What does this research mean for the field?

Numerically stable O(n) algorithms can efficiently and robustly evaluate all derivatives for barycentric representations and higher-order derivatives for Thiele continued fraction representations of rational functions. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish numerically stable and efficient algorithms for calculating derivatives of rational functions expressed in barycentric and Thiele continued fraction forms.

August 30, 2026Research in the Mathematical SciencesOpen Access

Stable evaluation of derivatives for barycentric and continued fraction representations of rational functions

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Authors

TDTobin A. Driscoll YZYuxing Zhou

Discussion

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Overview

Algorithmic analysis demonstrates stable linear-time derivative evaluation for barycentric and continued fraction representations, indicating robust efficiency for computational approximation.

Key Points

To establish numerically stable and efficient algorithms for calculating derivatives of rational functions expressed in barycentric and Thiele continued fraction forms.
Formulated an O(n) algorithm to compute all orders of derivatives in barycentric representation.
Extended an existing O(n) framework for first-order Thiele continued fraction derivatives to higher-order derivative calculations.
Validated the numerical stability, robustness, and execution efficiency across various experimental test cases.
Provided the first numerically stable methods for derivative evaluation in barycentric rational representations.
Demonstrated that both the barycentric and extended Thiele continued fraction methods maintain linear computational complexity across higher-order evaluations.
Confirmed through numerical experiments that the algorithms prevent error amplification and perform robustly.

Cite This Study

Driscoll et al. (2026) studied this question.

synapsesocial.com/papers/6a93efc06c1a8fb52e79bd62 https://doi.org/10.1007/s40687-026-00655-6

View Full Paper

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August 30, 2026Research in the Mathematical SciencesOpen Access

Stable evaluation of derivatives for barycentric and continued fraction representations of rational functions

View Full Paper

Ask AI

Bookmark

Authors

TDTobin A. Driscoll YZYuxing Zhou

Discussion

Member takes

Overview

Algorithmic analysis demonstrates stable linear-time derivative evaluation for barycentric and continued fraction representations, indicating robust efficiency for computational approximation.

Key Points

To establish numerically stable and efficient algorithms for calculating derivatives of rational functions expressed in barycentric and Thiele continued fraction forms.
Formulated an O(n) algorithm to compute all orders of derivatives in barycentric representation.
Extended an existing O(n) framework for first-order Thiele continued fraction derivatives to higher-order derivative calculations.
Validated the numerical stability, robustness, and execution efficiency across various experimental test cases.
Provided the first numerically stable methods for derivative evaluation in barycentric rational representations.
Demonstrated that both the barycentric and extended Thiele continued fraction methods maintain linear computational complexity across higher-order evaluations.
Confirmed through numerical experiments that the algorithms prevent error amplification and perform robustly.

Cite This Study

Driscoll et al. (2026) studied this question.

synapsesocial.com/papers/6a93efc06c1a8fb52e79bd62 https://doi.org/10.1007/s40687-026-00655-6

View Full Paper

Ask AI

Bookmark