zbMATH — the first resource for mathematics

Error bounds of certain Gaussian quadrature formulae. (English) Zbl 1205.41034
\[ \int_{-1}^{1} f(t)w(t)\,dt=\sum_{\nu=1}^{n}\lambda_{\nu}f(\tau_{\nu})+R_{n}(f) \tag{1} \]
the Gaussian quadrature formula for analytic functions on elliptical contours with foci \(\mp 1\) and the sum of semi-axes \(\rho>1\),
\[ \varepsilon_{\rho}=\left\{ z\in {\mathbb C}\mid z=\tfrac{1}{2}(\xi+\xi^{-1}),\;0\leq \theta\leq 2\pi\right\}, \quad \xi=\rho e^{i\theta}, \tag{2} \]
where \(w\) is a nonnegative and integrable function on the interval \((-1,1)\), which is exact for all algebraic polynomials of degree at most \(2n-1\). The nodes \(\tau_{\nu}\) in (1) are zeros of the orthogonal polynomials \(\pi_{n}\) with respect to the weight function \(w\).
Here, \(w\) represents the class of symmetric weight functions of Bernstein-Szegő type
\[ w(t)\equiv w_{\gamma}(t)=\frac{\sqrt{1-t^2}}{1-\frac{4\gamma}{(1+\gamma)^2}t^2},\quad t\in(-1,1),\;\gamma\in(-1,0). \]
Let \(\Gamma\) be a simple closed curve in the complex plane surrounding the interval \([-1,1]\) and \({\mathcal D}= \operatorname{int}\Gamma\) its interior. If the integrand \(f\) is analytic in \({\mathcal D}\) and continuous on \(\overline{\mathcal D}\), then the remainder term \( R_{n}(f)\) in (1) admits the contour integral representation
\[ R_{n}(f)=\frac{1}{2\pi i}\oint_{\Gamma} K_{n}(z)f(z)\,dz. \]
The kernel is given by
\[ K_{n}(z)\equiv K_{n}(z,w)=\frac{\rho(z)}{\pi_{n}(z)},\quad z \in[-1,1], \]
\[ \rho_{n}(z)\equiv\rho_{n,w}(z)=\int_{-1}^{1}\frac{\pi_{n}(t)}{z-t}w(t)\,dt. \]
The authors study the kernel \(K_{n}(z)\) in the remainder terms \(R_{n}(f)\) of the Gaussian quadrature formula. The location on the elliptic contour, where the modulus of the kernel attains its maximum value, is investigated.

41A55 Approximate quadratures
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
PDF BibTeX Cite
Full Text: DOI
[1] Gautschi, W.; Notaris, S.E., Gauss – kronrod quadrature formulae for weight function of bernstein – szegö type, J. comput. appl. math., J. comput. appl. math., 27, 429-224, (1989) · Zbl 0691.41028
[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] Hunter, D.B., Some error expansions for Gaussian quadrature, Bit, 35, 64-82, (1995) · Zbl 0824.41032
[4] Gautschi, W.; Tychopoulos, E.; Varga, R.S., A note on the contour integral representation of the remainder term for a gauss – chebyshev quadrature rule, SIAM J. numer. anal., 27, 219-224, (1990) · Zbl 0685.41019
[5] Schira, T., The remainder term for analytic functions of symmetric Gaussian quadratures, Math. comput., 66, 297-310, (1997) · Zbl 0854.41025
[6] Scherer, R.; Schira, T., Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems, Numer. math., 84, 497-518, (2000) · Zbl 0943.41014
[7] Stenger, F., Bounds on the error of Gauss-type quadratures, Numer. math., 8, 150-160, (1966) · Zbl 0149.12002
[8] Von Sydow, B., Error estimates for Gaussian quadrature formulae, Numer. math., 29, 59-64, (1977) · Zbl 0351.65005
[9] Notaris, S.E., The error norm of Gaussian quadrature formulae for weight functions of bernstein – szegö type, Numer. math., 57, 271-283, (1990) · Zbl 0676.41034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.