Error bounds for Gaussian quadrature rules using linear kernels

References

• I. Area, D.K. Dimitrov, E. Godoy, and A. Ronveaux. Zeros of Gegenbauer and Hermite polynomials and connection coefficients, Math. Comp. 73(248) (2004), pp. 1937–1951. doi: 10.1090/S0025-5718-04-01642-4
   Web of Science ®Google Scholar
• I. Area, J. Losada, and A. Manintchap. On some fractional Pearson equations, Fract. Calc. Appl. Anal. 18(5) (2015). doi:10.1515/fca-2015-0067
   Web of Science ®Google Scholar
• P.J. Davis and P. Rabinowitz, Methods of Numerical Integration, Computer Science and Applied Mathematics, 2nd ed., Academic Press Inc., Orlando, FL, 1984.
   Google Scholar
• B. de la Calle Ysern, Error bounds for rational quadrature formulae of analytic functions, Numer. Math. 101(2) (2005), pp. 251–271. doi: 10.1007/s00211-005-0575-8
   Web of Science ®Google Scholar
• J. Engelbrecht, I. Fedotov, T. Fedotova, and A. Harding, Error bounds for quadrature methods involving lower order derivatives, Int. J. Math. Ed. Sci. Technol. 34(6) (2003), pp. 831–846. doi: 10.1080/00207390310001595429
   Google Scholar
• I. Fedotov and S. S. Dragomir, On convergence of quadrature methods for the Lipschitz-continuous functions, Ital. J. Pure Appl. Math.(13) (2003), pp. 91–103.
   Google Scholar
• K.-J. Förster and K. Petras, Error estimates in Gaussian quadrature for functions of bounded variation, SIAM J. Numer. Anal. 28(3) (1991), pp. 880–889. doi: 10.1137/0728047
   Web of Science ®Google Scholar
• W. Gautschi, E. Tychopoulos, and R. S. Varga, A note on the contour integral representation of the remainder term for a Gauss–Chebyshev quadrature rule, SIAM J. Numer. Anal. 27(1) (1990), pp. 219–224. doi: 10.1137/0727015
   Web of Science ®Google Scholar
• W. Gautschi and R.S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 20(6) (1983), pp. 1170–1186. doi: 10.1137/0720087
   Web of Science ®Google Scholar
• A. Glaser, X. Liu, and V. Rokhlin, A fast algorithm for the calculation of the roots of special functions, SIAM J. Sci. Comput. 29(4) (2007), pp. 1420–1438. doi: 10.1137/06067016X
   Web of Science ®Google Scholar
• G.H. Golub and J.H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969), pp. 221–230. addendum, ibid., 23 (106, loose microfiche suppl): A1–A10, 1969. doi: 10.1090/S0025-5718-69-99647-1
   Web of Science ®Google Scholar
• G. Grüss, Über das Maximum des absoluten Betrages von 1b−a∫abf(x)g(x)dx−1(b−a)2∫abf(x)dx∫abg(x)dx, Math. Z. 39(1) (1935), pp. 215–226. doi: 10.1007/BF01201355
   Google Scholar
• D. B. Hunter, Some error expansions for Gaussian quadrature, BIT 35(1) (1995), pp. 64–82. doi: 10.1007/BF01732979
   Web of Science ®Google Scholar
• N. Hale and A. Townsend, Fast and accurate computation of Gauss–Legendre and Gauss–Jacobi quadrature nodes and weights, SIAM J. Sci. Comput. 35(2) (2013), pp. A652–A674. doi: 10.1137/120889873
   Web of Science ®Google Scholar
• D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice-Hall, Inc., Upper Saddle River, NJ, 1989.
   Google Scholar
• W. Koepf and M. Masjed-Jamei, A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it, Integral Transforms Spec. Funct. 17(8) (2006), pp. 559–576. doi: 10.1080/10652460600725234
   Web of Science ®Google Scholar
• J. Ma, V. Rokhlin, and S. Wandzura, Generalized Gaussian quadrature rules for systems of arbitrary functions, SIAM J. Numer. Anal. 33(3) (1996), pp. 971–996. doi: 10.1137/0733048
   Web of Science ®Google Scholar
• M. Masjed-Jamei, A basic class of symmetric orthogonal polynomials using the extended Sturm–Liouville theorem for symmetric functions, J. Math. Anal. Appl. 325(2) (2007), pp. 753–775. doi: 10.1016/j.jmaa.2006.02.007
   Web of Science ®Google Scholar
• G.V. Milovanović, M.M. Spalević, and M.S. Pranić, Bounds of the error of Gauss-Turán-type quadratures. II, Appl. Numer. Math. 60(1–2) (2010), pp. 1–9. doi: 10.1016/j.apnum.2009.08.002
   Web of Science ®Google Scholar
• S.E. Notaris, The error norm of Gaussian quadrature formulae for weight functions of Bernstein–Szegö type, Numer. Math. 57(3) (1990), pp. 271–283. doi: 10.1007/BF01386411
   Web of Science ®Google Scholar
• K. Petras, Error bounds for Gaussian and related quadrature and applications to R-convex functions, SIAM J. Numer. Anal. 29(2) (1992), pp. 578–585. doi: 10.1137/0729037
   Web of Science ®Google Scholar
• R. Scherer and T. Schira, Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems, Numer. Math. 84(3) (2000), pp. 497–518. doi: 10.1007/s002110050007
   Web of Science ®Google Scholar
• F. Stenger, Bounds on the error of Gauss-type quadratures, Numer. Math. 8 (1966), pp. 150–160. doi: 10.1007/BF02163184
   Web of Science ®Google Scholar
• J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd ed., Texts in Applied Mathematics, Vol. 12, Springer-Verlag, New York, 2002.
   Google Scholar
• B. von Sydow, Error estimates for Gaussian quadrature formulae, Numer. Math. 29(1) (1977/78), pp. 59–64. doi: 10.1007/BF01389313
   Web of Science ®Google Scholar