On generalized averaged gaussian formulas. Ii
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2017
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Recently, by following the results on characterization of positive quadrature formulae by Peherstorfer, we proposed a new (2l + 1)-point quadrature rule (G) over cap (2l + 1), referred to as a generalized averaged Gaussian quadrature rule. This rule has 2l + 1 nodes and the nodes of the corresponding Gauss rule G(l) with l nodes form a subset. This is similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l + 1 associated with G(l). An attractive feature of (G) over cap (2l + 1) is that it exists also when H2l + 1 does not. The numerical construction, on the basis of recently proposed effective numerical procedures, of (G) over cap (2l + 1) is simpler than the construction of H2l + 1. A disadvantage might be that the algebraic degree of precision of (G) over cap (2l + 1) is 2l + 2, while the one of H2l + 1 is 3l + 1. Consider a (nonnegative) measure ds with support in the bounded interval [a, b] such that the respective orthogonal polynomials, above a specific index r, sa...tisfy a three-term recurrence relation with constant coefficients. For l >= 2r -1, we show that (G) over cap (2l + 1) has algebraic degree of precision at least 3l + 1, and therefore it is in fact H2l + 1 associated with G(l). We derive some interesting equalities for the corresponding orthogonal polynomials.
Кључне речи:
Gauss-Kronrod quadrature / Gauss quadrature / averaged Gauss rulesИзвор:
Mathematics of Computation, 2017, 86, 306, 1877-1885Издавач:
- Amer Mathematical Soc, Providence
Финансирање / пројекти:
- Ministry of Science and Technological Development
DOI: 10.1090/mcom/3225
ISSN: 0025-5718
WoS: 000398823400015
Scopus: 2-s2.0-85016211070
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Институција/група
Mašinski fakultetTY - JOUR AU - Spalević, Miodrag PY - 2017 UR - https://machinery.mas.bg.ac.rs/handle/123456789/2566 AB - Recently, by following the results on characterization of positive quadrature formulae by Peherstorfer, we proposed a new (2l + 1)-point quadrature rule (G) over cap (2l + 1), referred to as a generalized averaged Gaussian quadrature rule. This rule has 2l + 1 nodes and the nodes of the corresponding Gauss rule G(l) with l nodes form a subset. This is similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l + 1 associated with G(l). An attractive feature of (G) over cap (2l + 1) is that it exists also when H2l + 1 does not. The numerical construction, on the basis of recently proposed effective numerical procedures, of (G) over cap (2l + 1) is simpler than the construction of H2l + 1. A disadvantage might be that the algebraic degree of precision of (G) over cap (2l + 1) is 2l + 2, while the one of H2l + 1 is 3l + 1. Consider a (nonnegative) measure ds with support in the bounded interval [a, b] such that the respective orthogonal polynomials, above a specific index r, satisfy a three-term recurrence relation with constant coefficients. For l >= 2r -1, we show that (G) over cap (2l + 1) has algebraic degree of precision at least 3l + 1, and therefore it is in fact H2l + 1 associated with G(l). We derive some interesting equalities for the corresponding orthogonal polynomials. PB - Amer Mathematical Soc, Providence T2 - Mathematics of Computation T1 - On generalized averaged gaussian formulas. Ii EP - 1885 IS - 306 SP - 1877 VL - 86 DO - 10.1090/mcom/3225 ER -
@article{ author = "Spalević, Miodrag", year = "2017", abstract = "Recently, by following the results on characterization of positive quadrature formulae by Peherstorfer, we proposed a new (2l + 1)-point quadrature rule (G) over cap (2l + 1), referred to as a generalized averaged Gaussian quadrature rule. This rule has 2l + 1 nodes and the nodes of the corresponding Gauss rule G(l) with l nodes form a subset. This is similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l + 1 associated with G(l). An attractive feature of (G) over cap (2l + 1) is that it exists also when H2l + 1 does not. The numerical construction, on the basis of recently proposed effective numerical procedures, of (G) over cap (2l + 1) is simpler than the construction of H2l + 1. A disadvantage might be that the algebraic degree of precision of (G) over cap (2l + 1) is 2l + 2, while the one of H2l + 1 is 3l + 1. Consider a (nonnegative) measure ds with support in the bounded interval [a, b] such that the respective orthogonal polynomials, above a specific index r, satisfy a three-term recurrence relation with constant coefficients. For l >= 2r -1, we show that (G) over cap (2l + 1) has algebraic degree of precision at least 3l + 1, and therefore it is in fact H2l + 1 associated with G(l). We derive some interesting equalities for the corresponding orthogonal polynomials.", publisher = "Amer Mathematical Soc, Providence", journal = "Mathematics of Computation", title = "On generalized averaged gaussian formulas. Ii", pages = "1885-1877", number = "306", volume = "86", doi = "10.1090/mcom/3225" }
Spalević, M.. (2017). On generalized averaged gaussian formulas. Ii. in Mathematics of Computation Amer Mathematical Soc, Providence., 86(306), 1877-1885. https://doi.org/10.1090/mcom/3225
Spalević M. On generalized averaged gaussian formulas. Ii. in Mathematics of Computation. 2017;86(306):1877-1885. doi:10.1090/mcom/3225 .
Spalević, Miodrag, "On generalized averaged gaussian formulas. Ii" in Mathematics of Computation, 86, no. 306 (2017):1877-1885, https://doi.org/10.1090/mcom/3225 . .