C Functions

1. Example: Alternative to pow(x, y)
   Background
   Here is an example of a C program to compute x^y without using pow(a, b).

   #include <stdio.h> #include <math.h> float power(float x, float y) { return exp(y*log(x)); } int main() { float x,y; printf("Enter x and y separated by space ="); scanf("%f %f",&x,&y); if (x<0) { printf("x must be positive !!\n"); return 0; } printf("%f to power of exponent %f is %f.\n", x, y, power(x,y)); return 0; }

2. Variables used in a function are local, i.e. they do not retain the values outside the function.

   #include <stdio.h> float f(int n) { int i; float sum=0; for (i=1;i < n;i++) sum=sum+1.0/i; return sum; } int main() { float sum=0; int i; for (i=1;i < 20;i++) sum=sum+1.0/i/i; printf("%f %f\n", sum, f(20)); return 0; }

   #include <stdio.h> int i=20; void f() { printf("%d\n",i); } int main() { int i=30; printf("%d\n", i); f(); return 0; }
3. C functions are recursive (i.e. a function can call its own. Examples follow.)
   • Fibonacci numbers
     
     from Da Vinci Code.
     The Fibonacci numbers are defined as
     

     an = an−1+ an−2,     a1 = 1,     a2 = 1.

     This is related to the problem of how fast rabbits could breed in ideal circumstances studied by Fibonacci (Click this link).
     
     Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...
     How many pairs will there be in two years (a₂₄ = ?) ?
     Code:

     #include <stdio.h> int fibo(int n) { if (n==1) return 1; if (n==2) return 1; return fibo(n-1) + fibo(n-2); } int main() { int i; printf("Enter n = "); scanf("%d", &i); printf("%d\n", fibo(i)); return 0; }

     The explicit form of the Fibonacci sequence is given by
     
     

     an= ⎛ ⎝ 1 2 (1+√5) ⎞ ⎠ n   − ⎛ ⎝ 1 2 (1−√5) ⎞ ⎠ n   √5 .
   • Alternative to 1 + 2 + 3 + 4 + …+ n.
     The following program computes
     

     1+2+3+ …+ n.

     #include <stdio.h> int mysum(int n) { if (n==0) return 0; return n + mysum(n-1); } int main() { int n; printf("Enter n = "); scanf("%d", &n); printf("1+2+....+%d =%d \n", n, mysum(n)); return 0; }
   • Another example of recursive algorithm:
     If a sequence of numbers is generated by the following rules, what is a₂₀ ?
     
     

     an = 4an−1 − 6 an−2,     a1 = 1,     a2 = 3.

     (1)

     #include <stdio.h> int mysequence(int n) { if (n==1) return 1; if (n==2) return 3; return 4*mysequence(n-1)-6*mysequence(n-2); } int main() { int n; printf("Enter n = "); scanf("%d", &n); printf("%d\n",mysequence(n)); return 0; }