SKEDSOFT

The exact root satisfies the equation α = φ(α).

If a1 ≠ 0 that is, φ′(α) ≠ 0, then the method is of order 1 or has linear convergence. For the general iteration method, which is of first order, we have derived that the condition of convergenceis 

                                                                     | φ′(x) | < 1 for all x in the interval (a, b) in which the root lies.

Note that in this method,
 

                                                                     | φ′(x) | ≠ 0 for all x in the neighborhood of the root α.

If a1 = φ′(α) = 0, and a2 = (1/2)φ″(α) ≠ 0,

then from equation                                       ε_{k 1} = a₁ε_k a₂ε_k² ...                        the method is of order 2 or has quadratic convergence.

Let us verify this result for the Newton-Raphson method. For the Newton-Raphson method

Then,

and,

since f(α) = 0 and f ′(α) ≠ 0 (α is a simple root).

When, x_k → α, f (x_k) → 0, we have | φ′(x_k) | < 1, k = 1, 2,... and → 0 as n → ∞.

Now,

and,

Therefore, a₂ ≠ 0 and the second order convergence of the Newton’s method is verified.