Error bounds for Gauss-Turán quadrature formulae of analytic functions

Milovanović, Gradimir; Spalević, Miodrag

doi:10.1090/S0025-5718-03-01544-8

Error bounds for Gauss-Turán quadrature formulae of analytic functions
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by Gradimir V. Milovanović and Miodrag M. Spalević PDF

Math. Comp. 72 (2003), 1855-1872 Request permission

Abstract:

We study the kernels of the remainder term $R_{n,s}(f)$ of Gauss-Turán quadrature formulas \[ \int _{-1}^1f(t)w(t) dt=\sum _{\nu =1}^n \sum _{i=0}^{2s}A_{i,\nu }f^{(i)}(\tau _\nu ) +R_{n,s}(f)\qquad (n\in \mathbb {N}; s\in \mathbb {N}_0)\] for classes of analytic functions on elliptical contours with foci at $\pm 1$, when the weight $w$ is one of the special Jacobi weights $w^{(\alpha ,\beta )}(t)=(1-t)^\alpha (1+t)^\beta$ $(\alpha =\beta =-1/2$; $\alpha =\beta =1/2+s$; $\alpha =-1/2$, $\beta =1/2+s$; $\alpha =1/2+s$, $\beta =-1/2)$. We investigate the location on the contour where the modulus of the kernel attains its maximum value. Some numerical examples are included.

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