On the accurate computation of the Newton form of the Lagrange interpolant
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866916357748031488 |
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| author | Khiar, Yasmina Mainar, Esmeralda Royo-Amondarain, Eduardo Rubio, Beatriz |
| author_facet | Khiar, Yasmina Mainar, Esmeralda Royo-Amondarain, Eduardo Rubio, Beatriz |
| contents | In recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great precision, even for large ill-conditioned matrices. In this framework, the present work provides the factorization of the collocation matrices of Newton bases -- of relevance when considering the Lagrange interpolation problem -- together with an algorithm that allows to numerically compute it to high relative accuracy. This further allows to determine the coefficients of the interpolating polynomial and to compute the singular values and the inverse of the collocation matrix. Conditions that guarantee high relative accuracy for these methods and, in the former case, for the classical recursion formula of divided differences, are determined. Numerical errors due to imprecise computer arithmetic or perturbed input data in the computation of the factorization are analyzed. Finally, numerical experiments illustrate the accuracy and effectiveness of the proposed methods with several algebraic problems, in stark contrast with traditional approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14483 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the accurate computation of the Newton form of the Lagrange interpolant Khiar, Yasmina Mainar, Esmeralda Royo-Amondarain, Eduardo Rubio, Beatriz Numerical Analysis 65F15, 65F05, 15A23, 15A18, 15A06 In recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great precision, even for large ill-conditioned matrices. In this framework, the present work provides the factorization of the collocation matrices of Newton bases -- of relevance when considering the Lagrange interpolation problem -- together with an algorithm that allows to numerically compute it to high relative accuracy. This further allows to determine the coefficients of the interpolating polynomial and to compute the singular values and the inverse of the collocation matrix. Conditions that guarantee high relative accuracy for these methods and, in the former case, for the classical recursion formula of divided differences, are determined. Numerical errors due to imprecise computer arithmetic or perturbed input data in the computation of the factorization are analyzed. Finally, numerical experiments illustrate the accuracy and effectiveness of the proposed methods with several algebraic problems, in stark contrast with traditional approaches. |
| title | On the accurate computation of the Newton form of the Lagrange interpolant |
| topic | Numerical Analysis 65F15, 65F05, 15A23, 15A18, 15A06 |
| url | https://arxiv.org/abs/2312.14483 |