RATE OF CONVERGENCE

The rate of convergence may be estimated from equation (31). It can be rewritten

$$\vert T_N-\tilde{T}_N \vert < {x^4 \over 8Nv^4 T^2_N (T_N+\tilde{T}_N)}.$$ (32)

Because

$$T_N+\tilde{T}_N \gt 2T_N$$ (33)

we can obtain this estimate of the number of iterations:  

$$N \gt {x^4/v^4 \over 8 (t^2-x^2/v^2)^{3/2}\epsilon},$$ (34)

where $\epsilon$ is the required precision.

The number of iterations estimated from equation (34) is higher than the real number of iterations needed to achieve the desired precision. For a low number of iterations (up to about 10), inequality (34) gives a good estimate of the actual number of iterations. Curves of the numbers of iterations required to achieve particular precisions are shown on Figure c. These curves are not computed from equation (34), but they are the numbers of iterations really needed to achieve the required precision.