Newton-raphson Method

Example 1.4   Derive the Newton’s method for finding 1/N, where N > 0. Hence, find 1/17, using the initial approximation as  0.05.Do the iterations converge ?

Solution :

Newton’s method gives

With N = 17, and x0 = 0.05, we obtain the sequence of approximations

Since, | x4 – x3 | = 0, the iterations converge to the root. The required root is 0.058823.

Example 1.5   Using Newton-Raphson method solve x log₁₀ x = 12.34 with x₀ = 10.

Solution : Define f(x) = x log10 x – 12.34.

Then

Using the Newton-Raphson method, we obtain

With x₀ = 10, we obtain the following results.

We have | x3 – x2 | = | 11.594854 – 11.594870 | = 0.000016.
We may take x ≈ 11.594854 as the root correct to four decimal places.