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Error bounds for approximation in Chebyshev points. (English) Zbl 1201.65040
{\it Lloyd N. Trefethen} [SIAM Rev. 50, No. 1, 67--87 (2008; Zbl 1141.65018)] has compared the convergence behavior of the Gauss quadrature with the Clenshaw-Curtis quadrature [cf. {\it C. W. Clenshaw} and {\it A. R. Curtis}, Numer. Math. 2, 197--205 (1960; Zbl 0093.14006)] and the experiments show that the supposed factor-of-2 advantage of the Gauss quadrature is rarely realized and backed by the corresponding theorems explaining this effect. These results are employed in this paper to consider new error estimates for approximations of $f$ in the Chebyshev points. It is demonstrated that polynomial interpolation in the Chebyshev points of the 1st and 2nd kind should be regarded as equally valuable and fundamental. Error bounds for Gauss, Clenshaw-Curtis and Fejér’s first quadratures are improved by using new error estimates for polynomial interpolation in the Chebyshev points. Numerical results (for highly oscillatory integrals) demonstrate that the improved error bounds are reasonably sharp. These results can be employed for approximate solutions of integral equations appearing in identification and control theory of nonlinear dynamical systems [{\it D. N. Sidorov}, Sib. Zh. Ind. Mat. 3, No. 1, 182--194 (2000; Zbl 0951.93021)].

MSC:
65D32Quadrature and cubature formulas (numerical methods)
65D30Numerical integration
41A50Best approximation, Chebyshev systems
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Full Text: DOI
References:
[1] Adam G.H., Nobile A.: Product integration rules at Clenshaw-Curtis and related points: a robust implementation. IMA J. Numer. Math. 11, 271--296 (1991) · Zbl 0726.65020 · doi:10.1093/imanum/11.2.271
[2] Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C. (1964) · Zbl 0171.38503
[3] Bernstein S.N.: Sur l’ordre de la meilleure approximation des fonctions continues par les polyn?mes de degré donné. Mem. Cl. Sci. Acad. Roy. Belg. 4, 1--103 (1912)
[4] Berrut J.P., Trefethen L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501--517 (2004) · Zbl 1061.65006 · doi:10.1137/S0036144502417715
[5] Boyd J.P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2000)
[6] Clenshaw C.W., Curtis A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197--205 (1960) · Zbl 0093.14006 · doi:10.1007/BF01386223
[7] Dahlquist G., Björck A.: Numerical Methods in Scientific Computing. SIAM, Philadelphia (2007)
[8] Davis P.J., Rabinowitz P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984) · Zbl 0537.65020
[9] Deaño A., Huybrechs D.: Complex Gaussian quadrature of oscillatory integrals. Numer. Math. 112, 197--219 (2009) · Zbl 1162.65011 · doi:10.1007/s00211-008-0209-z
[10] Evans G.A.: Practical Numerical Integration. Wiley, Chichester (1993)
[11] Evans G.A., Webster J.R.: A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112, 55--69 (1999) · Zbl 0947.65148 · doi:10.1016/S0377-0427(99)00213-7
[12] Glaser A., Liu X., Rokhlin V.: A fast algorithm for the calculation of the roots of special functions. SIAM J. Sci. Comput. 29, 1420--1438 (2007) · Zbl 1145.65015 · doi:10.1137/06067016X
[13] Gradshteyn I.S., Ryzhik I.M.: Tables of Integrals, Series, and Products, 6th edn. Academic Press, San Diego (2000) · Zbl 0981.65001
[14] O’Hara H., Smith F.J.: Error estimation in the Clenshaw-Curtis quadrature formula. Comput. J. 11, 213--219 (1968) · Zbl 0165.17901
[15] Hascelik A.I.: On numerical computation of integrals with integrands of the form f(x) sin(w/x r ) on [0,1]. J. Comput. Appl. Math. 223, 399--408 (2009) · Zbl 1155.65023 · doi:10.1016/j.cam.2008.01.018
[16] Iserles A., Nørsett S.P.: Efficient quadrature of highly-oscillatory integrals using derivatives. Proc. R. Soc. A 461, 1383--1399 (2005) · Zbl 1145.65309 · doi:10.1098/rspa.2004.1401
[17] Kussmaul R.: Clenshaw-Curtis quadrature with a weighting function. Computing 9, 159--164 (1972) · Zbl 0241.65019 · doi:10.1007/BF02236965
[18] Littlewood R.K., Zakian V.: Numerical evaluation of Fourier integrals. J. Inst. Math. Appl. 18, 331--339 (1976) · Zbl 0341.65087 · doi:10.1093/imamat/18.3.331
[19] Mason J.C., Handscomb D.C.: Chebyshev Polynomials. CRC Press, New York (2003)
[20] Paterson T.N.L.: On high precision methods for the evaluation of Fourier integrals with finite and infinite limits. Numer. Math. 27, 41--52 (1976) · Zbl 0319.65077 · doi:10.1007/BF01399083
[21] Piessens R., Branders M.: Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23, 370--381 (1983) · Zbl 0514.65008 · doi:10.1007/BF01934465
[22] Piessens R., Poleunis F.: A numerical method for the integration of oscillatory functions. BIT 11, 317--327 (1971) · Zbl 0234.65026 · doi:10.1007/BF01931813
[23] Powell M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981) · Zbl 0453.41001
[24] Sloan I.H.: On the numerical evaluation of singular integrals. BIT 18, 91--102 (1978) · Zbl 0386.65002 · doi:10.1007/BF01947747
[25] Sloan I.H., Smith W.E.: Product-integration with the Clenshaw-Curtis and related points. Numer. Math. 30, 415--428 (1978) · Zbl 0367.41015 · doi:10.1007/BF01398509
[26] Sloan I.H., Smith W.E.: Product integration with the Clenshaw-Curtis points: implementation and error estimates. Numer. Math. 34, 387--401 (1980) · Zbl 0416.65014 · doi:10.1007/BF01403676
[27] Stein E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993) · Zbl 0821.42001
[28] Szegö G.: Orthogonal Polynomial. American Mathematical Society, Providence, Rhode Island (1939) · Zbl 65.0278.03
[29] Trefethen L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000) · Zbl 0953.68643
[30] Trefethen L.N.: Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50, 67--87 (2008) · Zbl 1141.65018 · doi:10.1137/060659831
[31] Waldvogel J.: Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46, 195--202 (2006) · Zbl 1091.65028 · doi:10.1007/s10543-006-0045-4
[32] Xiang S.: Efficient Filon-type methods for ${\int_a^bf(x)e^{i\omega g(x)}dx}$ . Numer. Math. 105, 633--658 (2007) · Zbl 1158.65020 · doi:10.1007/s00211-006-0051-0