Bounds of the error of Gauss-Turán-type quadratures. II. (English) Zbl 1200.41034
This paper is the continuation of the paper of G. V. Milovanović and M. M. Spalević [J. Comput. Appl. Math. 178, No. 1–2, 333–346 (2005; Zbl 1071.41028)] and is concerned with the remainder term in Gauss-Turán quadrature rule
\[ \int^1_{-1} w(t) f(t)\,dt= \sum^n_{v=1} \sum^{2s}_{i=0} \lambda_{i,v}f^{(i)}(\tau_v) + R_{n,s}(f) \tag{1} \]
with the weight functions
\[ w_{n,\mu}(t) = (U_{n-1}(t)/n)^{2\mu+1}(1-t^2)^\mu, \]
where \(\mu=l-1/2,\) \(l=1,2,\dots\) and \(U_n(\cos \Theta)=\frac{\sin(n+1)\Theta}{\sin \Theta} -\) is the Chebyshev polynomial of the second kind. An explicit representation of the remainder term in (1) on elliptic contours is obtained. Error bounds are found.
\[ \int^1_{-1} w(t) f(t)\,dt= \sum^n_{v=1} \sum^{2s}_{i=0} \lambda_{i,v}f^{(i)}(\tau_v) + R_{n,s}(f) \tag{1} \]
with the weight functions
\[ w_{n,\mu}(t) = (U_{n-1}(t)/n)^{2\mu+1}(1-t^2)^\mu, \]
where \(\mu=l-1/2,\) \(l=1,2,\dots\) and \(U_n(\cos \Theta)=\frac{\sin(n+1)\Theta}{\sin \Theta} -\) is the Chebyshev polynomial of the second kind. An explicit representation of the remainder term in (1) on elliptic contours is obtained. Error bounds are found.
Reviewer: Alla Boikova (Penza)
MSC:
| 41A55 | Approximate quadratures |
Keywords:
Gauss-Turán quadrature formula; error bound; remainder term for analytic functions; contour integral representationCitations:
Zbl 1071.41028References:
| [1] | Bernstein, S., Sur les polynomes orthogonaux relatifs à un segment fini, J. Math. Pures Appl., 127-177 (1930) · JFM 56.0947.04 |
| [2] | Gautschi, W.; Varga, R. S., Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal., 20, 1170-1186 (1983) · Zbl 0545.41040 |
| [3] | Gautschi, W.; Tychopoulos, E.; Varga, R. S., A note on the contour integral representation of the remainder term for Gauss-Chebyshev quadrature rule, SIAM J. Numer. Anal., 27, 219-224 (1990) · Zbl 0685.41019 |
| [4] | Gori, L.; Micchelli, C. A., On weight functions which admit explicit Gauss-Turán quadrature formulas, Math. Comp., 65, 1567-1581 (1996) · Zbl 0853.65027 |
| [5] | Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (2000), Academic Press: Academic Press San Diego, (A. Jeffrey, D. Zwillinger (Eds.)) · Zbl 0981.65001 |
| [6] | Hunter, D. B., Some error expansions for Gaussian quadrature, BIT, 35, 64-82 (1995) · Zbl 0824.41032 |
| [7] | Kroó, A.; Peherstorher, F., Asymptotic representation of \(L_p\)-minimal polynomials, \(1 < p < \infty \), Constr. Approx., 25, 29-39 (2007) · Zbl 1103.41006 |
| [8] | Milovanović, G. V.; Spalević, M. M., Error bounds for Gauss-Turán quadrature formulae of analytic functions, Math. Comp., 72, 1855-1872 (2003) · Zbl 1030.41018 |
| [9] | Milovanović, G. V.; Spalević, M. M., An error expansion for Gauss-Turán quadratures and \(L^1\)-estimates of the remainder term, BIT, 45, 117-136 (2005) · Zbl 1082.41022 |
| [10] | Milovanović, G. V.; Spalević, M. M., Bounds of the error of Gauss-Turán-type quadratures, J. Comput. Appl. Math., 178, 333-346 (2005) · Zbl 1071.41028 |
| [11] | Milovanović, G. V.; Spalević, M. M., A note on the bounds of the error of Gauss-Turán-type quadratures, J. Comput. Appl. Math., 200, 276-282 (2007) · Zbl 1105.41025 |
| [12] | Milovanović, G. V.; Spalević, M. M.; Cvetković, A. S., Calculation of Gaussian type quadratures with multiple nodes, Math. Comput. Modell., 39, 325-347 (2004) · Zbl 1049.65019 |
| [13] | Milovanović, G. V.; Spalević, M. M.; Pranić, M. S., Maximum of the modulus of kernels in Gauss-Turán quadratures, Math. Comp., 77, 985-994 (2008) · Zbl 1149.41011 |
| [14] | Ossicini, A.; Rosati, F., Funzioni caratteristiche nelle formule di quadratura gaussiane con nodi multipli, Bull. Un. Mat. Ital., 11, 224-237 (1975) · Zbl 0311.41024 |
| [15] | Peherstorfer, F., Gauss-Turán quadrature formulas: Asymptotics of weights, SIAM J. Numer. Anal., 47, 2638-2659 (2009) · Zbl 1204.65028 |
| [16] | Shi, Y. G., Christoffel type functions for \(m\)-orthogonal polynomials, J. Approx. Theory, 137, 57-88 (2005) · Zbl 1097.41026 |
| [17] | Shi, Y. G.; Xu, G., Construction of \(σ\)-orthogonal polynomials and Gaussian quadrature formulas, Adv. Comput. Math., 27, 79-94 (2007) · Zbl 1122.65026 |
| [18] | Yang, S., On a quadrature formula of Gori and Micchelli, J. Comput. Appl. Math., 176, 35-43 (2005) · Zbl 1074.41028 |
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