y/x. double ceil(double x) -- Get smallest integral value that exceeds x. double cos(double x) -- Compute cosine of angle in radians. double cosh(double x) -- Compute the hyperbolic cosine of x. div_t div(int number, int denom) -- Divide one integer by another. double exp(double x -- Compute exponential of x double fabs (double x ) -- Compute absolute value of x. double floor(double x) -- Get largest integral value less than x. double fmod(double x, double y) -- Divide x by y with integral quotient and return remainder. double frexp(double x, int *expptr) -- Breaks down x into mantissa and exponent of no. labs(long n) -- Find absolute value of long integer n. double ldexp(double x, int exp) -- Reconstructs x out of mantissa and exponent of two. ldiv_t ldiv(long number, long denom) -- Divide one long integer by another. double log(double x) -- Compute log(x). double log10 (double x ) -- Compute log to the base 10 of x. double modf(double x, double *intptr) -- Breaks x into fractional and integer parts. double pow (double x, double y) -- Compute x raised to the power y. double sin(double x) -- Compute sine of angle in radians. double sinh(double x) - Compute the hyperbolic sine of x. double sqrt(double x) -- Compute the square root of x. void srand(unsigned seed) -- Set a new seed for the random number generator (rand). double tan(double x) -- Compute tangent of angle in radians. double tanh(double x) -- Compute the hyperbolic tangent of x.
http://www.cs.cf.ac.uk/Dave/C/node17.html#SECTION001710000000000000000
Autumn Semester 2009 Programming and Data Structure 13
Function Prototypes
• Function prototype
– Function name– Parameters - what the function takes in– Return type - data type function returns (default int)– Used to validate functions– Prototype only needed if function definition comes after
use in programint maximum( int, int, int );
• Takes in 3 ints
• Returns an int
• Promotion rules and conversions– Converting to lower types can lead to errors
Autumn Semester 2009 Programming and Data Structure 14
Header Files
• Header files– contain function prototypes for library functions– <stdlib.h> , <math.h> , etc– Load with #include <filename>
#include <math.h>
• Custom header files– Create file with functions – Save as filename.h– Load in other files with #include "filename.h"– Reuse functions
Autumn Semester 2009 Programming and Data Structure 15
/* Finding the maximum of three integers */
#include <stdio.h>
int maximum( int, int, int ); /* function prototype */
int main(){ int a, b, c;
printf( "Enter three integers: " ); scanf( "%d%d%d", &a, &b, &c ); printf( "Maximum is: %d\n", maximum( a, b, c ) );
return 0;}
/* Function maximum definition */int maximum( int x, int y, int z ){ int max = x;
if ( y > max ) max = y;
if ( z > max ) max = z;
return max;}
Autumn Semester 2009 Programming and Data Structure 16
Calling Functions: Call by Value and Call by Reference
• Used when invoking functions• Call by value
– Copy of argument passed to function– Changes in function do not affect original– Use when function does not need to modify argument
• Avoids accidental changes
• Call by reference – Passes original argument– Changes in function affect original– Only used with trusted functions
• For now, we focus on call by value
Autumn Semester 2009 Programming and Data Structure 17
An Example: Random Number Generation
• rand function– Prototype defined in <stdlib.h>– Returns "random" number between 0 and RAND_MAX (at
least 32767)i = rand();
– Pseudorandom• Preset sequence of "random" numbers• Same sequence for every function call
• Scaling– To get a random number between 1 and n1 + ( rand() % n )• rand % n returns a number between 0 and n-1• Add 1 to make random number between 1 and n1 + ( rand() % 6) // number between 1 and 6
Autumn Semester 2009 Programming and Data Structure 18
Random Number Generation: Contd.
• srand function– Prototype defined in <stdlib.h>– Takes an integer seed - jumps to location in
"random" sequence
srand( seed );
Autumn Semester 2009 Programming and Data Structure 19
1 /* A programming example
2 Randomizing die-rolling program */
3 #include <stdlib.h>
4 #include <stdio.h>
5
6 int main()
7 {
8 int i;
9 unsigned seed;
10
11 printf( "Enter seed: " );
12 scanf( "%u", &seed );
13 srand( seed );
14
15 for ( i = 1; i <= 10; i++ ) {
16 printf( "%10d ", 1 + ( rand() % 6 ) );
17
18 if ( i % 5 == 0 )
19 printf( "\n" );
20 }
21
22 return 0;
23 }
Algorithm1. Initialize seed
2. Input value for seed
2.1 Use srand to change random sequence
2.2 Define Loop
3. Generate and output random numbers
Autumn Semester 2009 Programming and Data Structure 20
Program Output
Enter seed: 867 2 4 6 1 6 1 1 3 6 2
Enter seed: 67 6 1 4 6 2 1 6 1 6 4
Enter seed: 67 6 1 4 6 2 1 6 1 6 4
Autumn Semester 2009 Programming and Data Structure 21
Another Example: A Game of Chance
• Rules– Roll two dice
• 7 or 11 on first throw, player wins• 2, 3, or 12 on first throw, player loses• 4, 5, 6, 8, 9, 10 - value becomes player's "point"
– Player must roll his point before rolling 7 to win
Autumn Semester 2009 Programming and Data Structure 22
Programme Structure
• 1. rollDice prototype•
1.1 Initialize variables•
1.2 Seed srand
• 2. Define switch statement for win/loss/continue
• 2.1 Loop
Autumn Semester 2009 Programming and Data Structure 23
1 /* A Game of Chance2 Rolling Two Dice*/3 #include <stdio.h>4 #include <stdlib.h>5 #include <time.h>67 int rollDice( void );89 int main()10 {11 int gameStatus, sum, myPoint;1213 srand( time( NULL ) ); /* Randomizing seed by time(NULL) */14 sum = rollDice(); /* first roll of the dice */1516 switch ( sum ) {17 case 7: case 11: /* win on first roll */18 gameStatus = 1;19 break;20 case 2: case 3: case 12: /* lose on first roll */21 gameStatus = 2;22 break;23 default: /* remember point */24 gameStatus = 0;25 myPoint = sum;26 printf( "Point is %d\n", myPoint );27 break;28 }2930 while ( gameStatus == 0 ) { /* keep rolling */31 sum = rollDice();32
Autumn Semester 2009 Programming and Data Structure 24
2.2 Print win/loss
Program Output
33 if ( sum == myPoint ) /* win by making point */
34 gameStatus = 1;
35 else
36 if ( sum == 7 ) /* lose by rolling 7 */
37 gameStatus = 2;
38 }
39
40 if ( gameStatus == 1 )
41 printf( "Player wins\n" );
42 else
43 printf( "Player loses\n" );
44
45 return 0;
46 }
47
48 int rollDice( void )
49 {
50 int die1, die2, workSum;
51
52 die1 = 1 + ( rand() % 6 );
53 die2 = 1 + ( rand() % 6 );
54 workSum = die1 + die2;
55 printf( "Player rolled %d + %d = %d\n", die1, die2, workSum );
56 return workSum;
57 }
Autumn Semester 2009 Programming and Data Structure 25
Program Output
Player rolled 6 + 6 = 12Player loses
Player rolled 4 + 6 = 10Point is 10Player rolled 2 + 4 = 6Player rolled 6 + 5 = 11Player rolled 3 + 3 = 6Player rolled 6 + 4 = 10Player wins
Player rolled 1 + 3 = 4Point is 4Player rolled 1 + 4 = 5Player rolled 5 + 4 = 9Player rolled 4 + 6 = 10Player rolled 6 + 3 = 9Player rolled 1 + 2 = 3Player rolled 5 + 2 = 7Player loses
Player rolled 6 + 5 = 11Player wins
Autumn Semester 2009 Programming and Data Structure 26
Recursion
• A process by which a function calls itself repeatedly.– Either directly.
• X calls X.
– Or cyclically in a chain.• X calls Y, and Y calls X.
• Used for repetitive computations in which each action is stated in terms of a previous result.– fact(n) = n * fact (n-1)
Autumn Semester 2009 Programming and Data Structure 27
Contd.
• For a problem to be written in recursive form, two conditions are to be satisfied:– It should be possible to express the problem in
recursive form.– The problem statement must include a
stopping conditionfact(n) = 1, if n = 0
= n * fact(n-1), if n > 0
Autumn Semester 2009 Programming and Data Structure 28
Example 1 :: Factorial
long int fact (n)int n;{ if (n = = 0) return (1); else return (n * fact(n-1));}
Autumn Semester 2009 Programming and Data Structure 29
Mechanism of Execution
• When a recursive program is executed, the recursive function calls are not executed immediately.– They are kept aside (on a stack) until the
stopping condition is encountered.– The function calls are then executed in reverse
order.
Autumn Semester 2009 Programming and Data Structure 30
Example :: Calculating fact(4)
– First, the function calls will be processed:fact(4) = 4 * fact(3)
fact(3) = 3 * fact(2)
fact(2) = 2 * fact(1)
fact(1) = 1 * fact(0)
– The actual values return in the reverse order:fact(0) = 1
fact(1) = 1 * 1 = 1
fact(2) = 2 * 1 = 2
fact(3) = 3 * 2 = 6
fact(4) = 4 * 6 = 24
Autumn Semester 2009 Programming and Data Structure 31
Factorials: Contd.
• Example: factorials– 5! = 5 * 4 * 3 * 2 * 1
– Notice that• 5! = 5 * 4!• 4! = 4 * 3! ...
– Can compute factorials recursively – Solve base case (1! = 0! = 1) then plug in
• 2! = 2 * 1! = 2 * 1 = 2;• 3! = 3 * 2! = 3 * 2 = 6;
Autumn Semester 2009 Programming and Data Structure 32
Recursive evaluation of 5!.
Autumn Semester 2009 Programming and Data Structure 33
1 /* An example */
2 /* Recursive factorial function */
3 #include <stdio.h>
4
5 long factorial( long number ); /* function prototype */
6
7 /* function main begins program execution */
8 int main( void )
9 {
10 int i; /* counter */
11
12 /* loop 11 times; during each iteration, calculate
13 factorial( i ) and display result */
14 for ( i = 0; i <= 10; i++ ) {
15 printf( "%2d! = %ld\n", i, factorial( i ) );
16 } /* end for */
17
18 return 0; /* indicates successful termination */
19
20 } /* end main */
21
22 /* recursive definition of function factorial */
23 long factorial( long number )
24 {
25 /* base case */
26 if ( number <= 1 ) {
27 return 1;
28 } /* end if */
29 else { /* recursive step */
30 return ( number * factorial( number - 1 ) );
31 } /* end else */
32
33 } /* end function factorial */ 0! = 1 1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40320 9! = 362880 10! = 3628800
TerminatingCriterion
Autumn Semester 2009 Programming and Data Structure 34
Another Example :: Fibonacci number
• Fibonacci number f(n) can be defined as: f(0) = 0
f(1) = 1
f(n) = f(n-1) + f(n-2), if n > 1– The successive Fibonacci numbers are:
0, 1, 1, 2, 3, 5, 8, 13, 21, …..
• Function definition:int f (int n){ if (n < 2) return (n); else return (f(n-1) + f(n-2));}
Autumn Semester 2009 Programming and Data Structure 35
Tracing Execution
• How many times the function is called when evaluating f(4) ?
• Inefficiency:– Same thing is
computed several times.
f(4)
f(3)
f(0)f(1) f(1)f(2)
f(2)
f(1) f(0)
9 times
Autumn Semester 2009 Programming and Data Structure 36
Example Codes: fibonacci()
– Code for the fibonacci functionlong fibonacci( long n ){if (n == 0 || n == 1) // base case
return n;else
return fibonacci( n - 1) + fibonacci( n – 2 );
}
Autumn Semester 2009 Programming and Data Structure 37
1 /* Another Example:
2 Recursive fibonacci function */
3 #include <stdio.h>
4
5 long fibonacci( long n ); /* function prototype */
6
7 /* function main begins program execution */
8 int main( void )
9 {
10 long result; /* fibonacci value */
11 long number; /* number input by user */
12
13 /* obtain integer from user */
14 printf( "Enter an integer: " );
15 scanf( "%ld", &number );
16
17 /* calculate fibonacci value for number input by user */
18 result = fibonacci( number );
19
20 /* display result */
21 printf( "Fibonacci( %ld ) = %ld\n", number, result );
22
23 return 0; /* indicates successful termination */
24
25 } /* end main */
26
27 /* Recursive definition of function fibonacci */
28 long fibonacci( long n )
29 {
30 /* base case */
31 if ( n == 0 || n == 1 ) {
32 return n;
33 } /* end if */
34 else { /* recursive step */
35 return fibonacci( n - 1 ) + fibonacci( n - 2 );
36 } /* end else */
37
38 } /* end function fibonacci */
Enter an integer: 0Fibonacci( 0 ) = 0
Enter an integer: 1Fibonacci( 1 ) = 1
Enter an integer: 2Fibonacci( 2 ) = 1 (continued on next slide… )
Autumn Semester 2009 Programming and Data Structure 38
(continued from previous slide…) Enter an integer: 3 Fibonacci( 3 ) = 2 Enter an integer: 4 Fibonacci( 4 ) = 3 Enter an integer: 5 Fibonacci( 5 ) = 5 Enter an integer: 6 Fibonacci( 6 ) = 8 )
Autumn Semester 2009 Programming and Data Structure 39
Set of recursive calls for fibonacci(3).
Autumn Semester 2009 Programming and Data Structure 40
Performance Tip
• Avoid Fibonacci-style recursive programs which result in an exponential “explosion” of calls.
Autumn Semester 2009 Programming and Data Structure 41
Recursion and activation records: an example
#include <stdio.h>main(){ int a, b; b = 7; a = xyz (b); printf ("a=%d, b=%d \n", a, b);}
int xyz (int n){ if (n <= 0) return (2); else { n = n - 2; return (xyz(n-1) * xyz(n-2) + 5); }}
What is the output?How many times xyz()Called?
Autumn Semester 2009 Programming and Data Structure 42
XYZ(7)
XYZ(4) XYZ(3)
XYZ(1) XYZ(0) XYZ(0) XYZ(-1)
XYZ(-2) XYZ(-3)
2 2
9 2 2 2
239
23*9+5=212
if (n <= 0) return (2); else { n = n - 2; return (xyz(n-1) * xyz(n-2) + 5);}
Autumn Semester 2009 Programming and Data Structure 43
Trace of the recursive calls
Autumn Semester 2009 Programming and Data Structure 44
RA .. main
-
b = 7
RA .. main
-
b = 7
RA .. xyz
-
n = 4
RA .. xyz
-
n = 1
RA .. main
-
b = 7
RA .. xyz
-
n = 4
RA .. xyz
2
n = -2
RA .. xyz
-
n = 1
RA .. main
-
b = 7
RA .. xyz
-
n = 4
RA .. xyz
2
n = -3
RA .. xyz
-
n = 1
RA .. main
-
b = 7
RA .. xyz
-
n = 4RA .. xyz
9
n = 1
RA .. main
-
b = 7
RA .. xyz
-
n = 4
RA .. main
-
b = 7
RA .. xyz
23
n = 4
RA .. main
-
b = 7
RA .. xyz
-
n = 3
Autumn Semester 2009 Programming and Data Structure 45
What is the value of a and b?
How many times the recursive function called?
(i) a=212, b=7
(i) xyz( ) called 9 times
Autumn Semester 2009 Programming and Data Structure 46
Recursion vs. Iteration
• Repetition– Iteration: explicit loop– Recursion: repeated function calls
• Termination– Iteration: loop condition fails– Recursion: base case recognized
• Both can have infinite loops• Balance
– Choice between performance (iteration) and good software engineering (recursion)
Autumn Semester 2009 Programming and Data Structure 48
Performance Tip
• Avoid using recursion in performance situations. Recursive calls take time and consume additional memory.