SKEDSOFT

We define the error of approximation at the k_th iterate as ε_k = x_k – α, k = 0, 1, 2,...

Subtracting (1.8) from (1.9), we obtain

                                                                                 x_{k 1} – α  =  φ(x_k) – φ(α).....................2.0

        = (xk – α)φ′(tk)          (using the mean value theorem)........................ 2.1     

or

......................2.3

Setting k = k – 1, we get

                                                                       .........................2.4                                                                                            

Hence,

                                                                       ...........................2.5

Using (2.1) recursively, we get

                                                                         ............................2.6

The initial error ε₀ is known and is a constant. We have                  

                                                       ...............................2.7

Let

Then,

                                                                      .......................2.8  

For convergence, we require that | ε_{k 1} | → 0 as k → ∞. This result is possible, if and only if c < 1. Therefore, the iteration method (1.9) converges, if and only if 

or,

                                                      | φ′(x) | ≤ c < 1, for all x in the interval (a, b).  

We can test this condition using x₀, the initial approximation, before the computations are done.

Let us now check whether the methods (1.5), (1.6), (1.7) converge to a root in (0, 1) of the equation f(x) = x³ – 5x 1 = 0.