SKEDSOFT

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 x₀ be an approximation of a root of the given equation f (x) = 0, Let x₁ = x₀ h be the exact approximation of the root. Then f (x₀ 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 x³ using Newton-Raphson Method.

Solution: Let f (x) = sin x − 1 − x³, then f ′(x) = cos x − 3x². 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 x₀ = −1.1 be the initial approximation. then successive iteration from above equations are;

Similarly, x₂ = −1.249746, x₃ = −1.2490526, x₄ = −1.2490522. In x₂ and x₃ 6 decimal places are same. i.e. approximated root up to six decimal place is -1.249052.