Numerical estimation of quantum channel capacity is an important problem both theoretically and practically. Oto et al. showed an effective approximation algorithm to compute a capacity of a one-qubit quantum channel. In this paper, we show an explicit upper bound for the error of their algorithm. The algorithm might be repeatedly used in the verification of a certain equation or inequality. The bound of the error we give is a criterion to decide when to stop that iteration, and eventually it can improve the performance of the whole process of the calculation.