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## Newton’s Method

**Intro:-** Newton-Raphson method also called as **Newton’s Method** is used to find simple real roots of a polynomial equation.

It has the fastest rate of convergence. The method is quite sensitive to the starting value.

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It may also diverge if the *first derivative* i.e.** f'(x) **of the function is near zero during the iterative cycle.

**see more**

## C++ Program Newton Raphson Method

```
//Newton raphson method c++
//techindetail.com
#include<iostream.h>
#include<math.h>
float fun(float x)//we are finding the root of x^4-x-10
{
return x*x*x*x-x-10;
}
float diff(float x) differential of x^4-x-10;
{
return 4*x*x*x-1;
}
int main()
{
int itr,maxitr;
float h,x0,x1,aerr;
cout<<"Enter x0,allowed error, maximum iterations"<<endl;
cin>>x0>>aerr>>maxitr;
for(itr=1;itr<=maxitr;itr++)
{
h=fun(x0)/diff(x0);
x1=x0-h;
if(fabs(h)<aerr)
{
cout<<"after "<<itr<<" root = "<<x1<<endl;
return 0;
}
x0=x1;
}
cout<<"Iterations not sufficient, Solution does not converge"<<endl;
return 1;
}
```

Code language: C++ (cpp)

**Algorithm for Newton Raphson method c/c++:**

**Read****x0**, e, n, N where**x0**is the initial guess of the root, e the allowed error, n the order of the polynomial, and N the total number of iterations.**for i=0 to n in steps of 1 do Read bi end for.****for i=0 to n-1 in steps of 1 do Read bi end for.****P=an****bn-1=an****S=bn-1****for k=1 to N in steps of 1 do****for i=1 to n-1 in steps of 1 do****b**_{n-(i+1)}=a_{n-i}+**x0**b_{n-i}**S=b**_{n-(i+1)}+**x0**S

endfor**P=a****0**+ b**0**x0**x1**=**x0**-(P/S)**if |****x1**–**x0**/**x1**| ≤ e goto step 18

else**x0**=**x1**

endif

endfor**write “root not found in N iterations”****write S, P,****x1**,**x0****stop****write “root found in k iterations”****x0**=**x1****write****x0**, S, P**stop**

#### Some observations about Newton Raphson method c:

- Newton’s method is useful in cases of large values of f'(x) e.e.
- when the graph of f(x) while crossing the x-axis in nearly vertical.

For if f'(x) is small in the vicinity of the root, then by*h=-f(x)/f'(x),* - h will be large and the computation of the root is slow or may not be possible.
- Thus this method is not suitable in those cases where the graph of f(x) is nearly horizontal while crossing the x-axis.
- Newton’s method is applicable to both algebraic and transcendental equations.
- Newton’s method is useful when x0 is chosen sufficiently close to the root.
- Newton’s Method has second-order convergence.