On the conditioning of polynomial histopolation
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910071641866240 |
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| author | Bruno, Ludovico Bruni Serra-Capizzano, Stefano |
| author_facet | Bruno, Ludovico Bruni Serra-Capizzano, Stefano |
| contents | Histopolation is the approximation procedure that associates a degree $ d-1 $ polynomial $ p_{d-1} \in \mathscr{P}_{d-1} (I) $ with a locally integrable function $ f $ imposing that the integral (or, equivalently, the average) of $p$ coincides with that of $f$ on a collection of $ d $ distinct segments $s_i$. In this work we discuss unisolvence and conditioning of the associated matrices, in an asymptotic linear algebra perspective, i.e., when the matrix-size $d$ tends to infinity. While the unisolvence is a rather sparse topic, the conditioning in the unisolvent setting has a uniform behavior: as for the case of standard Vandermonde matrix-sequences with real nodes, the conditioning is inherently exponential as a function of $d$ when the monomial basis is chosen. In contrast, for an appropriate selection of supports, the Chebyshev polynomials of second kind exhibit a bounded conditioning. A linear behavior is also observed in the Frobenius norm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_15395 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the conditioning of polynomial histopolation Bruno, Ludovico Bruni Serra-Capizzano, Stefano Numerical Analysis Histopolation is the approximation procedure that associates a degree $ d-1 $ polynomial $ p_{d-1} \in \mathscr{P}_{d-1} (I) $ with a locally integrable function $ f $ imposing that the integral (or, equivalently, the average) of $p$ coincides with that of $f$ on a collection of $ d $ distinct segments $s_i$. In this work we discuss unisolvence and conditioning of the associated matrices, in an asymptotic linear algebra perspective, i.e., when the matrix-size $d$ tends to infinity. While the unisolvence is a rather sparse topic, the conditioning in the unisolvent setting has a uniform behavior: as for the case of standard Vandermonde matrix-sequences with real nodes, the conditioning is inherently exponential as a function of $d$ when the monomial basis is chosen. In contrast, for an appropriate selection of supports, the Chebyshev polynomials of second kind exhibit a bounded conditioning. A linear behavior is also observed in the Frobenius norm. |
| title | On the conditioning of polynomial histopolation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2511.15395 |