On numerical stability of continued fractions - V. Hladun; V. Кravtsiv; M. Dmytryshyn; R. Rusyn (Математичні Студії, 2024)
The paper considers the numerical stability of the backward recurrence algorithm (BR-algorithm) for computing approximants of the continued fraction with complex elements. The new method establishes sufficient conditions for the numerical stability of this algorithm and the error bounds of the calculation of the $n$th approximant of the continued fraction with complex elements. It follows from the obtained conditions that the numerical stability of the algorithm depends not only on the rounding errors of the elements and errors of machine operations but also on the value sets and the element sets of the continued fraction. The obtained results were used to study the numerical stability of the BR-algorithm for computing the approximants of the continued fraction expansion of the ratio of Horn's confluent functions $H_7$. Bidisc and bicardioid regions are established, which guarantee the numerical stability of the BR-algorithm.