On numerical stability of continued fractions
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 calcu...
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| Format: | Article |
| Language: | German |
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Ivan Franko National University of Lviv
2024-12-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/558 |
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| author | V. Hladun V. Кravtsiv M. Dmytryshyn R. Rusyn |
| author_facet | V. Hladun V. Кravtsiv M. Dmytryshyn R. Rusyn |
| author_sort | V. Hladun |
| collection | DOAJ |
| description | 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 $\mathrm{H}_7$. Bidisc and bicardioid regions are established, which guarantee the numerical stability of the BR-algorithm.
The obtained result is applied to the study of the numerical stability of computing approximants of the continued fraction expansion of the ratio of Horn's confluent function $\mathrm{H}_7$ with complex parameters. In addition, the analysis of the relative errors arising from the computation of approximants using the backward recurrence algorithm, the forward recurrence algorithm, and Lenz's algorithm is given.
The method for studying the numerical stability of the BR-algorithm proposed in the paper can be used to study the numerical stability of the branched continued fraction expansions and numerical branched continued fractions with elements in angular and parabolic domains. |
| format | Article |
| id | doaj-art-8f0876f66f514cd18c0a01c04ab35bb1 |
| institution | Matheson Library |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2024-12-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-8f0876f66f514cd18c0a01c04ab35bb12025-07-08T09:02:48ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202024-12-0162216818310.30970/ms.62.2.168-183558On numerical stability of continued fractionsV. Hladun0V. Кravtsiv1M. Dmytryshyn2R. Rusyn3Lviv Polytechnic National University, Lviv, UkraineVasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, UkraineWest Ukrainian National University, Ternopil, UkraineVasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, UkraineThe 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 $\mathrm{H}_7$. Bidisc and bicardioid regions are established, which guarantee the numerical stability of the BR-algorithm. The obtained result is applied to the study of the numerical stability of computing approximants of the continued fraction expansion of the ratio of Horn's confluent function $\mathrm{H}_7$ with complex parameters. In addition, the analysis of the relative errors arising from the computation of approximants using the backward recurrence algorithm, the forward recurrence algorithm, and Lenz's algorithm is given. The method for studying the numerical stability of the BR-algorithm proposed in the paper can be used to study the numerical stability of the branched continued fraction expansions and numerical branched continued fractions with elements in angular and parabolic domains.http://matstud.org.ua/ojs/index.php/matstud/article/view/558continued fractionnumerical approximationroundoff error |
| spellingShingle | V. Hladun V. Кravtsiv M. Dmytryshyn R. Rusyn On numerical stability of continued fractions Математичні Студії continued fraction numerical approximation roundoff error |
| title | On numerical stability of continued fractions |
| title_full | On numerical stability of continued fractions |
| title_fullStr | On numerical stability of continued fractions |
| title_full_unstemmed | On numerical stability of continued fractions |
| title_short | On numerical stability of continued fractions |
| title_sort | on numerical stability of continued fractions |
| topic | continued fraction numerical approximation roundoff error |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/558 |
| work_keys_str_mv | AT vhladun onnumericalstabilityofcontinuedfractions AT vkravtsiv onnumericalstabilityofcontinuedfractions AT mdmytryshyn onnumericalstabilityofcontinuedfractions AT rrusyn onnumericalstabilityofcontinuedfractions |