## Newton Raphson Method

**Newton Raphson Method:**

When an approximate value of a root of an equation is given, this method is used to obtain better and closer approximation to the root. Let x_{0} be an approximation of a root of the given equation f (x) = 0, Let x_{1} = x_{0} h be the exact approximation of the root. Then f (x_{0} h) = 0.

By Taylor’s theorem, we have

Since h is small, we can neglect second, third and higher degree terms in h and thus we get

We may iterate the process to refine the root. In general, we may write

This result is known as **Newton-Raphson formula.**

**Example: **Solve sinx = 1 x^{3} using Newton-Raphson Method.

**Solution:** Let f (x) = sin x − 1 − x^{3}, then f ′(x) = cos x − 3x^{2}. Then Newton-Raphson formula for this problem reduces to;

Now, since f (−1) < 0 and f (−2) > 0, root lies in between −1 and −2. Let x_{0 }= −1.1 be the initial approximation. then successive iteration from above equations are;

Similarly, x_{2} = −1.249746, x_{3} = −1.2490526, x_{4} = −1.2490522. In x_{2} and x_{3 }6 decimal places are same. i.e. approximated root up to six decimal place is -1.249052.