CPNM LAB PROGRAM -10

10. Implement Newton Raphson method to determine a root of polynomial equation.

                                                                       FLOW CHART

Algorithm: 

Step – 1 : Start 

Step – 2 : Declare x, epsilon, slope, xnew, rerror 

Step – 3 : Print enter initial approximations to root 

Step – 4 : Enter accuracy and limit on allowed slope 

Step – 5 : If(f’(x) < slope) if true go to Step–6 otherwise go to Step–11 

Step – 6 : Print slope is too small 

Step – 7 : Calculate xnew = x - (f(x) / f1(x)) 

Step – 8 : Calculate rerror = fabs((xnew - x) / xnew) 

Step – 9 : Calculate x = xnew 

Step – 10 : Check whether rerror >= epsilon 

Step – 11 : Print the root as xnew 

Step – 12 : Stop 

               Step – 1 : Call function f(float x) 

             Step – 2 : return(x * x * x - 2 * x - 5)

               Step – 3 : Call function f1(float x) 

               Step – 4 : return(3 * x * x - 2)             

Program:-

#include<stdio.h>

#include<conio.h>

#include<stdlib.h>

#include<math.h>

float f(float x)

{

return(x * x * x - 2 * x - 5);

}

float f1(float x)

{

return(3 * x * x - 2);

}

 main() 

{

float x, epsilon, slope, xnew, rerror; 

clrscr();

printf("\nEnter Initial approximation to root : \n"); 

scanf("%f",&x);

printf("\nEnter accuracy and limit on allowed slope : \n"); 

scanf("%f,%f",&epsilon, &slope); 

do {

if(f1(x) < slope)

{

printf("slope is too small \n"); 

break; }

xnew = x - (f(x) / f1(x)); 

rerror = fabs((xnew - x) / xnew); 

x = xnew; }

while(rerror >= epsilon); 

printf("Root = %f ", xnew); 

getch(); 

return 0;

}

 

Output:-

 

Run 1:

 

Enter Initial approximation to root : 3

Enter accuracy and limit on allowed slope :

0.001,0

Root = 2.094552

 

Run 2:

Enter Initial approximation to root :1

 

Enter accuracy and limit on allowed slope :

0.001,2 slope is too small

Root = 0.000000