Question

use C programing to solve the following exercise.

Compute a root of the equation 4. (20 points) e-3 cos(x)-o using (a) Bisection Method between 0 and I. (b) Newton Method using an initial guess of I. Use e0.00001 Show that Newton Method has a faster convergence than Bisection Method

Answer 1

Part-A:

the program is:

#include<stdio.h>
#include<math.h>

double func(double x)
{
return exp(x)-3*cos(x);
}

double e=0.00001;
double c;

void bisection(double a,double b)
{
if(func(a) * func(b) >= 0)
{
printf("Incorrect a and b");
return;
}

c = a;

while ((b-a) >= e)
{
c = (a+b)/2;
if (func(c) == 0.0){
printf("Root = %lf\n",c);
break;
}
else if (func(c)*func(a) < 0){
printf("Root = %lf\n",c);
b = c;
}
else{
printf("Root = %lf\n",c);
a = c;
}
}
}

int main()
{
double a,b;
a=0;
b=1;

printf("The function used is e^x-3cos(x)\n");
printf("a = %lf\n",a);
printf("b = %lf\n",b);
bisection(a,b);
printf("\n");
printf("Accurate Root calculated is = %lf\n",c);

return 0;
}

The output is:

The function used is e^x-3cos(x)

a = 0.000000
b = 1.000000
Root = 0.500000
Root = 0.750000
Root = 0.875000
Root = 0.812500
Root = 0.781250
Root = 0.765625
Root = 0.773438

Accurate Root calculated is = 0.773438

Part-B:

#include<stdio.h>
#include<math.h>
float f(float x)
{
return exp(x)-cos(x);
}
float df (float x)
{
return exp(x)-cos(x);
}
main()
{
int itr, maxmitr;
float h, x0, x1, allerr;
printf("\nEnter x0, error and iterations\n");
scanf("%f %f %d", &x0, &allerr, &maxmitr);
for (itr=1; itr<=maxmitr; itr++)
{
h=f(x0)/df(x0);
x1=x0-h;
printf(" Iteration no. %d, x = %f\n", itr, x1);
if (fabs(h) < allerr)
{
printf("After %d iterations, root = %f\n", itr, x1);
return 0;
}
x0=x1;
}
printf(" The required solution does not converge \n");
return 1;
}

Output:

Enter x0, allowed error and maximum iterations
1
0.0001
80
At Iteration no. 1, x = 0.000000
At Iteration no. 2, x = -1.#IND00
At Iteration no. 3, x = -1.#IND00
At Iteration no. 4, x = -1.#IND00
At Iteration no. 5, x = -1.#IND00
At Iteration no. 6, x = -1.#IND00
At Iteration no. 7, x = -1.#IND00
At Iteration no. 8, x = -1.#IND00
At Iteration no. 9, x = -1.#IND00
At Iteration no. 10, x = -1.#IND00
At Iteration no. 11, x = -1.#IND00
At Iteration no. 12, x = -1.#IND00
At Iteration no. 13, x = -1.#IND00
At Iteration no. 14, x = -1.#IND00
At Iteration no. 15, x = -1.#IND00
At Iteration no. 16, x = -1.#IND00
At Iteration no. 17, x = -1.#IND00
At Iteration no. 18, x = -1.#IND00
At Iteration no. 19, x = -1.#IND00
At Iteration no. 20, x = -1.#IND00
At Iteration no. 21, x = -1.#IND00
At Iteration no. 22, x = -1.#IND00
At Iteration no. 23, x = -1.#IND00
At Iteration no. 24, x = -1.#IND00
At Iteration no. 25, x = -1.#IND00
At Iteration no. 26, x = -1.#IND00
At Iteration no. 27, x = -1.#IND00
At Iteration no. 28, x = -1.#IND00
At Iteration no. 29, x = -1.#IND00
At Iteration no. 30, x = -1.#IND00
At Iteration no. 31, x = -1.#IND00
At Iteration no. 32, x = -1.#IND00
At Iteration no. 33, x = -1.#IND00
At Iteration no. 34, x = -1.#IND00
At Iteration no. 35, x = -1.#IND00
At Iteration no. 36, x = -1.#IND00
At Iteration no. 37, x = -1.#IND00
At Iteration no. 38, x = -1.#IND00
At Iteration no. 39, x = -1.#IND00
At Iteration no. 40, x = -1.#IND00
At Iteration no. 41, x = -1.#IND00
At Iteration no. 42, x = -1.#IND00
At Iteration no. 43, x = -1.#IND00
At Iteration no. 44, x = -1.#IND00
At Iteration no. 45, x = -1.#IND00
At Iteration no. 46, x = -1.#IND00
At Iteration no. 47, x = -1.#IND00
At Iteration no. 48, x = -1.#IND00
At Iteration no. 49, x = -1.#IND00
At Iteration no. 50, x = -1.#IND00
At Iteration no. 51, x = -1.#IND00
At Iteration no. 52, x = -1.#IND00
At Iteration no. 53, x = -1.#IND00
At Iteration no. 54, x = -1.#IND00
At Iteration no. 55, x = -1.#IND00
At Iteration no. 56, x = -1.#IND00
At Iteration no. 57, x = -1.#IND00
At Iteration no. 58, x = -1.#IND00
At Iteration no. 59, x = -1.#IND00
At Iteration no. 60, x = -1.#IND00
At Iteration no. 61, x = -1.#IND00
At Iteration no. 62, x = -1.#IND00
At Iteration no. 63, x = -1.#IND00
At Iteration no. 64, x = -1.#IND00
At Iteration no. 65, x = -1.#IND00
At Iteration no. 66, x = -1.#IND00
At Iteration no. 67, x = -1.#IND00
At Iteration no. 68, x = -1.#IND00
At Iteration no. 69, x = -1.#IND00
At Iteration no. 70, x = -1.#IND00
At Iteration no. 71, x = -1.#IND00
At Iteration no. 72, x = -1.#IND00
At Iteration no. 73, x = -1.#IND00
At Iteration no. 74, x = -1.#IND00
At Iteration no. 75, x = -1.#IND00
At Iteration no. 76, x = -1.#IND00
At Iteration no. 77, x = -1.#IND00
At Iteration no. 78, x = -1.#IND00
At Iteration no. 79, x = -1.#IND00
At Iteration no. 80, x = -1.#IND00
The required solution does not converge or iterations are insufficient