Termination Criterion and Numerical Convergence.

Let $x_n \to x$ be a sequence which is generated by an algorithm as an approximation of the real number x, and let $\tilde x_n (L)$ be an approximate value for $x_n$, computed on an L-digit computer, with the property $\lim _{L \to \infty } \tilde x_n (L) = x_n $. A termination criterion $N(L)$ is given such that, for any fixed L, it is sufficient to compute only the values $\tilde x_0 (L),\tilde x_1 (L), \cdots ,\tilde x_{N(L)} (L)$ and that, under very weak assumptions, \[\mathop {\lim }\limits_{L \to \infty } \tilde x_{N(L)} (L) = x;\]i.e., $\tilde x_{N(L)} $ “converges numerically.”