Gauss-type quadrature rules for variable-sign weight functions
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2023
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When the Gauss quadrature formula $G_n$ is applied, it is often assumed that the weight function (or the measure) is non-negative on the integration interval $[a,b]$. In the present paper, we introduce a Gauss-type quadrature formula $Q_n$ for weight functions that change the sign in the interior of $[a,b]$. Construction of $Q_n$ is based on the idea to transform the given integral into a sum of one integral that does not cause a quadrature error and the other integral with a property that the points from the interior of $[a,b]$ at which the weight function changes sign are the zeros of its integrand. It proves that all nodes of $Q_n$ are pairwise distinct and contained in the interior of $[a,b]$. Moreover, $G_n$ (with a non-negative weight function) turns out to be a special case of $Q_n$. Obtained results on the remainder term of $Q_n$ suggest that the application of $Q_n$ makes sense both when the points from the interior of $[a,b]$ at which the weight function changes sign are know...n exactly, as well as when those points are known approximately. The accuracy of $Q_n$ is confirmed by numerical examples.
Кључне речи:
Gauss quadrature formula / Variable-sign weight function / Modifier function / Vandermonde matrix / Maximum normИзвор:
Journal of Computational and Applied Mathematics, 2023Издавач:
- Elsevier
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Mašinski fakultetTY - JOUR AU - Tomanović, Jelena PY - 2023 UR - https://machinery.mas.bg.ac.rs/handle/123456789/6963 AB - When the Gauss quadrature formula $G_n$ is applied, it is often assumed that the weight function (or the measure) is non-negative on the integration interval $[a,b]$. In the present paper, we introduce a Gauss-type quadrature formula $Q_n$ for weight functions that change the sign in the interior of $[a,b]$. Construction of $Q_n$ is based on the idea to transform the given integral into a sum of one integral that does not cause a quadrature error and the other integral with a property that the points from the interior of $[a,b]$ at which the weight function changes sign are the zeros of its integrand. It proves that all nodes of $Q_n$ are pairwise distinct and contained in the interior of $[a,b]$. Moreover, $G_n$ (with a non-negative weight function) turns out to be a special case of $Q_n$. Obtained results on the remainder term of $Q_n$ suggest that the application of $Q_n$ makes sense both when the points from the interior of $[a,b]$ at which the weight function changes sign are known exactly, as well as when those points are known approximately. The accuracy of $Q_n$ is confirmed by numerical examples. PB - Elsevier T2 - Journal of Computational and Applied Mathematics T1 - Gauss-type quadrature rules for variable-sign weight functions DO - 10.1016/j.cam.2023.115477 ER -
@article{ author = "Tomanović, Jelena", year = "2023", abstract = "When the Gauss quadrature formula $G_n$ is applied, it is often assumed that the weight function (or the measure) is non-negative on the integration interval $[a,b]$. In the present paper, we introduce a Gauss-type quadrature formula $Q_n$ for weight functions that change the sign in the interior of $[a,b]$. Construction of $Q_n$ is based on the idea to transform the given integral into a sum of one integral that does not cause a quadrature error and the other integral with a property that the points from the interior of $[a,b]$ at which the weight function changes sign are the zeros of its integrand. It proves that all nodes of $Q_n$ are pairwise distinct and contained in the interior of $[a,b]$. Moreover, $G_n$ (with a non-negative weight function) turns out to be a special case of $Q_n$. Obtained results on the remainder term of $Q_n$ suggest that the application of $Q_n$ makes sense both when the points from the interior of $[a,b]$ at which the weight function changes sign are known exactly, as well as when those points are known approximately. The accuracy of $Q_n$ is confirmed by numerical examples.", publisher = "Elsevier", journal = "Journal of Computational and Applied Mathematics", title = "Gauss-type quadrature rules for variable-sign weight functions", doi = "10.1016/j.cam.2023.115477" }
Tomanović, J.. (2023). Gauss-type quadrature rules for variable-sign weight functions. in Journal of Computational and Applied Mathematics Elsevier.. https://doi.org/10.1016/j.cam.2023.115477
Tomanović J. Gauss-type quadrature rules for variable-sign weight functions. in Journal of Computational and Applied Mathematics. 2023;. doi:10.1016/j.cam.2023.115477 .
Tomanović, Jelena, "Gauss-type quadrature rules for variable-sign weight functions" in Journal of Computational and Applied Mathematics (2023), https://doi.org/10.1016/j.cam.2023.115477 . .