Newton-Raphson Method

This method is also called "Tangent Method".
ALGORITHM:

1. START.
2. Find the first order derivative f'(x) of the given non-linear equation f(x).
3. Assume initial guess as x0.
4. Find the next value from N-R formula as x=x0-f(x0)/f'(x0). 
5. If |x-x0|<accuracy, root = x, Else set x0=x and go to step 4.
6. STOP.

PROGRAM in C:

#include<stdio.h>

#include<math.h>

#include<conio.h>

float f(float x)

{

  return(3*x-cos(x)-1); /* 3x=cos(x)+1 */

}

float g(float y)

{

  return(3+sin(y)); /* 3+sin(y) */

}

void main()

{

  float x0,x=0,eps=0.001,tmp;

  int iteration=0;

  printf("\nEnter guess:");

  scanf("%f",&x0);

  do

  {

    tmp=x;

    x=x0-(f(x0)/g(x0));

    iteration++;

    x0=x;

  }while(fabs(tmp-x)>eps);

  printf("Root = %.4f Iteration = %d",x,iteration);

  getch();

}

Secant Method >>