On computing rational Gauss-Chebyshev quadrature formulas

Van Deun, Joris; Bultheel, Adhemar; González Vera, Pablo

doi:10.1090/S0025-5718-05-01774-6

On computing rational Gauss-Chebyshev quadrature formulas
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by Joris Van Deun, Adhemar Bultheel and Pablo González Vera;

Math. Comp. 75 (2006), 307-326

DOI: https://doi.org/10.1090/S0025-5718-05-01774-6

Published electronically: October 4, 2005

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Abstract:

We provide an algorithm to compute the nodes and weights for Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside $[-1,1]$. Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order $O(n)$. This method is based on the derivation of explicit expressions for Chebyshev orthogonal rational functions, which are (thus far) the only examples of explicitly known orthogonal rational functions on $[-1,1]$ with arbitrary real poles outside this interval.

References

A. Bultheel, C. Díaz-Mendoza, P. González-Vera, and R. Orive, On the convergence of certain Gauss-type quadrature formulas for unbounded intervals, Math. Comp. 69 (2000), no. 230, 721–747. MR 1651743, DOI 10.1090/S0025-5718-99-01107-2
Adhemar Bultheel, Pablo González-Vera, Erik Hendriksen, and Olav Njåstad, Orthogonal rational functions, Cambridge Monographs on Applied and Computational Mathematics, vol. 5, Cambridge University Press, Cambridge, 1999. MR 1676258, DOI 10.1017/CBO9780511530050
A. Bultheel, P. González-Vera, E. Hendriksen, and Olav Njåstad, Orthogonal rational functions and quadrature on the real half line, J. Complexity 19 (2003), no. 3, 212–230. Numerical integration and its complexity (Oberwolfach, 2001). MR 1984110, DOI 10.1016/S0885-064X(03)00002-5
A. Bultheel, P. González-Vera, E. Hendriksen, and O. Njåstad, Orthogonal rational functions and tridiagonal matrices, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), 2003, pp. 89–97. MR 1985681, DOI 10.1016/S0377-0427(02)00602-7
C. Díaz Mendoza, P. González Vera, and M. Jiménez Paiz, Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation, Math. Comp. 74 (2005), 1843–1870.
C. Díaz Mendoza, P. González Vera, M. Jiménez Paiz, and F. Cala Rodríguez, Orthogonal Laurent polynomials corresponding to certain strong Stieltjes distributions with applications to numerical quadratures, Math. Comp., posted on September 9, 2005, PII S 0025-5718(05)01781-3 (to appear in print).
Walter Gautschi, Construction of Gauss-Christoffel quadrature formulas, Math. Comp. 22 (1968), 251–270. MR 228171, DOI 10.1090/S0025-5718-1968-0228171-0
Walter Gautschi, On the construction of Gaussian quadrature rules from modified moments, Math. Comp. 24 (1970), 245–260. MR 285117, DOI 10.1090/S0025-5718-1970-0285117-6
Walter Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982), no. 3, 289–317. MR 667829, DOI 10.1137/0903018
Walter Gautschi, Orthogonal polynomials: applications and computation, Acta numerica, 1996, Acta Numer., vol. 5, Cambridge Univ. Press, Cambridge, 1996, pp. 45–119. MR 1624591, DOI 10.1017/S0962492900002622
—, Algorithm 793: GQRAT - Gauss Quadrature for Rational Functions, ACM Trans. Math. Software 25 (1999), 113–139.
Walter Gautschi, Laura Gori, and M. Laura Lo Cascio, Quadrature rules for rational functions, Numer. Math. 86 (2000), no. 4, 617–633. MR 1794345, DOI 10.1007/PL00005412
A. Sri Ranga, Another quadrature rule of highest algebraic degree of precision, Numer. Math. 68 (1994), no. 2, 283–294. MR 1283343, DOI 10.1007/s002110050062
Paul N. Swarztrauber, On computing the points and weights for Gauss-Legendre quadrature, SIAM J. Sci. Comput. 24 (2002), no. 3, 945–954. MR 1950519, DOI 10.1137/S1064827500379690
Gábor Szegő, Orthogonal polynomials, 3rd ed., American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, RI, 1967. MR 310533
Walter Van Assche and Ingrid Vanherwegen, Quadrature formulas based on rational interpolation, Math. Comp. 61 (1993), no. 204, 765–783. MR 1195424, DOI 10.1090/S0025-5718-1993-1195424-6
J. Van Deun and A. Bultheel, Orthogonal rational functions and quadrature on an interval, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), 2003, pp. 487–495. MR 1985717, DOI 10.1016/S0377-0427(02)00598-8
J. Van Deun and A. Bultheel, Ratio asymptotics for orthogonal rational functions on an interval, J. Approx. Theory 123 (2003), no. 2, 162–172. MR 1990094, DOI 10.1016/S0021-9045(03)00116-3
Patrick Van Gucht and Adhemar Bultheel, A relation between orthogonal rational functions on the unit circle and the interval $[-1,1]$, Commun. Anal. Theory Contin. Fract. 8 (2000), 170–182. MR 1789681