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Error bounds for rational quadrature formulae of analytic functions. (English) Zbl 1083.65032
It is well known that rational quadrature rules are exact in a vectorial space of rational functions. The paper is concerned with the errors commited when one uses such rules to approximate the integral of a function which is analytic on some neighborhood of the set of integration. A strict bound for the error is obtained (Theorem 1). Numerical examples are also presented.

MSC:
65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
Software:
BRENT; GQRAT
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References:
[1] Achieser, N.I.: Theory of Approximation. Dover, New York 1992 · Zbl 0072.28403
[2] Ambroladze, A., Wallin, H.: Extremal polynomials with preassigned zeros and rational approximants. Constr. Approx. 14, 209–229 (1998) · Zbl 0891.42013
[3] van Assche, W., Vanherwegen, I.: Quadrature formulas based on rational interpolation. Math. Comp. 61, 765–783 (1993) · Zbl 0791.65011
[4] Bello Hernández, M., de la Calle Ysern, B., López Lagomasino, G.: Generalized Stieltjes polynomials and rational Gauss-Kronrod quadrature. Constr. Approx. 20, 249–265 (2004) · Zbl 1056.42017
[5] Brent, R.P.: Algorithms for Minimization Without Derivatives. Dover, New York 2002 · Zbl 1009.90133
[6] Bultheel, A., Díaz-Mendoza, C., González-Vera, P., Orive, R.: On the convergence of certain Gauss-type quadrature formulas for unbounded intervals. Math. Comp. 69, 721–747 (2000) · Zbl 0941.41015
[7] Cala Rodríguez, F., López Lagomasino, G.: Multipoint rational approximants with preassigned poles. J. Math. Anal. Appl. 256, 142–161 (2001) · Zbl 1160.41305
[8] Calvetti, D., Golub, G.H., Gragg, W.B., Reichel, L.: Computation of Gauss-Kronrod quadrature rules. Math. Comp. 69, 1035–1052 (2000) · Zbl 0947.65022
[9] Conway, J.B.: Functions of One Complex Variable I. Springer-Verlag, New York 1978 · Zbl 0424.68009
[10] Daruis, L., González-Vera, P.: Interpolatory quadrature formulas on the unit circle for Chebyshev weight functions. Numer. Math. 90, 641–664 (2002) · Zbl 0993.41017
[11] Ehrich, S.: Gauss-Kronrod quadrature error estimates for analytic functions. Zeitschr. f. Angew. Mathematik und Mechanik. 74, T691–T693 (1995)
[12] Gautschi, W.: Gauss-type quadrature rules for rational functions. In: Numerical Integration IV, H. Brass, G. Hämmerlin, (eds.), 112, International Series of Numerical Mathematics Birkhäuser, Basel 1993, pp. 111–130 · Zbl 0802.65019
[13] Gautschi, W.: Algorithm 793: GQRAT – Gauss quadrature for rational functions. ACM Trans. Math. Software 25, 213–239 (1999) · Zbl 0961.65019
[14] Gautschi, W., Gori, L., Lo Cascio, M.L.: Quadrature rules for rational functions. Numer. Math. 86, 617–633 (2000) · Zbl 0968.65014
[15] González-Vera, P., Jiménez Paiz, M., Orive, R., López Lagomasino, G.: On the convergence of quadrature formulas connected with multipoint Padé-type approximation. J. Math. Anal. Appl. 202, 747–775 (1996) · Zbl 0856.41027
[16] Laurie, D.P.: Calculation of Gauss-Kronrod quadrature rules. Math. Comp. 66, 1133–1145 (1997) · Zbl 0870.65018
[17] López Lagomasino, G., Illán González, J.: A note on generalized quadrature formulas of Gauss-Jacobi type. In: Constructive Theory of Functions ’84, V. Sendov, V. Popov, eds., Publ. House Bulgarian Acad. Sci., Sofia, 1984, pp. 513–518
[18] Min, G.: Lagrange interpolation and quadrature formula in rational systems. J. Approx. Theory 95, 123–145 (1998) · Zbl 0912.41003
[19] Notaris, S.E.: Error bounds for Gauss-Kronrod quadrature of analytic functions. Numer. Math. 64, 371–380 (1993) · Zbl 0793.41024
[20] Rivlin, T.J.: The Chebyshev Polynomials. John Wiley & Sons, New York 1974 · Zbl 0299.41015
[21] Rovba, E.A.: Quadrature formulae of interpolatory rational type. Dokl. Nats. Akad. Nauk Belarusi 40, (1996) 42–46 (in Russian) · Zbl 1057.41503
[22] Rudin, W.: Real and Complex Analysis. Mc Graw Hill, New York 1966 · Zbl 0142.01701
[23] von Sydow, B.: Error estimates for Gaussian quadrature formulae. Numer. Math 29, 59–64 (1977) · Zbl 0351.65005
[24] Walsh, J.: Interpolation and Approximation by Rational Functions in the Complex Domain. Coll. Publ. XX, Amer. Math. Soc. Providence, Rhode Island 1969
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